The Rytov Approximation

Why the Rytov Approximation?

The Born approximation fails for objects that are large compared to the wavelength, even when the contrast is small β€” because the accumulated phase shift can be large. The Rytov approximation linearizes the complex phase (log-amplitude and phase) rather than the field itself, making it better suited for phase objects such as biological tissue or building materials at RF frequencies.

,

Definition:

The Complex Phase Representation

Write the total field as

u(r)=uinc(r) eΟ•(r),u(\mathbf{r}) = u^{\text{inc}}(\mathbf{r})\,e^{\phi(\mathbf{r})},

where Ο•(r)=Ο•1(r)+Ο•2(r)+β‹―\phi(\mathbf{r}) = \phi_1(\mathbf{r}) + \phi_2(\mathbf{r}) + \cdots is the complex phase perturbation. The real part of Ο•\phi gives the log-amplitude change; the imaginary part gives the phase change relative to the incident field.

For the unperturbed case (Ο‡=0\chi = 0), Ο•=0\phi = 0 and u=uincu = u^{\text{inc}}.

Theorem: The Rytov Integral Equation

Under the first Rytov approximation, the complex phase perturbation satisfies:

Ο•1(r)=1uinc(r)∫DG(r,rβ€²) κ02Ο‡(rβ€²) uinc(rβ€²) drβ€².\phi_1(\mathbf{r}) = \frac{1}{u^{\text{inc}}(\mathbf{r})} \int_D G(\mathbf{r}, \mathbf{r}')\,\kappa_{0}^{2}\chi(\mathbf{r}')\, u^{\text{inc}}(\mathbf{r}')\,d\mathbf{r}'.

Equivalently:

Ο•1(r)=uBornsca(r)uinc(r),\phi_1(\mathbf{r}) = \frac{u^{\text{sca}}_{\text{Born}}(\mathbf{r})}{u^{\text{inc}}(\mathbf{r})},

where uBornscau^{\text{sca}}_{\text{Born}} is the Born scattered field. Thus the Rytov complex phase equals the Born scattered field normalized by the incident field.

Proof
Derivation from the Helmholtz equation

Substituting u=uinceΟ•u = u^{\text{inc}} e^{\phi} into βˆ‡2u+ΞΊ02(1+Ο‡)u=0\nabla^2 u + \kappa_{0}^{2}(1+\chi)u = 0, dividing by uinceΟ•u^{\text{inc}} e^{\phi}, and using βˆ‡2uinc+ΞΊ02uinc=0\nabla^2 u^{\text{inc}} + \kappa_{0}^{2} u^{\text{inc}} = 0:

βˆ‡2Ο•+2βˆ‡uincuincβ‹…βˆ‡Ο•+βˆ£βˆ‡Ο•βˆ£2+ΞΊ02Ο‡=0.\nabla^2\phi + 2\frac{\nabla u^{\text{inc}}}{u^{\text{inc}}} \cdot \nabla\phi + |\nabla\phi|^2 + \kappa_{0}^{2}\chi = 0.

Dropping the βˆ£βˆ‡Ο•1∣2|\nabla\phi_1|^2 term (small phase gradient) yields a linear equation for Ο•1\phi_1.

Connection to Born

The Rytov phase equation is equivalent to the Born equation for the auxiliary field ψ=uincΟ•1\psi = u^{\text{inc}}\phi_1. The solution ψ=uBornsca\psi = u^{\text{sca}}_{\text{Born}} gives Ο•1=ψ/uinc=uBornsca/uinc\phi_1 = \psi / u^{\text{inc}} = u^{\text{sca}}_{\text{Born}} / u^{\text{inc}}. β– \blacksquare

,

Definition:

Rytov Validity Conditions

The Rytov approximation is valid when the phase gradient is small relative to the wavenumber:

βˆ£βˆ‡Ο•1∣β‰ͺΞΊ0.|\nabla\phi_1| \ll \kappa_{0}.

For a uniform slab of thickness LL and contrast Ο‡0\chi_0, this requires:

βˆ£Ο‡0∣β‰ͺ1.|\chi_0| \ll 1.

The Rytov approximation requires weak contrast but does NOT require the object to be electrically small. This is its key advantage over Born for large, weakly scattering objects.

,

Born vs Rytov Approximation

CriterionBornRytov
Validity conditionΞΊ0aβˆ£Ο‡0∣β‰ͺ1\kappa_{0} a |\chi_0| \ll 1βˆ£Ο‡0∣β‰ͺ1|\chi_0| \ll 1
Object size restrictionMust be small or low contrastCan be large if low contrast
What is approximatedScattered fieldComplex phase
Best suited forSmall scatterers, moderate contrastLarge phase objects, tissue, atmosphere
Fails forLarge, moderate-contrast objectsHigh-contrast objects (any size)

Example: Rytov Approximation in Biomedical Imaging

In microwave breast imaging at f=3f = 3 GHz (Ξ»=10\lambda = 10 cm):

  • Normal tissue contrast: Ο‡0β‰ˆ0.1\chi_0 \approx 0.1--0.30.3.
  • Tumor diameter: aβ‰ˆ1a \approx 1--22 cm.

Compare Born and Rytov validity for this scenario.

Solution
Born criterion

ΞΊ0a=(2Ο€/0.1)β‹…0.02β‰ˆ1.3\kappa_{0} a = (2\pi/0.1) \cdot 0.02 \approx 1.3 for a=2a = 2 cm. ΞΊ0aβˆ£Ο‡0βˆ£β‰ˆ0.13\kappa_{0} a |\chi_0| \approx 0.13--0.40.4. Born is marginal for large tumors with higher contrast.

Rytov criterion

βˆ£Ο‡0βˆ£β‰ˆ0.1|\chi_0| \approx 0.1--0.30.3. Rytov is valid for normal tissue (βˆ£Ο‡0βˆ£β‰ˆ0.1|\chi_0| \approx 0.1) but marginal for high-contrast tumors (βˆ£Ο‡0βˆ£β‰ˆ0.3|\chi_0| \approx 0.3).

In practice, both approximations serve as initializations for iterative refinement (Section 5.6).

Rytov for RF Propagation Through Building Materials

In RF imaging scenarios involving propagation through walls and building materials, the Rytov approximation is often more appropriate than Born. A typical drywall panel has Ο‡β‰ˆ1.5\chi \approx 1.5 and thickness L\appro1L \appro 1 cm at f=5f = 5 GHz:

  • Born: ΞΊ0Lβˆ£Ο‡0βˆ£β‰ˆ1.6\kappa_{0} L |\chi_0| \approx 1.6 β€” fails.
  • Rytov: βˆ£Ο‡0∣=1.5|\chi_0| = 1.5 β€” also fails (not weak contrast).

This motivates the extended Born/Rytov approximations and iterative methods of the next sections. The xPRA-LM method by Ross Murch et al. extends Rytov to lossy media with moderate contrast.

Born vs Rytov Comparison

Compares Born and Rytov approximations for a dielectric slab against the exact transmission coefficient.

As slab thickness increases, Born fails (accumulated phase error) while Rytov remains accurate for small contrast. Both fail for high contrast.

Parameters
0.1
2

Rytov approximation

A first-order approximation that linearizes the complex phase Ο•=log⁑(u/uinc)\phi = \log(u/u^{\text{inc}}) rather than the field itself. Valid when βˆ£Ο‡0∣β‰ͺ1|\chi_0| \ll 1 regardless of object size, making it preferred over Born for large, weakly scattering objects.

Related: Born approximation

Quick Check

An object has Ο‡0=0.05\chi_0 = 0.05 and electrical size ΞΊ0a=20\kappa_{0} a = 20. Which approximation is more appropriate?

Born β€” because the contrast is small.

Rytov β€” because the contrast is small but the object is large.

Neither β€” both fail for ΞΊ0a>10\kappa_{0} a > 10.

Both are equally valid.

Correction:
Rytov β€” because the contrast is small but the object is large.

Rytov requires only βˆ£Ο‡0∣β‰ͺ1|\chi_0| \ll 1, which is satisfied. Born requires ΞΊ0aβˆ£Ο‡0∣β‰ͺ1\kappa_{0} a |\chi_0| \ll 1, which is not.

Key Takeaway

The Rytov approximation linearizes the complex phase Ο•=log⁑(u/uinc)\phi = \log(u / u^{\text{inc}}) rather than the field. The Rytov phase equals uBornsca/uincu^{\text{sca}}_{\text{Born}} / u^{\text{inc}} β€” the same Green's function integral, differently interpreted. Rytov validity: βˆ£Ο‡0∣β‰ͺ1|\chi_0| \ll 1 regardless of object size, vs Born's ΞΊ0aβˆ£Ο‡0∣β‰ͺ1\kappa_{0} a |\chi_0| \ll 1. Rytov is preferred for large, weakly scattering objects; Born is preferred for small objects. Neither first-order approximation handles high contrast AND large electrical size β€” iterative methods are needed.

The Born ApproximationBeyond First-Order: Iterative Methods