= Solution
Write the observation point as $r\widehat r$, put $q=k_0(\widehat r-\widehat r_0)$, and take $\psi_0(r\widehat r)=e^{ik_0r\widehat r_0\cdot\widehat r}$. For a bounded medium, the <far-field approximation for an outgoing source> uses
$$
|r\widehat r-r'|=r-\widehat r\cdot r'+O(\ell^2/r),\qquad G_0\simeq\frac{e^{ik_0r}}{4\pi r}e^{-ik_0\widehat r\cdot r'}.
$$
For finite-distance accuracy require $r\gg\ell$ and $k_0\ell^2/r\ll1$ as well as being in the radiation region. Define the <Fourier transform> sample $S(q)=\int_DV(r')e^{-iq\cdot r'}\,d^3r'$ and the known factor
$$
B(r,\widehat r)=\frac{e^{ik_0r(1-\widehat r_0\cdot\widehat r)}}{4\pi r}.
$$
Then the required total <Born approximation for scalar wave scattering> and total <Rytov approximation> are
$$
\boxed{\psi_B(r\widehat r)\simeq e^{ik_0r\widehat r_0\cdot\widehat r}+\frac{e^{ik_0r}}{4\pi r}S(q),\qquad \psi_R(r\widehat r)\simeq e^{ik_0r\widehat r_0\cdot\widehat r}\exp\{B S(q)\}.}
$$
The first <Born approximation> <far-field pattern> is $S(q)/(4\pi)$. In both formulas the incident field is retained; an outgoing scattered term alone would not be a total-field answer.
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