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The leastsquare (generalized) inverse of operator (1) has
the famous form
 
(12) 
where the adjoint operator is defined by the dotproduct
test:
 
(13) 
With a specified definition of the dotproduct, the generalized
inverse minimizes the following quantity, which is the squared L_{2}
norm of the residual:
 
(14) 
In the case of integral operators, a natural definition of the dotproduct
is the double integral
 
(15) 
 
(16) 
What is the adjoint of the integral operator (1) in this
case? In the discrete world, where stacking is represented by a row
vector, the adjoint (transpose) of a summation matrix is a column
vector. In other words, the adjoint of collecting the input data along
the stacking curve trajectory and summing it into an individual output
bin is dividing the output bin into a number of portions sprayed along
the specified trajectory. Claerbout 1995a calls the stacking
operator a ``pull'' and its adjoint a ``push''.
The relationship between forward and adjoint operators is different in
the continuous world. Let us substitute the definition of the stacking
operator (1) into the dot product
(13), as follows:
 
(17) 
Changing the integration variable t to , we can
rewrite (17) in the form
 
(18) 
where has the same meaning as in equation
(7), and
 
(19) 
Comparing formulas (18) and (13), we conclude that the adjoint
operator is defined by the equality
 
(20) 
Thus we have proven that in the continuous world the adjoint of a
stacking operator is another stacking operator. The adjoint operator
has the same summation path as the asymptotic inverse (7),
which guarantees the correct reconstruction of the kinematics of the
input wavefield. The amplitude (weighting function) of the adjoint
operator is directly proportional to the forward weighting according
to equation (19). The coefficient of proportionality is the
Jacobian of the transformation of the variables z and t.
Similar results have been published for particular cases of
stacking operators: velocity transform
Jedlicka (1989); Thorson (1984), Kirchhoff
constantvelocity migration Ji (1994b), and NMO Crawley (1995).
To exemplify the application of a ``pull'' adjoint to inversion, let
us consider the case of the Radon transform from the preceding
section. Forming the product for this case leads
to the double integral
 

 
 (21) 
Applying Fourier transform with respect to z, we can rewrite
equation (21) in the frequency domain as
 
(22) 
where
 
(23) 
 (24) 
The inner integral in equation (22) reduces to the mdimensional
delta function:
 
(25) 
As follows from the properties of delta function,
 
(26) 
Inverting (26) for M, we conclude that
 
(27) 
Substituting equation (27) into (12) produces the result precisely
equivalent to Radon's inversion (4).
The SEPlib canonical library contains various examples of stacking
operators coupled with their adjoint counterparts. In practice,
discrete ``push'' adjoints provide the machineprecise accuracy of the
discrete dotproduct test. The ``pull'' adjoints defined in this
section cannot compete in precision because of roundoff
errors. However, their practical use can be justified for the purpose
of a ``smoother'' output. Claerbout 1995a and Crawley
1995 discuss this possibility in more detail.
The notion of the adjoint operator completely depends on the
arbitrarily chosen definition of the dot product and norm in the model
and data spaces. A simple way to change those definitions is to find
some positive weights W_{M}(z,x) in the model space and W_{S}(t,y) in
the data space that define the dot products as follows:
 
(28) 
 (29) 
Next: ASYMPTOTIC PSEUDOUNITARY OPERATOR
Up: Fomel: Stacking operators
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Stanford Exploration Project
11/12/1997