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Solving the system

The second relation of system (14) can be isolated because it simply adds two relations and two unknowns (the coordinates of ${\bf x}_0$). Similarly, the third equation can be substituted into the others by replacing ts (or tg) with tsg - tg (or tsg - ts).

Thus, we obtain a reduced system of four relations (six equations):  
 \begin{displaymath}
\left\{ \begin{array}
{ccc}

 \xi(p_g, t_g)\frac{{\bf p}_g}{...
 ... p}_s + {\bf p}_g) \cos \theta(p_0,t_0)

 \end{array} \right. .\end{displaymath} (15)
and two additional relations to compute the remaining unknowns:  
 \begin{displaymath}
{\bf x}_0 = \xi(p_0,t_0)\frac{{\bf p}_0}{p_0}
 - \frac{1}{2}...
 ...c{{\bf p}_s}{p_s}
 + \xi(p_g,t_g)\frac{{\bf p}_g}{p_g} \right) \end{displaymath} (16)

 
ts = tsg - tg . (17)

The six unknowns of system (15) are t0, tg (or ts), ${\bf p}_s$ [including the two unknowns (pxs,pys) or $(p_s,\phi_s)$], and ${\bf p}_g$ [(pxg,pyg) or $(p_g,\phi_g)$]. The parameters are tsg, ${\bf p}_0$[including the two parameters (px0,py0) or $(p_0,\phi_0)$], ${\bf x}_s$, and ${\bf x}_g$. This system, like its 2-D counterpart, can be solved using the Newton-Raphson algorithm Press et al. (1986).


previous up next print clean
Next: AGREEMENT WITH THE EQUATIONS Up: DERIVING THE SYSTEM OF Previous: Sets of equations and
Stanford Exploration Project
11/17/1997