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ESTIMATING MEASUREMENT PRECISION BY MEANS OF MEASUREMENT DIFFERENCES
V. Casella
DIET - Università degli Studi di Pavia, Pavia, Italy
email: easel 1 a@uni pv.it
ISPRS Commission VI, Working Group 3
KEYWORDS: Precision estimation, variance estimation, estimators, statistics
ABSTRACT
When it is necessary to determine the precision of a certain instrument B, devoted to point position measurement for
instance, but not necessarily, one way is to measure a certain number of points with B as well as with another instru
ment A, whose precision is known. Forming the differences between the B and A measurements, it is possible to calcu
late B precision by means of the dispersion of the differentiated quantities. The paper investigates how the precision of
A influences the estimation of the precision of B. Even if the subject is basic, it is not easy to find in textbooks; for this
reason the paper could have a certain interest for teaching purposes.
1. INTRODUCTION
Let's assume there are n points in the space and two in
struments A and B able to measure their positions. The
instrument called A has well known characteristics, while
B has an unknown precision, which we want to determine
by comparison between the measurements given by it and
the ones given by A. This scheme is very common and
applies, for instance, to the estimation of the precision of
a DPW (Digital Photogrammetric Workstation) by means
of the comparison with an analytical stereoplotter; the
same scheme could also be applied to the study of the
performances of a quick GPS mode, based on static
measurements.
Our discussion will be limited to only one component,
and will assume that the instruments have always the
same precision, regardless of the position of the measured
point. The measurement of the z-th point by the A instru
ment can be represented by a normal random variable (rv
starting from now)
where the x with the small line above it represents true
values, while A and B represent the measured posi
tions. The variance of A is supposed to be known, while
the variance of B is unknown. The equation (1) and (2)
also contains the hypothesis that both the instruments
have no biases; this hypothesis should be checked in an
actual case, but it doesn't damage the general value of our
discussion: it only makes the job simpler.
It is possible to form the differences between the meas
urements given by the two instruments:
t>i= x Bi- a ( 3 )
which are formally extracted by n different but identical
normal rvs (we will follow the convention of indicating
one rv with an uppercase letter and extractions from it
with the same letter, lowercased)
X i =N\x i ,a]
(1)
A,- = N[o, o 2 a +g 2 b
and the measurement of the B instrument can be repre- « differences can also be thought of as multiple ex
sented by the following normal rv tractl0ns from a umc l ue " * at wl11 be called
X.
,=tf[x„CT
(2)
A = a[o, o 2 a +o 2 b
(4)
and this allows the estimation of the dispersion of :
