ing 
2) 
3) 
get 
nnt 
the 
ate 
ds). 
4) 
na 
Nn 
the 
5b) 
9a) 
)b) 
Joz Wu 
  
The nxn matrix ¢,Q contains the measurement variances and covariances (=0), so Q is a diagonal matrix. 
The xn Q;, uxu Q, and nxn Q, (not fully expressed) are the scaled covariance matrices, arising from the 
l-, x-, and v-vector error propagations, respectively. v/Q'v isa quadratic form of the estimated 
measurement residuals; n—u denotes the degree of freedom. Eq. (7) then leads to the a posteriori reference 
variance estimate ó, . 
3. STATISTICALPARAMETER ANALYSIS 
Statistical testings are very useful algorithmic routines toevaluate the least-squares parameter and 
measurement estimates, see Eqs. (6a, 6b). The quadratic form (7) of the estimated measurement residuals can 
be used to build a chi-square test statistic, involving the a priori reference variance (Leick, 1995; Wang et al., 
1998). By specifying a significance level, usually at 596, the lower and upper critical chi-square distributed 
values can be derived from a statistical look-up table. If the chi-square test statistic fails to fall within the 
confidence interval, it can be stated that, with a 95% confidence level, the functional and stochastical modeling 
equation (5a) is incorrect. 
3.1 Parameter-Significance Test 
According to Zhong (1997), an F-test statisticis given as the left-hand term in the following inequality 
equation: 
—l 
XQ. X 
22 
Oo 
  
5 Fa nu (8) 
where x is a component parameter in the x-solution equation (6a); q, stands for the corresponding cofactor 
(scaled variance) of x; ó ? is the estimated reference variance, as from Eq. (7). F_a-1n-u 1S the critical value 
—u 
that results from an F-distribution of (1, n—u) degrees of freedom and at a 1— confidence level. If the F-test 
(8) is passed, the parameter x is not significant, and should be deleted from the model-parameter set. Thus, a 
new model is formed, the appearance of which is the same as expressed in Eq. (5). According to Eq. (6), the 
new (u—1)-vecto x and n-vector v solutions are readily obtained. 
3.2 Optimization Criteria 
In order to distinguish an old model having an ux1 x-vector of parameter corrections and the new/alternate 
model holding the (u-1)-component x, their respective n-component measurement residuals are employed to 
produce the corresponding v-quadratic forms. The next minimum criteria will be used as the optimization 
indices: i.e., 
62 =vIQ v/in-u — min (9a) 
V, (6$ — min (9b) 
AlC2nln((v/Q !v) x2u — min (9c) 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part Bl. Amsterdam 2000. 349