ARM 9 
MATHEMATICAL FORMULATION & DIGITAL ANALYSIS 1365 
in which 4 is the matrix of unknown parameters. Wong?? has shown that, for a direct linear 
least-squares solution in which the observation equations are exactly linear, itis theoretically 
valid to compute the variance-covariance matrix (04) of the A-matrix by the following expres- 
sion: 
0, = GEN * (15) 
where 02 is the variance-of-unit-weight. Most of the least-squares adjustment problems 
encountered in analytical photogrammetry require the use of an iterative procedure because 
of the nonlinearity of the observation or condition equations. In this type of iterative least- 
squares solution, equation 15 is theoretically valid only if the unknown corrections in the 
A-matrix converge to zero in the last iteration. 
However, experimental results by Wong showed that equation 15 could provide reliable 
estimates to the root-mean-square (RMS) errors of the computed parameters as long as the 
correction parameters converge to a value which is less than the computed estimate of the 
corresponding RMS error. If the corrections in the last iteration exceed the computed RMS 
errors in the last iteration, then the variance-and-covariance matrix is not a reliable estimator 
of the adjustment accuracy even though the solution has stabilized. Experimental evidence 
showed that the formulation in equation 15 could not reflect any rapid accumulation of 
systematic effects which are caused by the random errors in the measurements. 
ERROR ELLIPSE AND ELLIPSOID 
Error ellipse and ellipsoid are used to evaluate the accuracy of photogrammetric determina- 
tion of positions in two- and three-dimensional spaces respectively. For example, let the 
variance-and-covariance matrix (0p) of the computed X and Y coordinates of a point be given as 
follows: 
02 On 
Op. = (16) 
Ozy OF 
Then, the family of error ellipses about the computed position (X,Y) is given by the following 
expressions: 
2 2 
E xo ec, (17) 
Ox, Oy 
où = (g2 +07) + Va; - 0,2)2 + 4g (18) 
  
  
x, 2 
2 (0,7 tog? TT oM m. 9,2)2 + 4074? (19) 
Vt 
and 
20, 
tan20 = Ta S (20) 
in which x, is the major axis of the ellipse; y, is the minor axis; VC30;, and VC 10m, are the 
semi-major and semi-minor axes respectively; 0 is the angle which the x,-axis makes with the 
X-axis; and C, is a constant. Figure 7 illustrates these parameters. 
Statistically, there is 38 per cent chance that the true position ofthe point will fall within the 
error ellipse defined by C, = 1. This ellipse is called the standard ellipse. Similarly, there is 90 
per cent and 99 per cent probability that the true position would fall within the ellipse defined 
by C, = 4.6 and C, = 9.2 respectively. 
In geodetic surveying, the survey network and measurement procedures are designed so 
that the semi-major and semi-minor axes of the ellipse about each unknown point are approx- 
imately equal and that their magnitudes fall within the accuracy specification of the survey. 
The computation of the parameters of an error ellipsoid in three-dimensional space is 
considerably more involved. Let the variance-and-covariance matrix (9a) of a pointj which has 
the coordinates X;, Y;, and Z; be defined as follows: 
02 Op Og 
0 =} Ogy Gy? Oyz (21) 
O yz Oyz oz