LEAST-SQUARES ADJUSTMENTS - GPS Satellite Surveying - page 95

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4.1.3 Observational Errors

Fi eld observations are not perfect, and neither are the recordings and management of
ob servations. The measurement process suffers from several error sources. Repeated
m easurements do not yield identical numerical values because of random measure-
m ent errors. These errors are usually small, and the probability of a positive or a
ne gative error of a given magnitude is the same (equal frequency of occurrence). Ran-
do m errors are inherent in the nature of measurements and can never be completely
ov ercome. Random errors are dealt with in least-squares adjustment.
Systematic errors are errors that vary systematically in sign and/or magnitude.
Ex amples are a tape that is 10 cm too short or the failure to correct for vertical or lat-
er al refraction in angular measurement. Systematic errors are particularly dangerous
be cause they tend to accumulate. Adequate instrument calibration, care when observ-
in g, such as double centering, and observing under various external conditions help
av oid systematic errors. If the errors are known, the observations can be corrected
be fore making the adjustment; otherwise, one might attempt to model and estimate
th ese errors. Discovering and dealing with systematic errors requires a great deal of
ex perience with the data. Success is not at all guaranteed.
Blunders are usually large errors resulting from carelessness. Examples of blun-
de rs are counting errors in a whole tape length, transposing digits when recording field
ob servations, continuing measurements after upsetting the tripod, and so on. Blun-
de rs can largely be avoided through careful observation, although there can never
be absolute certainty that all blunders have been avoided or eliminated. Therefore,
an important part of least-squares adjustment is to discover and remove remaining
bl unders in the observations.

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4. 1.4 Accuracy and Precision

Accuracy refers to the closeness of the observations (or the quantities derived from
the observations) to the true value. Precision refers to the closeness of repeated
observations (or quantities derived from repeated sets of observations) to the sample
mean. Figure 4.1 shows four density functions that represent four distinctly different
measurement processes of the same quantity. Curves 1 and 2 are symmetric with
respect to the true value x T . These measurements have a high accuracy, because the
sample mean coincides or is very close to the true value. However, the shapes of
both curves are quite different. Curve 1 is tall and narrow, whereas curve 2 is short
and broad. The observations of process 1 are clustered closely around the mean (true
value), whereas the spread of observations around the mean is larger for process 2.
Larger deviations from the true value occur more frequently for process 2 than for
process 1. Thus, process 1 is more precise than process 2; however, both processes
are equally accurate. Curves 3 and 4 are symmetric with respect to the sample mean
x S , which differs from the true value x T . Both sequences have equally low accuracy,
but the precision of process 3 is higher than that of process 4. The difference x T − x S

is caused by a systematic error. An increase in the number of observations does not
reduce this difference.