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Here we review the historic ``transformation method''
of deriving the parabolic wave equation.
A vertically downgoing plane wave is represented mathematically
by the equation
 
(54) 
In this expression, P_{0} is absolutely constant.
A small departure from vertical incidence can be modeled
by replacing the constant P_{0} with something,
say, Q(x,z), which is not strictly constant but varies slowly.
 
(55) 
Inserting (55) into the scalar wave
equation yields
 

 (56) 
The wave equation has been reexpressed in terms of Q(x,z).
So far no approximations have been made.
To require the wavefield to be
near to a plane wave, Q(x,z) must be near to a constant.
The appropriate means
(which caused some controversy when it was first introduced)
is to drop the highest depth derivative of Q,
namely, Q_{zz}.
This leaves us with the
parabolic wave equation
 
(57) 
I called equation (57) the equation.
After using it for about a year I discovered a way to improve on it
by estimating the dropped term.
Differentiate equation (57) with respect to z
and substitute the result back into equation (56)
getting
 
(58) 
I named equation (58) the migration equation.
It is first order in ,so it requires only a single surface boundary condition,
however, downward continuation will require
something more complicated than equation (53).
The above approach,
the transformation approach,
was and is very useful.
But people were confused by the dropping and estimating of the derivative, and a philosophically more pleasing approach was
invented by Francis Muir,
a way of getting equations to extrapolate waves at wider angles
by fitting the dispersion relation of a semicircle
by polynomial ratios.
Next: Muir squareroot expansion
Up: HIGHER ANGLE ACCURACY
Previous: HIGHER ANGLE ACCURACY
Stanford Exploration Project
12/26/2000