.3 
GROSS ERRORS LOCATION BY TWO STEP ITERATIONS METHOD 
Prof. 
Wang Renxiang 
Xian Research Institute of Surveying And Mapping, China , Communssion III 
ABSTRICT 
In consideration of the capability and relibility about 
localizing gross errors are decreased by correlation of 
residuals seriously. From the strategical point view, 
the iterated weight least squares method is developed to 
so-called ' Two step iterations method '. In the first 
step of iterations, the observational weight is 
calculated by selected weight function in an usual way. 
In the second step, we start with statistical test and 
analysis of residual correlation. Based on convergence 
in the first step, obtain the possible gross error 
observation(s) and weighted zero to its. Then the 
second step iteration is performed. After that, gross 
errors localization is done by regirous statistical test 
according to the standardized residuals and with due 
regard for the magnitude of so-called ’ weighted zero 
residual’. The capability and relibility of localizing 
gross errors are improved by two step iterations method. 
The paper give some examples with simulated data for 
comparasion of the results about gross errors location 
by different step iterations methods. 
KEYWORDS: Gross Error Location, Standardized residual, 
Weighted zero residual, Qvv.P matrix. 
INTRODUCTION 
Gross errors localizing by iterated weight least squares 
method has been investigated for a long time. One of the 
key problems of this method is to select weight function 
There are many weight functions proposed by different 
authors in present application. Every weight function 
has its own properties. Among these functions, the types 
of function, the parameters and the statistical 
quantities as well as the critical values are somewhath 
different to each other. However a common property is 
that the function is an inverse measure of residual in 
absolute. Therefore the magnitute of main diagonal 
element relating the observation with rather large 
residual in absolute in Qvv.p matrix will be increased, 
aftre iteration with the weight function and will be 
capable of localizing gross errors (Wang Renxiang, 1986a) 
The present author points out in (Wang Renxiang, 1988b, 
appendix) that gross errors localization is unreliable by 
iteration with weight function, when some residuals are 
of strong correlation. The paper has proposed an idea 
so-called  ' Two step iteration method ’ in order to 
improve the capability and reliability about localizing 
gross errors. 
1. THE TWO STEP ITERATIONS METHOD 
The two step iterations method is proposed based on the 
properties of so-called weighted zero residual ( 
appendix--3.) and the "'cheking correlation of residual 
programm '( Wang Renxiang 1998b ). From the strategical 
point of review, the iterated weight least squares 
method has been contrived in two step iterations. The 
first. step is to perform least squares iterations with 
weight function untill convergence. The second step is 
to analyse the correlation of residuals in which the 
standardized value is large than the critical value. 
Because of the first step iterated convergence, the 
searching areas of gross error observations are limlted 
in the observations in which the standardized residual 
is of large or strong correlation with another large 
standardized residual. 
1.1 The First Step Iterations 
  
In a general way, all the weight functions used in 
iterated least squares method or robust estimate method 
can be taken in the first step iterations. However the 
present author emphasizes that standardized residuals 
have to be used in every iteration at least in the last 
one for statistical test. Therfore Qvv.P matrix should 
be calculated in every iterations. The papers (Shan Jie 
1988,Wang Renxiang 1990 in Chinese, appendix ) have give 
the fast recurive algorithm for computation of Qvv.P 
matrix. The time comsuming for calculatng Qvv.P matrix 
have been overcome. As an experiment in this paper, 
the author gives a weight function modified from ( Wang 
Renxiang 1989c) and used in this step iterations as 
fellows 
  
  
I. i or CL 
p = 1/,8 ; à «C for 1,2 iteration 
lb 5 i > C after 2 iterations 
where 
a, Pl >} 
a= | 80 2 50 
^i ,  eleswhere 
; | vi ; 
i = -—— = enlarged residual 
80 Ki 
7c 4 = standardized residual 
89 K2 
Ki “VE = enlarged factor 
Kf Ez standardized factor 
€ 22.0, a = 2.5, b = 3.0--4.0 
1.2 The Second Step Iterations 
  
In-the first step, the mistakes of localizing gross 
errors are from two major circumstances. The first is 
the gross error that can not be detected by statistical 
test with the standard critical value, because the 
magnitute of main diagonal element related to gross 
error observation in Qvv.P matrix and correspondene to 
the main component coefficient of the standardized 
residual MCCV ( appendix----3.2 ) are too small. It 
is impossible to overcome by any iterationl method 
determined by the design matrix. The second, gross error 
revealed in the standardized residual is - dispersed by 
the correlation coefficient of residuals and make wrong 
decision with statistical test. This problem can 
possibly be overcome by disassembling the correlation 
of residuals. The properties of weighted zero residual 
( appendix----3) as fellows plays important part in this 
discussion. 
1.2.1 Gross error can be revealed in its weighted 
zero residual completly. 
1.2.2 After iteration, any two observations are