e 
  
Transform image coordinates 
" from comparator to photo system for 
oto 1 > ; ; ; 
p (x, Yız) and photo 2 (X55; Y54) 
| 
Compute trans. parameters 
for photo 2 to photo 1 
| 
Compute trans. parameters 
for photo 1 to object-space 
Transform photo 2 coordinates (x, 
  
  
  
  
  
  
  
  
  
  
oy) 
X53:Y24) 
  
to photo 1 coordinate system ( 
| 
Loop to A for all (x 
  
  
  
  
! ny 
then all (Kai 
| 
Compute preliminary object- 
space coordinates (a 5» Yig) 
  
  
  
  
  
  
  
Compute tj 
| 
Recompute object-space 
coordinates*(X. Y.) using Z.. 
1J iJ 1J 
| 
Apply refraction corrections 
  
  
  
  
  
  
  
  
  
  
  
  
Compute x and y 
components of 
Displacement vectors 
  
  
  
Figure 4. General Steps In the Analytical Solution 
for three rotation parameters of. of and Kf, and three 
translation parameters xs, Y^ and 2 of photo 1. 
Eleven control points are used in the solution. 
The parameters wf, 4j, KT» xt. Y, and Z$ are then 
used to compute loe Object space coordinates of the 
image points in photo l. The three dimensional projec- 
tive transformation equations are used in a two-step 
solution. Let X13 and y1; represent the image coordi- 
nates of point jin phoit 1; and xj;, Y,; and Zi 
represent the corresponding object Space" coordinates 
of the same point. Because of the deflection of the 
glass plate, the cross-sectional surface of the model 
is no longer a planar surface and Z-coordinates will 
not be all equal to zero. In the first step, approxi- 
mate values of the X1j and Y4: coordinates are compu- 
ted by assuming that Z1; -Q* These approximate co- 
ordinates are then used lo compute the actual Z- 
deflection of the glass plate. This is accomplished 
by using an inverse-weighted distance function and the 
four nearest data points obtained from strain gauge 
measurements. The Z1; coordinate is then used to re- 
compute the X13 and Y1j coordinates of point j. 
15 
The object space coordinates X2j, Y2j and Zp; of the 
same point j after a certain movement of the construc- 
tion shield are then computed from photo 2. Let x5j 
and 24 represent the photo coordinates of point j 
measured from photo 2. These coordinates are first 
transformed to the photo 1 system using the orientation 
parameters Wp, $2, Ko, X2, Y7 and Z2. This step cor- 
rects for the effects of any movement of the camera 
between exposures. Let x4; and Y2j represent the trans- 
formed photo coordinates if point j. Then the corre- 
sponding object space coordinates Xois Yay and Z2j 
are computed using the orientation parameters oT, 
$T. «T, Xf, YT and Zf and the same procedure as that. 
described in hu above paragraph. The computed coordi- 
nates X15, Y]. X2; and Yo; are then corrected for 
refraction. The refraction correction, dX and dY, for 
the coordinates X1; and nj are computed from the 
following expressiüns: 
y = tan’! (xi X5)/01j - Yf)) (2) 
is 
a7 tan (X - PL + (YY - hp] Tj 
j) 
(3) 
8 - Sin" (Sin a/N) (4) 
d = T(tan o - tan 8) (5) 
dX s d.«- Sin (6) 
dY = d * Cos y (7) 
Xj 7 X4j * dX (8) 
Vig = Yi; + OY (9) 
where N is the index of refraction, and T is the : 
thickness of the glass plate. The movement of point j 
in the X and Y directions during the time interval 
between the two exposures are then computed as the 
changes in the corrected coordinates, i.e. 
AX = Xb; - Xi (10) 
AVE Yl 7 94 (11) 
Since the displacement vectors are in numerical form, 
many different graphical illustrations can be directly 
generated from the computer. Figure 5 is an example of 
a CALCOMP plot of the movement vectors at points around 
the tunnel model. Figure 6 is a contour plot of the 
movement in the X-direction during one test. Every 
point on the same contour line has the same amount of 
movement. Figure 7 is a contour plot of the movement 
in the Y-direction. Both of these contour plots were 
generated directly from the computed movement vectors 
by the computer. Figure 8 is a profile plot of the y- 
displacement at a distance R above the center lines of 
two parallel tunnels, where R is the radius of the 
tunnel. Figure 9 is a profile plot of the correspon- 
ding x-displacement. Contour plots can also be gene- 
rated to study the distribution of strain and stress 
throughout the cross-sectional area of the model. 
ACCURACY ANALYSIS 
In order to test the overall accuracy of the method, 
two photographs were taken of the tunnel model in its 
undisturbed position. These two photographs were then 
measured and analysed as a pair of "before" and "after" 
photographs. Figure 10 is a plot of the computed dis- 
placement vectors which, in fact, are measurement 
errors. If the system of measurement was totally free 
of error, there would have been zero displacement at 
anywhere within the model area. The average value of 
the 84 vectors in Figure 10 amounted to 0.04 mm, and 
the standard deviation was + 0.09 mm. 
Listed in Table 1 are the means and standard devia-