PULL ADJOINTS

The least-square (generalized) inverse of operator (1) has the famous form  

$$\widetilde{M}(z,x)={\bf \widetilde{A}}[S(t,y)]= {\bf \left(A^{T}\,A\right)^{-1}\,A^{T}}[S(t,y)]\;,$$ (12)

where the adjoint operator ${\bf A^{T}}$ is defined by the dot-product test:  

$$\left(S(t,y),{\bf A}[M(z,x)]\right) \equiv \left({\bf A^{T}}[S(t,y)],M(z,x)\right)\;.$$ (13)

With a specified definition of the dot-product, the generalized inverse minimizes the following quantity, which is the squared L₂ norm of the residual:

$$\left(S(t,y)-{\bf A}[M(z,x)], S(t,y)-{\bf A}[M(z,x)]\right)\;.$$ (14)

In the case of integral operators, a natural definition of the dot-product is the double integral

$$\left(S_1(t,y),S_2(t,y)\right) = \int\int\,S_1(t,y)\,S_2(t,y)\,dy\,dt\;,$$ (15)

$$\left(M_1(z,x),M_2(z,x)\right) = \int\int\,M_1(z,x)\,M_2(z,x)\,dx\,dz\;.$$ (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:  

$$\left(S(t,y),{\bf A}[M(z,x)]\right) = \int\int\int\,w(x;t,y)\,M(\theta(x;t,y),x)\,S(t,y)\,dx\,dy\,dt\;.$$ (17)

Changing the integration variable t to $z=\theta(x;t,y)$, we can rewrite (17) in the form  

$$\left(S(t,y),{\bf A}[M(z,x)]\right) = \int\int\int\,\widetilde{w}(y;z,x)\,M(z,x)\, S(\widehat{\theta}(y;z,x),x)\,dy\,dx\,dz\;,$$ (18)

where $\widehat{\theta}$ has the same meaning as in equation (7), and  

$$\widetilde{w}(y;z,x) = w(x;\widehat{\theta}(y;z,x),y)\, \left\vert\partial \widehat{\theta} \over \partial z\right\vert\;.$$ (19)

Comparing formulas (18) and (13), we conclude that the adjoint operator ${\bf A^{T}}$ is defined by the equality  

$${\bf A^{T}}[S(t,y)]= \int \widetilde{w}(y;z,x)\,S(\widehat{\theta}(y;z,x),y)\;dy\;.$$ (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 constant-velocity 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 ${\bf A^{T}\,A}$ for this case leads to the double integral

$$\begin{eqnarray} H(z,x) & = & {\bf (A^{T}\,A)}[M(z,x)] = \nonumber \ & = & \in... ...ber \ & = & \int\int\,M\left(z + y\,(\xi - x)\right)\,d\xi\,dy\;.\end{eqnarray}$$ (21)

Applying Fourier transform with respect to z, we can rewrite equation (21) in the frequency domain as  

$$\check{H}(\omega,x) = \int\,\check{M}(\omega,\xi)\,\int\, e^{i\omega\,y\,(\xi-x)}\,dy\,d\xi\;,$$ (22)

where

$$\begin{eqnarray} \check{H}(\omega,x) & = & \int\,H(z,x)\,e^{-i\omega\,z}\,dz\;, \ \check{M}(\omega,x) & = & \int\,M(z,x)\,e^{-i\omega\,z}\,dz\;.\end{eqnarray}$$ (23) (24)

The inner integral in equation (22) reduces to the m-dimensional delta function:

$$\check{H}(\omega,x) = (2\,\pi)^m\,\int\,\check{M}(\omega,\xi)\, \delta\left(\omega^m\,(\xi-x)\right)\,d\xi\;.$$ (25)

As follows from the properties of delta function,  

$$\check{H}(\omega,x) = {{(2\,\pi)^m} \over {\vert\omega\vert^... ...2\,\pi)^m} \over {\vert\omega\vert^m}}\, \check{M}(\omega,x)\;.$$ (26)

Inverting (26) for M, we conclude that  

$${\bf (A^{T}\,A)^{-1}} = {{\vert{\bf D}\vert^m} \over {(2\,\pi)^m}}\;.$$ (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 machine-precise accuracy of the discrete dot-product test. The ``pull'' adjoints defined in this section cannot compete in precision because of round-off 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:

$$\begin{eqnarray} \left(S_1(t,y),S_2(t,y)\right) & = & \int\int\,W_S(t,y)\,S_1(t,... ...,x)\right) & = & \int\int\,W_M(z,x)\,M_1(z,x)\,M_2(z,x)\,dx\,dz\;.\end{eqnarray}$$ (28) (29)