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The Born approximation for scalar wave scattering relates an incident plane wave and the transverse Fourier transform of its scattered field to samples of the three-dimensional Fourier transform of a scattering potential on a shifted Ewald sphere. One incident direction at one frequency samples a surface, rather than an entire three-dimensional frequency region. Additional illuminations, frequencies or structural constraints are needed for a full reconstruction.

The logarithmic wave perturbation measures relative complex amplitude and phase through $\psi ={\psi }_i{e}^{\chi }$. A continuous branch requires nonzero incident and total fields in the region considered. For the Helmholtz equation with scattering potential convention $L_0\psi =-V\psi$, its equation is $(\Delta +2\nabla \log {\psi }_i\cdot \nabla )\chi +\nabla \chi \cdot \nabla \chi =-V$. The dot product is bilinear, not a squared modulus.

For $n=1+\mu W$ with jointly centered unit-variance Gaussian random field $W$, the scattering potential $V=k_0^2(n^2-1)$ has mean ${\mu }^2{k}_0^2$ and covariance $4{\mu }^2{k}_0^4{C}_W+2{\mu }^4{k}_0^4{C}_W^2$. It is not Gaussian. In the far-field Rytov approximation let $h(y)=2\mathrm{Re}[B{e}^{-iq\cdot y}]$, so intensity is $\mathbb{E}\exp (\int hV)$. Expansion through order ${\mu }^2$ includes both the mean term and the covariance term shown. The remainder starts at order ${\mu }^4$ under suitable exponential integrability. Keeping a covariance correction while omitting the same-order mean term is inconsistent. The exact quadratic model uses the Gaussian quadratic exponential moment.

Use the scattering potential convention $V=k_0^2(n^2-1)$. With time dependence $e^{-i\omega t}$ the outgoing Green function is $G_0(r,r^{\prime })=e^{i{k}_0|r-r^{\prime }|}/(4\pi |r-r^{\prime }|)$ and satisfies $(\Delta +k_0^2)G_0=-\delta$. The total-field Lippmann-Schwinger equation isReplacing the field in the integral by the incident field gives the first Born approximation for scalar wave scattering:For the first Rytov approximation, write $\psi ={\psi }_0{e}^{\chi }$ on a region where the incident field is nonzero and a continuous complex logarithm can be chosen. The logarithmic wave perturbation equation isThe last dot product is bilinear, not a squared modulus. Dropping it leaves a linear equation for ${\chi }_1$. Multiplication by ${\psi }_0$ shows that ${\psi }_0{\chi }_1$ satisfies the same outgoing inhomogeneous Helmholtz equation as ${\psi }_1$. HenceIntroduce a small multiplier $\lambda$ on $V$. Then ${\psi }_1=O(\lambda )$ andThe two total-field approximations agree to first order. Exponentiating the first logarithmic wave perturbation does not supply the generally missing second-order Born series contribution.

The zero-mean fluctuation is $W$, not the refractive index $n$: $⟨n⟩=1$. Also the full scattering potential has a nonzero mean and is not Gaussian:The usual Gaussian closure requires $W$ to be a jointly Gaussian random field, not merely to have a Gaussian distribution at each point. We first give the intended leading weak-fluctuation result with the linearized potential $V_1=2\mu k_0^{2}W$, then account for the quadratic term.

Keep the definitions $q,B$ from the preceding solution and let ${\chi }_1=B{\int }_D{V}_1(y)e^{-iq\cdot y}dy$. Because the incident field has unit modulus,The exponent is a real Gaussian linear functional $Z={\int }_{D}h{V}_1$. If $C_W(y-z)=⟨W(y)W(z)⟩$, the linearized potential has autocorrelation function of a random field $C_1(s)=4{\mu }^2{k}_0^4{C}_W(s)$. Complete the square in the one-dimensional Gaussian density of $Z$: $⟨e^Z⟩=\exp (⟨Z⟩+\mathrm{Var}Z/2)$. Here its mean is zero, soThis is Gaussian intensity in the first Rytov approximation. In a more explicit complex notation, putThenThe second term is the pseudo-covariance of a complex random variable contribution. It does not vanish for an arbitrary finite real medium. The real-kernel expression shows that the combined exponent is nonnegative, even though its two complex-form contributions need not be separately nonnegative.

For the full potential, Gaussian moment pairing theorem gives its centered covariance functionIts uncentered autocorrelation is $R_V(s)=C_V(s)+{\mu }^4{k}_0^4$. The first Rytov approximation is linear in $V$, but its intensity is an exponential average; applying a Gaussian formula exactly to this quadratic potential would be incorrect. A consistent expansion through order ${\mu }^2$ isThe first term is the coherent mean-potential correction. Odd powers vanish by the symmetry of the joint Gaussian field. The variance term can equally use $C_V=R_V-{\overline{V}}^2$ through this order. Thus the mean shift must not be discarded while retaining the order-${\mu }^2$ fluctuation intensity.

An exact expression within the far-field, first-Rytov approximation model is also available when the full $n^2$ is retained. Assume the restriction of $W$ to $D$ is a Gaussian element of the real Hilbert space $L^2(D)$, with covariance operator $C$ of kernel $C_W(y-z)$. Let $H$ denote multiplication by $h$, ${\ell }_h=2\mu k_0^{2}h$, $Q={\mu }^2{k}_0^2{C}^{1/2}H{C}^{1/2}$ and $b_h=C^{1/2}{\ell }_h$. The Gaussian quadratic exponential moment isHere the determinant is the Fredholm determinant; require $I-2Q$ strictly positive for finiteness. Diagonalize the self-adjoint trace-class operator $Q$ and complete the square in each Gaussian coordinate to derive the formula. On a bounded $D$, unit pointwise variance makes $\mathrm{tr}C=|D|$, so the covariance is trace class under the stated measurable $L^2$ hypothesis. Small enough $\mu$ satisfies the positivity condition. Expanding the determinant and inverse recovers the preceding mean and covariance terms. For small $\mu$ this is also determined by the potential autocorrelation: since $C_W\in [-1,1]$,using the branch near zero. This makes explicit the extra joint-Gaussian assumption behind an exact correlation-based answer.

Finally, Gaussian marginals alone are insufficient. Take $X\sim N(0,1)$ and independent random signs $S_j$. The stationary sequence $W_j=S_{j}X$ has Gaussian marginals and covariance $⟨W_i{W}_j⟩={\delta }_{ij}$, just like independent standard Gaussian variables. However $⟨e^{t(W_1+W_2)}⟩=(1+e^{2t^2})/2$, whereas for the independent Gaussian pair it is $e^{t^2}$. The different intensities show why the joint hypothesis is needed. The mean-zero hint is interpreted for $W$, and the linearized Gaussian formula is not claimed to be exact for $V=k_0^2(n^2-1)$.

Choose the time dependence $e^{-i\omega t}$ and measure distances in background-wavenumber units. Thus $L_0=\Delta +1$, the incident field satisfies $L_0{\psi }_i=0$, and the scattering potential convention is $V=n^2-1$, giving $L_0\psi =-V\psi$. To match the minus sign in the paper's integral, take the outgoing Green function to satisfy $L_{0}G=\delta$. This is the negative of the frequently used outgoing Helmholtz equation fundamental solution satisfying $L_{0}G=-\delta$. In physical coordinates one restores $k_0^2$ in the scattering potential, or absorbs it in the integral kernel; the sign convention must remain consistent.

On a region where the incident and total fields are nonzero, introduce the logarithmic wave perturbationChoose a continuous logarithm branch connected to the unperturbed field. Substitution into the Helmholtz equation givesThe quadratic term uses the complex bilinear dot product, not the squared modulus of the gradient. Write $V=\epsilon V_1$ and $\chi =\epsilon {\chi }^{[1]}+\cdots$. The first-order Rytov approximation discards that quadratic term. Since $L_0({\psi }_i\chi )={\psi }_i{\mathcal{L}}_i\chi$, the outgoing first correction isand henceThe validity of the first Rytov approximation concerns the omitted logarithmic correction. Its next contribution satisfies ${\mathcal{L}}_i{\chi }_2=-\nabla {\chi }_1\cdot \nabla {\chi }_1$, so a useful explicit criterion is that the outgoing solution ${\chi }_2$ be small in the region of interest. Weak refractive index contrast and small wave phase gradients on a wavelength scale provide the usual perturbative regime, with weak amplitude fluctuations and no strong focusing or zeros that destroy the logarithm. A sufficient local source comparison is $|\nabla {\chi }_1\cdot \nabla {\chi }_1|\ll |V|$ where the scattering potential is nonzero, together with control of propagation of that error. This is not a universal pointwise test at zeros of $V$.

Small accumulated wave phase is not required in the same way as in a linear field approximation: the exponential retains that wave phase accumulation. Large gradients, strong multiple-scattering amplitude effects or field zeros can still invalidate the Rytov approximation. The integrals also require a finite scattering region or appropriate convergence conditions.

Let $\delta {\psi }_B={\psi }_1^{(B)}-{\psi }_i$ be the first-order scattered correction in the Born approximation for scalar wave scattering, using the paper's Green function convention. Comparing its outgoing integral with the preceding logarithmic correction givesTherefore the exact algebraic relation between the two first-order approximations isIf $V=O(\epsilon )$, then ${\chi }_1=O(\epsilon )$ on a controlled fixed region, andThus Born and Rytov agree to first order in the scattering potential, but differ as finite approximations. The Rytov approximation exponentiates the first logarithmic correction; the Born approximation adds the first field correction. The exponential's higher powers are not a calculation of all higher multiple-scattering terms in the Born series. This distinction explains why a smooth, appreciable wave phase accumulation can be represented more naturally by the Rytov approximation even when the corresponding linear field expansion is inaccurate.

Let the stationary scattering potential fluctuation have autocorrelation function of a random fieldBecause $W$ is centered, this is also its covariance function. Multiplying the two real wave phase integrals and taking expectations givesThe two minus signs cancel. This is the wave phase variance, since the wave phase mean is zero. More generally the phase covariance in the first Rytov approximation replaces the first kernel by $b({\mathbf{r}}_1,{\mathbf{r}}^{\prime })$ and the second by $b({\mathbf{r}}_2,{\mathbf{r}}^{\prime \prime })$. Finite observation/scattering windows, or suitable weighted-integrability hypotheses, make these double integrals well-defined in a stationary infinite-medium model.

The correlation needed here is that of the scattering potential fluctuation. With the printed $V=n^2-1$, the covariance of a squared random field isIt involves a fourth moment of $n$, so its value is not generally determined by the ordinary two-point correlation $C_n=⟨n({\mathbf{r}}^{\prime })n({\mathbf{r}}^{\prime }+\mathbf{\rho })⟩$ alone. If one additionally assumes a zero-mean Gaussian random field, Isserlis theorem yields $C_W=2C_n^2$. That assumption is not printed and must not be inserted silently. Alternatively, for a physical weak fluctuation $n_{\mathrm{phys}}=1+\eta$, $W\simeq 2\eta$ gives $C_W\simeq 4C_{\eta }$. The general answer uses $C_W$; either reduction to a refractive index two-point correlation requires an extra assumption.

Assume a real refractive index and write the scattering potential as $V=⟨V⟩+W$, so $⟨W⟩=0$ by construction. With the real kernel components in the question, the logarithmic Rytov approximation separates intoInclude the incident wave phase and the mean-potential wave phase in the deterministic reference ${\varphi }_0$: at each observation point it is $\arg {\psi }_i-\int b⟨V⟩$. It need not be spatially constant. The amplitude is $|{\psi }_i|\exp [-\int aV]$. The phase covariance in the first Rytov approximation therefore starts from the fluctuating wave phaseUnder the integrability assumptions needed to interchange the expectation and integral,For a complex absorbing potential, the corresponding phase fluctuation is $-\int [b\mathrm{Re}W+a\mathrm{Im}W]$; the displayed scalar formula is the real-index case. The zero mean comes from centering $V$, not from setting the mean of $V$ equal to zero. In particular the printed $⟨n⟩=0$ does not imply $⟨V⟩=0$. A physical positive refractive index usually has a nonzero background mean; a zero-mean assumption normally refers to its fluctuation. The algebra above remains meaningful for a signed real random field and explicitly retains its mean scattering potential.

Take the same $e^{-i\omega t}$ convention and an outgoing field. Define the scattering potential with the positive-contrast sign,The outgoing Green function for the three-dimensional Helmholtz equation isThus the Lippmann-Schwinger equation has a plus sign:Replacing the total field inside the integral by the incident field gives the Born approximation for scalar wave scatteringMore generally, iterating the integral equation gives the Born series ${\psi }_s={\sum }_{j\ge 1}{T}^j{\psi }_i$; its first term is the boxed expression. A sufficient convergence condition is $\Vert T\Vert <1$ on the chosen space of fields in $V$, and accurate single-scattering truncation needs the neglected terms to be small. Weak refractive contrast is useful, but for a sufficiently large or resonant scatterer it is not by itself a guarantee of negligible multiple scattering.

Use the physical refractive index convention $k=k_{0}n$, with $k_0$ the background wavenumber and time dependence $e^{-i\omega t}$. Treat $B$ as a bounded region in ${\mathbb{R}}^3$; the printed $B\in {\mathbb{R}}^3$ should be $B\subset {\mathbb{R}}^3$. Define the contrast $q=n^2-1=2n_{ϵ}+n_{ϵ}^2$ and the outgoing Green functionThe Sommerfeld radiation condition selects this sign of the outgoing wave phase. Since $(\Delta +k_0^2)\psi =-k_0^{2}q\psi$, the Lippmann-Schwinger equation isThe positive sign in this integral follows from the minus sign in the defining Green function equation. One may instead define the scattering potential with the opposite sign, provided both equations change consistently.

If $\Vert T_q\Vert <1$ on the chosen field space inside $B$, its Neumann series converges. The Born series for the scattered field isFor observation points outside $B$, restrict the intermediate factors to $B$ and use the same outgoing integral for the final factor. For example, the first two terms areThe $m$th term describes $m$ successive scattering interactions. The first Born approximation for scalar wave scattering keeps one interaction and replaces the field inside the medium by the incident field. Higher terms describe multiple scattering and the resulting feedback on the internal field.

A concrete sufficient condition follows on $L^{\infty }(B)$. If $d_B$ is the diameter of $B$, then for $\mathbf{r}\in B$ the region lies in the ball of radius $d_B$ about $\mathbf{r}$, andThus $\boxed{\rho \ll 1}$ is a conservative sufficient condition for the first Born approximation for scalar wave scattering. When $\rho <1$, the omitted internal-field terms satisfyThe bound ignores cancellation in the oscillatory Green function, so it is sufficient, not necessary. A common physical small-contrast criterion for an extended weak medium is small accumulated extra wave phase, $k_0\ell \Vert n_{ϵ}{\Vert }_{\infty }\ll 1$, together with weak scattering and no resonant internal enhancement. Small local contrast alone is not enough for an arbitrarily large or resonant object.

Let $V=k_0^2(n^2-1)$ be the scattering potential supported in $D$. The background Helmholtz equation and its outgoing kernel give the scalar Lippmann-Schwinger equationReplacing the unknown field in the integral by the incident field gives the Born approximation for scalar wave scattering:For the Rytov approximation, work where ${\psi }_i\ne 0$ and set $\psi ={\psi }_i{e}^{\varphi }$. Dividing the wave equation by the total field and subtracting the incident equation givesNeglect the quadratic gradient term. If ${\varphi }_1={\psi }_{s,B}/{\psi }_i$, direct differentiation shows $\Delta {\varphi }_1+2\nabla \log {\psi }_i\cdot \nabla {\varphi }_1=-V$, soExpanding the exponential in the potential gives ${\psi }_R={\psi }_i+{\psi }_{s,B}+O(V^2)={\psi }_B+O(V^2)$. Thus both approximations have the same first-order field, although exponentiation retains a particular family of higher powers.

Since $n=1+W$ with small variance,To leading order, $V$ is therefore a centered stationary Gaussian random field. Let its autocorrelation function of a random field beThe $W^2$ contribution gives higher-order mean and non-Gaussian corrections and is consistently omitted at this order.

The far-field Rytov approximation from part ii has unit incident intensity andThe real random variable $X={\varphi }_1+{\varphi }_1^{\ast }$ is centered Gaussian. Its moment-generating function givesDefineThenand henceThis expression depends only on the two-point autocorrelation of the scattering potential. In the weak-fluctuation expansion it becomes

Use the sign convention in the question,Then the total field satisfiesThe outgoing free-space Green function iswith $({\nabla }^2+k_0^2)G_0=-\delta$. Hence the Lippmann-Schwinger equation isReplacing the unknown interior total field by the incident field gives the first Born approximation

For the Rytov approximation, put $\psi ={\psi }_i{e}^{\varphi }$. After division by $\psi$, the wave equation givesNeglecting the quadratic term $(\nabla \varphi {)}^2$ makes ${\psi }_i\varphi$ obey the same inhomogeneous equation as the first Born scattered field. ThusandIts power series beginsso the two approximations agree through first order in the scattering potential.

With the scattering potential $V(\mathbf{r})=k_0^2(1-n^2(\mathbf{r}))$, the total field obeysThe outgoing Green function is $G_0(\mathbf{r})=e^{i{k}_0|\mathbf{r}|}/(4\pi |\mathbf{r}|)$. The Lippmann-Schwinger equation and the first Born approximation giveFor $r\to \infty$,sowhere $\mathbf{q}=k_0(\hat{\mathbf{r}}-{\hat{\mathbf{x}}}_0)$ is the momentum transfer. Thus the far-field pattern is $-1/(4\pi )$ times the Fourier transform of $V$ at the measured transfer vectors. If sufficiently many incident directions and frequencies supply all $\mathbf{q}$, the formal reconstruction iswith constants adjusted to the chosen Fourier convention.

For a centered real scattering potential fluctuation $W$ and the imaginary part $b$ of the incident-field-weighted Green function, $\phi (\mathbf{r})=-\int b(\mathbf{r},{\mathbf{r}}^{\prime })W({\mathbf{r}}^{\prime })d{\mathbf{r}}^{\prime }$. Its mean vanishes, and its covariance function is the double integral of the two deterministic kernels against the covariance function of $W$. The correlation of the refractive index itself is insufficient when the scattering potential depends quadratically on that index.