ANALYSIS OF RESIDUALS FROM 7, NORM ESTIMATION 
John Marshall 
Surveying Engineering 
Purdue University 
U.S.A. 
marshalj@ce.ecn.purdue.edu 
James Bethel 
Surveying Engineering 
Purdue University 
U.S.A. 
bethel@ce.ecn.purdue.edu 
Key Words: Surveying, Statitistics, Adjustment, Design, Estimation. 
ABSTRACT 
The detection of erroneous observations is a challenging 
skill that requires the dedicated energies of photogram- 
metrists and surveyors on a day-to-day basis. This paper 
focuses on a robust estimator for its erroneous observation 
detection because robust estimators often perform satisfac- 
torily in the face of observations with blunders. Many cur- 
rent blunder detection methods are based on least squares 
which is not a robust estimator. The robust estimator ex- 
amined is L; norm minimization and some of its unique 
properties related to network adjustment are presented. 
The Li norm residual sampling distribution is described 
both theoretically and empirically to illustrate the foun- 
dation for statistical inference of erroneous observations. 
Network reliability issues are presented in light of unique 
properties surrounding the 1 norm. This analysis has ap- 
plication to many estimation problems such as those from 
geodesy, photogrammetry, and, in particular, close-range 
photogrammetry. 
1 INTRODUCTION 
An important asset to any photogrammetric or survey en- 
gineer is the ability to accurately detect erroneous observa- 
tions based on accepted statistical premises. Traditionally 
erroneous observations have been identified by examining 
least squares (L2 norm) residuals (Baarda, 1968), (Pope, 
1976), (Kavouras, 1982) and others. Least squares estima- 
tion maintains many useful properties when the error vec- 
tor e, is normally distributed with zero mean, e ~ N(0, e). 
However, in cases where erroneous observations exist and 
the relatively stringent normality assumptions surround- 
ing the La norm are violated, a robust estimator such as 
the L1 norm may be better suited to deal with these depar- 
tures from normality. In addition to effectively managing 
the departures from normality, the Z1 norm may serve as 
an additional means of ruling whether borderline observa- 
tions should be removed from a network adjustment. Used 
in this capacity, the L1 norm is especially appealing to an 
engineer given the nonscientific method of selecting the sig- 
nificance level for statistical inference. The strengths and 
weaknesses of [1 norm blunder detection are illustrated 
using Monte Carlo simulation. 
2 OBSERVATION WEIGHTS 
Accurate observation weights are necessary in surveying 
and photogrammetric applications, particularly when the 
38 
observations come from different sources. For L; estima- 
tion the definition of the weight itself does not change, but 
we incorporate the weights differently in the computations. 
The function to be minimized under the L; criterion is, 
tT |v] — minimum (1) 
where t is comprised of the diagonal elements of T, 
1/01 0 
1/02 
0 1/0» 
and v is the vector of residuals. In the actual computations 
we implement the weighting by premultiplying the matrix 
equations by T, i.e., 
BA =f (2) 
becomes 
TBA ~ Tf. (3) 
3 THEORETICAL SAMPLING DIS- 
TRIBUTION OF L; RESIDUALS 
3.1 Mathematical Model 
The theoretical sampling distribution of L; norm residuals 
forms the foundation for statistical inference concerning 
erroneous observations and it is based on a priori knowl- 
edge of network geometry, observation variances, and the 
method of estimation. In the case of L1 norm estimation, 
many new challenges arise for the engineer who has relied 
exclusively on the L2 norm for his erroneous observation 
detection because of the differing mathematical models as- 
sociated with the L4 and L2 norm estimators. In terms 
of residuals and residual sampling distributions, the two 
most significant differences between the L; and L2 esti- 
mators are, 
1. The L5 norm residuals can be expressed in terms of 
all the observations, whereas the L1 norm residuals 
are expressed in terms of subsets of observations. 
Do 
For the L2 norm, linear combinations of Gaussian 
random variables give rise to Gaussian combinations 
(Menke, 1989), whereas for the Li norm, the results 
are not strictly linear combinations. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part BS. Vienna 1996 
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