Global Positioning System Reference
In-Depth Information
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4.1 ELEMENTS
Objective quality control of observations is necessary when dealing with any kind
of measurements such as angles, distances, pseudoranges, carrier phases, and the
geopotential. It is best to separate conceptually quality control of observations and
precision or accuracy of parameters. It is unfortunate that least-squares adjustment is
most often associated only with high-precision surveying. It may be as important to
discover and remove a 10 m blunder in a low-precision survey asa1cmblunder in a
high-precision survey.
Least-squares adjustment allows the combination of different types of observa-
tions (such as angles, distances, and height differences) into one solution and permits
simultaneous statistical analysis. For example, there is no need to treat traverses, in-
tersections, and resections separately. Since these geometric figures consist of angle
and distance measurements, the least-square rules apply to all of them, regardless of
the specific arrangements of the observations or the geometric shape they represent.
Least-squares adjustment simulation is a useful tool to plan a survey and to ensure
that accuracy specifications will be met once the actual observations have been made.
Simulations allow the observation selection to be optimized when alternatives exist.
For example, should one primarily measure angles or rely on distances? Considering
the available instrumentation, what is the optimal use of the equipment under the
constraints of the project? Experienced surveyors often answer these questions intu-
itively. Even in these cases, an objective verification using least-squares simulation
and the concept of internal and external reliability of networks is a welcome assurance
to those who carry responsibility for the project.
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4.
1.1 Statistical Nature of Surveying Measurement
Assume that a distance of 100 m is measured repeatedly with a tape that has cen-
timeter divisions. A likely outcome of these measurements could be 99.99, 100.02,
100.00, 100.01, etc. Because of the centimeter subdivision of the tape, the surveyor
is likely to record the observations to two decimal places. The result therefore is a
series of numbers ending with two decimal places. One could wrongly conclude that
this measurement process belongs to the realm of discrete statistics yielding discrete
outcomes with two decimal places. In reality, however, the series is given two decimal
places because of the centimeter division of the tape and the fact that the surveyor did
not choose to estimate the millimeters. Imagining a reading device that allows us to
read the tape to as many decimal places as desired, we readily see that the process of
measuring a distance belongs to the realm of continuous statistics. The same is true
for other types of measurements typically used in positioning. A classic textbook case
for a discrete statistical process is the throwing of a die in which case the outcome is
limited to integers.
When measuring the distance we recognize that any value
x
i
could be obtained,
although experience tells us that values close to 100.00 are most likely. Values such
as 99.90 or 100.25 are very unlikely when measured with care. Assume that
n
