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The second relation of system (14) can be isolated
because it simply adds two relations and two unknowns (the
coordinates of ). Similarly, the third equation
can be substituted into the others by replacing t_{s} (or t_{g})
with t_{sg}  t_{g} (or t_{sg}  t_{s}).
Thus, we obtain a reduced system of four relations (six equations):
 
(15) 
and two additional relations to compute the remaining unknowns:
 
(16) 

t_{s} = t_{sg}  t_{g} .

(17) 
The six unknowns of system (15) are t_{0}, t_{g}
(or t_{s}), [including the two unknowns (p_{xs},p_{ys})
or ], and [(p_{xg},p_{yg}) or
]. The parameters are t_{sg}, [including the two parameters (p_{x0},p_{y0}) or ],
, and . This system, like its 2D
counterpart, can be solved using the NewtonRaphson algorithm
Press et al. (1986).
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Stanford Exploration Project
11/17/1997