The transformed normal set of equations are below: 



+ r. 



' ^13^*^2 " ( 2 



2 



r^^x 



2u 2, 2, 2 



^13^ + X - r^^x b3 = r^^x 



2 



-)x b^ + r^^x b^ X b3 = r^3X 



and 



where 



b = XCl - b, - b_ - bj 

 o ^ 12 3 



X = estimate of the mean bulk depth 

 X = estimate of the variance of the bulk depth. 



Note that although x cancels in the above equations, it is retained for clarity. 



Transects oriented in directions 1 and 3 should have equivalent parameters be- 

 cause of their symmetry about the uphill orientation (orientation 2). The following 

 changes have been made to insure this symmetry: 



r' = r' = fr + r 1/2. 

 11 13 ^11 13^^ 



Primes pertain to transformed new values. Theoretically, the serial correlation of 

 lag 2 should be the square of lag 1. To insure that the two matrix elements (1,3) 

 and (3,1) that have correlation of lag 2, (R21 influenced by this re- 



lation, the following change has been made: ^ 



(r^l ^ ^23)72 . [(r-^^ . . -'uV^J/^ - [(-21 ^ -23^/2^ 



where the second term is the mean of the squares of the three correlations of lag 1 

 Note that rj^^ and r|3 are the new values as expressed above. 



Solving the three simultaneous equations gives: 



where 



b^ - b3 = R^(l - - 2R^ + 1) 



^2 = ^12 - ^Vl 



^ = f^ll ^ ^13^/2 



R2 = (r^^ . r23)/2 . (2r2^ . rl^)/6. 



Rewriting for completeness 



b = X(l - b - b^ - b,) 



o 1 2.3 



where X is the mean bulk depth, 



33 



