- ST / /k1 T2-T1 \ 



(3) 



C. Trial and Error Regression Method 



It is a simple task to fit a straight line to a point subset by the method 

 of least squares. Assuming the intercept A and B the slope of the line of 

 regression on (m - n + 1) points (z:, Tj), (j = m, m + 1, . , n), then the solution 

 for A and B is given in matrix notation 



(4) 





A 



= 



(Q^- Q)"^ 



(Q^-H), 











.B. 











where Q = 



1 Zm 

 1 Zm+1 



, Q' = 



1 1 •• 

 Zm Zm+1- 



1 

 •^n 



, and H = 



Tm 

 Tm+l 





1 in 









_Tn _ 



A set of points may be tested for alignment within the thickness of a 

 stylus trace by trial and error. Given the equation 



T; = A + B 



(5) 



(6) 



then the points (zt, T:) align in the depth interval z^ ^ Zj 1 Zp' '^ 



for all values [ in the range m < j < n. 



It is required to identify data point subsets in which as many points as 

 possible align. This may be done likewise by trial and error, using the following 

 method. Proceeding from any point, P, in a set of consecutive profile data 

 points (A, B, C, . .), determine whether points P and Q align. If they do 

 (they always will), then determine whether points P, Q, and R align, and so on. 

 When a set of points is found that does not align, test the previous set for align- 

 ment with points preceding P. In this way a point set of maximum alignment 

 extension downward and upward may be identified with any profile data point. 

 Because this set of consecutive points is selected for negligible departure from a 

 straight line, let it be referred to as a "line set". 



