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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 :