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["Spreadsheet solution of 3-point problem as Simul Lin Eqns:  "]
[null,"--WF, 13vii04"]
[]
[null,"Directions:  "]
[null,null,"1) Enter x- & y-locations of three drillholes in cells B14:C16"]
[null,null,"2) Enter depths of horizon in cells F14:F16"]
[null,null,"3) Read the x- & y-slopes and z-intercept in cells I14:i16"]
[null,null,"4)  you can also predict the horizon depth at any location"]
[null,null,"      by entering the location in H22:I22, and reading I24"]
[]
["The exact fit case:  "]
[]
["  DH#: "," x-loc","  y-loc"," enter \"1\"",null," Obs depths: ",null,null," results:  "]
[1,1,0,1,null,6,null," Mx:",0]
[2,0,1,1,null,6,null," My:",0]
[3,1,1,1,null,6,null,"  Zo: ",6]
[]
[]
[{"$nulls":7},"Predicted depth"]
[null,null,"The inverse matrix:  ",{"$nulls":4},"  for DH at:"]
[null,null,0,-1,1,null,null,"   x-loc","  y-loc"]
[null,null,-1,0,1,null,null,2.5,3,1]
[null,null,1,1,-1]
[{"$nulls":7}," Z(pred) =",6]
[]
[]
[null,"The overdetermined problem:  "]
[null,"   Best fit of a plane to five drillholes"]
[null,"  (this form can be extended to any number of drillholes, but it could get tedious…) "]
[]
[]
[null,"  DH#: "," x-loc","  y-loc"," enter \"1\"",null,"Obs depths:",null,"Estimated Parameters:  "]
[null,1,1,0,1,null,6.1,null,"   Mx =",0.07156626506023911]
[null,2,0,1,1,null,5.9,null,"   My =",-0.10469879518072123]
[null,3,1,1,1,null,6,null,"   Zo =",6.023855421686749]
[null,4,2.5,3,1,null,5.9]
[null,5,7,4,1,null,6.1]
[{"$nulls":8},"Predicted depth"]
[{"$nulls":8},"  for DH at:"]
[{"$nulls":8},"   x-loc","  y-loc"]
[{"$nulls":8},2.5,3,1]
[]
[{"$nulls":8}," Z(pred) =",5.888674698795183]
[null,"Workspace:"]
[null,null," Matrix A-transpose:"]
[null,null,1,0,1,2.5,7]
[null,null,0,1,1,3,4]
[null,null,1,1,1,1,1]
[]
[null,null,"  A-trans * A:",null,null,null,"Inverse of [A-trans * A]:"]
[null,null,57.25,36.5,11.5,null,0.13012048192771086,-0.19036144578313258,0.043373493975903614]
[null,null,36.5,27,9,null,-0.1903614457831326,0.3710843373493977,-0.2301204819277109]
[null,null,11.5,9,5,null,0.04337349397590371,-0.230120481927711,0.5144578313253013]
[]
[null,null,"  At * z:"]
[null,null,69.55]
[null,null,54]
[null,null,30]

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