Inverse scattering problem

An inverse scattering problem reconstructs a medium or obstacle from measured scattered waves. Under the Born approximation, fixed-frequency far-field data sample the Fourier transform of the scattering potential on a restricted frequency surface.

For an outgoing field outside a sphere of radius $R$, write $u_s=k{\sum }_{n,m}{i}^{n+1}{a}_n^m{h}_n^{(1)}(kr)Y_n^m$. Its far-field pattern is $\sum a_n^m{Y}_n^m$. If its trace on that sphere lies in L2 space, Parseval's identity gives $k^2\sum |h_n^{(1)}(kR){|}^2|a_n^m{|}^2<\infty$. The fixed-argument growth of spherical Hankel functions yields the displayed equivalent range condition, with the low-order weights defined finitely. Square summability of the far-field coefficients alone does not ensure continuation to the chosen sphere.

Continuation of a far-field pattern to a spherical trace multiplies harmonic coefficients by the displayed factor, whose magnitude grows faster than exponentially with order. A data error of size $\delta$ in one normalized high-order harmonic therefore produces near-field error $\delta k|h_n^{(1)}(kR)|$, unbounded as its order increases. Some noisy square-summable far fields fail the range condition entirely. This proves unstable field continuation; bounds on actual obstacle geometry additionally require an admissible shape class and a stable boundary-extraction procedure.

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.

With $|{\mathbf{k}}_i|=k$, elastic scattering into a propagating wavevector ${\mathbf{k}}_s$ samples Fourier transfer $\mathbf{\kappa }={\mathbf{k}}_s-{\mathbf{k}}_i$. These transfers satisfy the displayed shifted sphere relation. This reciprocal-space construction appears in crystallographic diffraction and the Fourier diffraction theorem.

With the normal directed out of an obstacle $D$ into the fluid, a field solving $(\Delta +k^2)\psi =-Q$ outside $D$ has the representationHere $G_k$ is the outgoing Green function for the three-dimensional Helmholtz equation and the observation point lies outside the obstacle. The formula follows from Green second identity; the normal to the exterior fluid on its inner boundary is the negative of the stated obstacle normal. The surface term represents the scattered field.

For a scattered field satisfying ${\psi }_s=e^{ikr}{f}_{\infty }(\hat{\mathbf{r}})/r+O(r^{-2})$, the Kirchhoff–Helmholtz representation givesThe sign assumes the normal points out of the obstacle.

A Herglotz wave function is a superposition of incident plane waves,It solves the Helmholtz equation everywhere and is real analytic. The finite sphere area makes the density integrable, so each spatial derivative can be taken under the integral.

Pairing a far-field pattern with the conjugate density of a Herglotz wave function converts the angular integral into boundary data:This follows by substituting the surface representation of an obstacle far-field pattern and interchanging the integrals.

If measured far-field patterns determine a density $g$ and its Herglotz wave function equals a prescribed spherical wave on the obstacle boundary, that trace identity can locate the boundary. A star-shaped axisymmetric obstacle can be represented by one radial function $h(\theta )$. Numerically, expand $g$ and $h$ in suitable bases, match the far-field constraints, and minimize the complex boundary residual at angular collocation points, using regularization of an inverse problem for noise and small singular values. A single scalar pairing constraint alone does not determine an arbitrary density.

For a refractive index $n(\mathbf{r})$ and background wavenumber $k_0$, one sign convention defines the scattering potential by $V=k_0^2(1-n^2)$. The total field then satisfies $(\Delta +k_0^2)\psi =V\psi$.

The opposite convention $V=k_0^2(n^2-1)$ gives $(\Delta +k_0^2)\psi =-V\psi$ and reverses the sign attached to the corresponding Green-function integral. Either convention is valid when used consistently.

Using $V=k_0^2(n^2-1)$, the scalar Lippmann-Schwinger equation is $\psi ={\psi }_i+{\int }_D{G}_{k_0}V\psi$. The Born approximation replaces the total field inside the integral by the known incident field:For a unit plane wave, its far-field pattern is $(4\pi {)}^{-1}{\int }_{D}V({\mathbf{r}}^{\prime })e^{i{k}_0({\hat{\mathbf{r}}}_0-\hat{\mathbf{r}})\cdot {\mathbf{r}}^{\prime }}d{\mathbf{r}}^{\prime }$. Thus each incident direction samples a shifted sphere in the Fourier transform of the potential.

For real $h\in C_c^{\infty }(D)$, the displayed real compactly supported contrast has Fourier transform $\hat{V}(\mathbf{\kappa })=[|\mathbf{\kappa }{|}^4-4({\mathbf{k}}_i\cdot \mathbf{\kappa }{)}^2]\hat{h}(\mathbf{\kappa })$. This vanishes on the shifted Ewald sphere because $|\mathbf{\kappa }{|}^2+2{\mathbf{k}}_i\cdot \mathbf{\kappa }=0$ there. Thus nonzero contrasts can have zero fixed-frequency, fixed-incidence Born approximation data. A small amplitude preserves weak contrast and a positive refractive index.

The Born series is the Neumann series expansion of the Lippmann-Schwinger equation. Its terms describe successive orders of wave scattering; convergence is ensured, for example, by a scattering operator norm smaller than one.

The Rytov approximation writes a total wave field as $\psi ={\psi }_i{e}^{\varphi }$ and linearizes the equation for the complex logarithmic perturbation $\varphi$. At first order, ${\psi }_i\varphi$ equals the first Born approximation scattered field, soIt can remain useful when phase accumulation is appreciable but local scattering is weak.

If the first logarithmic perturbation ${\chi }_1$ is a centered linear functional of a Gaussian random field, the complex Gaussian exponential moment gives the displayed mean of the first Rytov approximation ${\psi }_R={\psi }_i{e}^{\mu {\chi }_1}$. This is the mean of the truncated logarithmic model. A nonzero mean of the omitted second logarithmic perturbation can change the true field mean at order ${\mu }^2$.

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.

The Rytov approximation omits the quadratic gradient source in the logarithmic wave perturbation equation. Its next correction solves ${\mathcal{L}}_i{\chi }_2=-\nabla {\chi }_1\cdot \nabla {\chi }_1$. Control of the propagated correction, weak amplitude fluctuations and avoidance of field zeros are needed. An appreciable smooth accumulated phase can be retained by exponentiation; small total phase is not the same condition as small local phase gradients.

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.

Let ${\psi }_R={\psi }_i{e}^{{\varphi }_1}$ with unit-modulus incidence and ${\varphi }_1={\int }_{D}K(\mathbf{r},{\mathbf{r}}^{\prime })V({\mathbf{r}}^{\prime })d{\mathbf{r}}^{\prime }$. For a centered real Gaussian random field $V$, the exponential moment of a Gaussian linear functional givesHere $C_V$ is the autocorrelation function of a random field. Both the covariance $\mathbb{E}|{\varphi }_1{|}^2$ and the generally nonzero term $\mathrm{Re}\mathbb{E}{\varphi }_1^2$ enter the exponent; dropping the latter is not justified for an arbitrary finite real random medium.

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.

The far-field pattern is the directional coefficient $f_{\infty }(\hat{\mathbf{r}})$ in an outgoing asymptotic field $u_s(\mathbf{r})=e^{ikr}{f}_{\infty }(\hat{\mathbf{r}})/r+O(r^{-2})$.

For a bounded source region and fixed positive wavenumber, expand $|R\mathbf{\theta }-{\mathbf{r}}^{\prime }|=R-\mathbf{\theta }\cdot {\mathbf{r}}^{\prime }+O(R^{-1})$. In the outgoing Green function for the three-dimensional Helmholtz equation, this gives $G=e^{ikR}{e}^{-ik\mathbf{\theta }\cdot {\mathbf{r}}^{\prime }}/(4\pi R)+O(R^{-2})$. Volume integration gives the outgoing field's far-field pattern with an $O(R^{-2})$ remainder. The estimate is an asymptotic statement for fixed source size and frequency.

The Sommerfeld radiation condition selects outgoing solutions of the exterior Helmholtz equation. In three dimensions it requires $r({\partial }_{r}u-iku)\to 0$ as $r\to \infty$ uniformly in direction.