tan" 



,fWc 



(B4) 



The unknowns f c , x, and <{> are determined with the use of targets at known 

 image and ground coordinates, known as ground control points. An iterative 

 procedure is used to calculate these unknown parameters. If two targets are 

 used, then a solution can be obtained, but additional targets will provide more 

 accurate estimations by computing a least-squares solution of the camera 

 geometry. Lippmann and Holman (1989) found typical errors in estimates of 

 t, (j), and/ c to be less than 0.25°, 0.5°, and 0.5 percent, respectively. 



At this point the functions F x andF 2 can be evaluated, but we are still left 



with three unknowns in Equations Bl and B2. One unknown can be 

 eliminated if there is a line across the beach at a known angle fx, described by 

 the line 



Y L = Y 0L + X L t3n V 



(B5) 



where the subscript L indicates points on the line and Y QL is the intercept of the 

 Y-axis. Using this, Equations Bl and B2 reduce to a problem with two 



unknowns 



X L = Z L F,(a £ , y L ) 



(B6) 



Y l = Z L F 2 K' Yz) 

 rearranging in terms of the unknowns X L and Z L , gives 



(B7) 



X L ~ Y 0L 



' F 2 (a L , J L ) 

 - tan u. 



{ Ffrv Yz) 



(B8) 



X, 



f Md Y £ ) 



(B9) 



One application would be the measurement of a cross-shore profile ( tan /x 

 = 0) at a known longshore coordinate (Y 0L = Yj), say for instance a light beam 

 cast across the beach. Functions F, and F 2 are computed from a measured a L 

 and Yl> th en *h e other two ground coordinates (X L and Yj) are found using 

 Equations B6 and B7. Repeating this for all video pixels (oc L and y L values) 



Appendix B Photogrammetry 



B3 



