86 



Amplitudes and Phases The amplitudes and phases of a particular record can 

 be found by rearranging the equations as a linear least squares problem by separating 

 the cosine and sine components as was done in chapter 3: 



T]c = a cos {kscX + kyU — ujt + a) 



= 6 cos {kxX + kyU — ut) + csin ik^x + kyV — ujt) 

 a - V62 + c2 

 a = arctan {—c/b) 



for the single wave method, and 



Tjc = ai cos {kx^ix + ky^iy - cot + ai) + a2 cos {k^^2X + ky^2y — ujt + Q2) 

 = 61 cos {k^^ix + ky^iy — Lot) + Ci sin (Aj^jX + ky^iy — ujt) 

 +62 cos {k:,:^2X + ky^2y — i^t) + C2 sin {k:,^2X + ky^2y — i^t) 



«i = yH + cl (5.19) 



02 = \Jh + 4 



Qi = arctan (—Ci/61) 

 02= arctan (—C2/62) 



for the two wave method. The system is determined if 'qd't) is defined at at least two 

 points in time for the single wave method, and at four points in time for the two wave 

 method. The system is solved in the least squared sense in the case of more points. 



Refining the Linear Estimates These procedures result in very rough estimates 

 for the parameters of two intersecting linear waves. The estimates are then refined 

 by optimizing for a best linear wave theory fit to the record: 



N M 



minimizeO(X) = Y.Y. (^^'-■" " ^^ (X; x„,2/„,i„))' (5.20) 



n—l m=X 



where: 



/'' (X;a;„,y„,im) = acos(A;^x„ + kyy^ - utm + a) 



