Stochastic Scattering Models

When and Why Statistical Scattering Models Matter

Telecommunications uses stochastic channel models (Rayleigh, Rician, clustered delay-line) to characterize propagation statistically. These models are fundamental to communication system design but cannot replace deterministic forward models for imaging. The reason is simple: imaging recovers the specific realization of the scene, while stochastic models average over realizations.

Nevertheless, statistical models play a specific and important role in RF imaging: they describe target fluctuation (Swerling models), speckle statistics, and clutter characteristics.

Why Stochastic Models Fail for Imaging

The fundamental mismatch:

Imaging recovers the specific realization of the scene; stochastic models average over realizations.

Concretely:

  1. Scene specificity: Imaging needs the exact A\mathbf{A} for the specific scene geometry. A stochastic model provides only the statistics of A\mathbf{A}, not its realization.

  2. Spatial structure: Stochastic models describe a single link. Imaging needs the joint channel across all Tx-Rx-frequency triples, preserving spatial coherence.

  3. Phase information: Coherent imaging exploits measurement phase. Stochastic models typically characterize only power, discarding the phase relationships essential for focusing.

  4. Channel is the signal: In telecom the channel is a nuisance to equalize; in imaging the channel is the signal we want to recover.

Definition:

Swerling Target Fluctuation Models

The Swerling models describe the statistical fluctuation of a target's radar cross-section (RCS) across measurements. They model the scattering coefficient ci,j,qc_{i,j,q} as a random variable.

Model PDF of ∣ci,j,q∣2|c_{i,j,q}|^2 Decorrelation Physical basis
Swerling I Exponential Scan-to-scan Many equal scatterers (Rayleigh envelope)
Swerling II Exponential Pulse-to-pulse Same as I, faster fluctuation
Swerling III Chi-squared, k=2k=2 Scan-to-scan One dominant + many small (Rician-like)
Swerling IV Chi-squared, k=2k=2 Pulse-to-pulse Same as III, faster fluctuation

Swerling I (most common): The RCS Οƒ\sigma follows an exponential distribution:

f(Οƒ)=1ΟƒΛ‰exp⁑(βˆ’ΟƒΟƒΛ‰),Οƒβ‰₯0,f(\sigma) = \frac{1}{\bar{\sigma}}\exp\left(-\frac{\sigma}{\bar{\sigma}}\right), \quad \sigma \geq 0,

where ΟƒΛ‰=E[Οƒ]\bar{\sigma} = \mathbb{E}[\sigma] is the average RCS. Equivalently, the scattering amplitude ∣ci,j,q∣|c_{i,j,q}| is Rayleigh distributed.

Swerling III: The RCS follows a chi-squared distribution with 4 degrees of freedom (2 complex Gaussian components, one dominant):

f(Οƒ)=4σσˉ2exp⁑(βˆ’2σσˉ).f(\sigma) = \frac{4\sigma}{\bar{\sigma}^2}\exp\left(-\frac{2\sigma}{\bar{\sigma}}\right).

Theorem: Rayleigh Speckle Statistics

When a resolution cell contains many (N≫1N \gg 1) independent scatterers of comparable strength, the complex reflectivity at that cell is:

cq=βˆ‘n=1NΞ±n ejΟ•n,c_q = \sum_{n=1}^{N} \alpha_n\,e^{j\phi_n},

where {Ο•n}\{\phi_n\} are i.i.d. uniform on [0,2Ο€)[0, 2\pi) and {∣αn∣}\{|\alpha_n|\} are comparable. By the central limit theorem, Re(cq)\text{Re}(c_q) and Im(cq)\text{Im}(c_q) are approximately Gaussian. Therefore:

  1. The amplitude ∣cq∣|c_q| follows a Rayleigh distribution.
  2. The phase arg⁑(cq)\arg(c_q) is uniform on [0,2Ο€)[0, 2\pi).
  3. The intensity ∣cq∣2|c_q|^2 follows an exponential distribution.

This is the origin of speckle in coherent imaging.

Speckle is the coherent-imaging analogue of noise: it arises from the random constructive/destructive interference of many unresolved scatterers within a single pixel. Unlike thermal noise, speckle is multiplicative and signal-dependent.

Proof
CLT application

Write cq=X+jYc_q = X + jY where X=βˆ‘n∣αn∣cos⁑ϕnX = \sum_n |\alpha_n|\cos\phi_n and Y=βˆ‘n∣αn∣sin⁑ϕnY = \sum_n |\alpha_n|\sin\phi_n. Since Ο•n\phi_n are i.i.d. uniform, E[cos⁑ϕn]=E[sin⁑ϕn]=0\mathbb{E}[\cos\phi_n] = \mathbb{E}[\sin\phi_n] = 0 and Cov(cos⁑ϕn,sin⁑ϕn)=0\text{Cov}(\cos\phi_n, \sin\phi_n) = 0. By the CLT for N≫1N \gg 1:

X∼N(0,Οƒs2),Y∼N(0,Οƒs2),X \sim \mathcal{N}(0, \sigma_s^2), \quad Y \sim \mathcal{N}(0, \sigma_s^2),

where Οƒs2=12βˆ‘n∣αn∣2\sigma_s^2 = \frac{1}{2}\sum_n |\alpha_n|^2.

Rayleigh envelope

Since XβŠ₯YX \perp Y and both are zero-mean Gaussian with equal variance, ∣cq∣=X2+Y2|c_q| = \sqrt{X^2 + Y^2} has the Rayleigh distribution with parameter Οƒs\sigma_s:

f∣cq∣(r)=rΟƒs2exp⁑(βˆ’r22Οƒs2).f_{|c_q|}(r) = \frac{r}{\sigma_s^2}\exp\left(-\frac{r^2}{2\sigma_s^2}\right).

Exponential intensity

The intensity I=∣cq∣2=X2+Y2I = |c_q|^2 = X^2 + Y^2 is the sum of two squared i.i.d. Gaussians, hence exponentially distributed:

fI(i)=12Οƒs2exp⁑(βˆ’i2Οƒs2).β– f_I(i) = \frac{1}{2\sigma_s^2}\exp\left(-\frac{i}{2\sigma_s^2}\right). \quad\blacksquare

Definition:

Rician Speckle Model

When one dominant scatterer (with fixed amplitude A0A_0 and phase Ο•0\phi_0) coexists with many small random scatterers, the reflectivity becomes:

cq=A0ejΟ•0+βˆ‘n=1NΞ±n ejΟ•n,c_q = A_0 e^{j\phi_0} + \sum_{n=1}^{N} \alpha_n\,e^{j\phi_n},

and ∣cq∣|c_q| follows a Rician distribution with K-factor:

K=A022Οƒs2,K = \frac{A_0^2}{2\sigma_s^2},

where Οƒs2\sigma_s^2 is the diffuse scattering power. The PDF is:

f∣cq∣(r)=rΟƒs2exp⁑(βˆ’r2+A022Οƒs2)I0(rA0Οƒs2),f_{|c_q|}(r) = \frac{r}{\sigma_s^2}\exp\left(-\frac{r^2 + A_0^2}{2\sigma_s^2}\right) I_0\left(\frac{r A_0}{\sigma_s^2}\right),

where I0(β‹…)I_0(\cdot) is the zeroth-order modified Bessel function.

Physical interpretation: Kβ†’0K \to 0 gives pure Rayleigh (no dominant scatterer). Kβ†’βˆžK \to \infty gives a deterministic target. Typical values: K=0K = 0-33 dB for urban clutter, K=10K = 10-2020 dB for a strong isolated target.

Definition:

Spatial Correlation of Reflectivity

For extended targets, the reflectivity at nearby pixels is correlated. The spatial correlation function is:

Rc(Ξ”p)=E[c(p) cβˆ—(p+Ξ”p)].R_c(\Delta\mathbf{p}) = \mathbb{E}[c(\mathbf{p})\,c^*(\mathbf{p} + \Delta\mathbf{p})].

For Gaussian random scenes, the reflectivity is fully characterized by its mean and this correlation function. The covariance matrix of the discretized reflectivity vector is:

[Ξ£c]q,qβ€²=Rc(pqβˆ’pqβ€²).[\boldsymbol{\Sigma}_{c}]_{q,q'} = R_c(\mathbf{p}_{q} - \mathbf{p}_{q'}).

Common models:

  • Uncorrelated: Rc(Ξ”p)=Ξ³i,j,qq δ(Ξ”p)R_c(\Delta\mathbf{p}) = {\gamma_{i,j,q}}_{q}\,\delta(\Delta\mathbf{p}). Each pixel is independent. This is the Swerling I assumption.
  • Exponential: Rc(Ξ”p)=Οƒc2 eβˆ’βˆ₯Ξ”pβˆ₯/β„“cR_c(\Delta\mathbf{p}) = \sigma_c^2\,e^{-\|\Delta\mathbf{p}\|/\ell_c}, with correlation length β„“c\ell_c.
  • Gaussian (squared exponential): Rc(Ξ”p)=Οƒc2 eβˆ’βˆ₯Ξ”pβˆ₯2/(2β„“c2)R_c(\Delta\mathbf{p}) = \sigma_c^2\,e^{-\|\Delta\mathbf{p}\|^2/(2\ell_c^2)}.

Spatial correlation introduces structure in Ξ£c\boldsymbol{\Sigma}_{c} that can be exploited as a prior in Bayesian reconstruction (Part III).

Definition:

Clutter Statistical Models

Clutter consists of unwanted returns that obscure targets. The measurement model with clutter is:

y=Actarget+cc+w,\mathbf{y} = \mathbf{A}\mathbf{c}_{\text{target}} + \mathbf{c}_{c} + \mathbf{w},

where cc\mathbf{c}_{c} is the clutter contribution.

Common clutter distributions:

Type Distribution Physical origin
Thermal noise CN(0,Οƒ2I)\mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I}) Receiver electronics
Homogeneous clutter CN(0,Οƒc2I)\mathcal{CN}(\mathbf{0}, \sigma_c^2\mathbf{I}) Many weak scatterers (CLT)
K-distributed Compound Gaussian, Ο„βˆΌGamma\tau \sim \text{Gamma} Sea surface, vegetation
Log-normal Compound Gaussian, lnβ‘Ο„βˆΌN\ln\tau \sim \mathcal{N} High-resolution ground clutter
Pareto Heavy-tailed Urban clutter, interference

The signal-to-clutter ratio (SCR) determines imaging quality:

SCR=βˆ₯Actargetβˆ₯2E[βˆ₯ccβˆ₯2].\text{SCR} = \frac{\|\mathbf{A}\mathbf{c}_{\text{target}}\|^2}{\mathbb{E}[\|\mathbf{c}_{c}\|^2]}.

Swerling Model RCS Realizations

Generate and compare RCS realizations from the four Swerling models. The histogram shows the PDF of the RCS amplitude, and the time series shows the fluctuation pattern (scan-to-scan vs. pulse-to-pulse decorrelation).

Parameters
10

Speckle Statistics β€” Rayleigh vs. Rician

Visualize the amplitude distribution of speckle for varying numbers of scatterers per cell and Rician K-factor. As NN increases, the distribution converges to Rayleigh. Increasing KK shifts the distribution toward a deterministic peak.

Parameters
20
0

Example: Same Environment, Different Models: Indoor at 60 GHz

Consider an indoor environment at 60 GHz with furniture, walls, and a moving person.

(a) How would a telecom engineer model this channel?

(b) How would an imaging engineer model the same environment?

(c) Why does the imaging model need deterministic A\mathbf{A} while the telecom model uses stochastic fading?

Solution
Telecom perspective

The telecom engineer uses a Saleh-Valenzuela clustered model: 4 clusters, 10 rays each, Ο„rms=15\tau_{\text{rms}} = 15 ns, KK-factor =5= 5 dB. The channel is estimated per coherence interval for equalization/beamforming. Phase across snapshots is not preserved.

Imaging perspective

The imaging engineer constructs the reflectivity map c(p)c(\mathbf{p}) on a 2D grid. The sensing matrix A\mathbf{A} is computed from ray tracing with exact wall positions, material properties, and antenna locations. Phase is essential for coherent processing across all measurements.

Why the difference

The telecom model captures average channel behavior (adequate for system design: throughput, BER). The imaging model captures the specific realization of how each scene point contributes to each measurement (required for reconstruction).

In telecom, the channel is a nuisance parameter to be estimated and compensated. In imaging, the channel structure is the information we want to extract.

Quick Check

In Swerling Model I, the RCS amplitude ∣ci,j,q∣|c_{i,j,q}| follows which distribution?

Gaussian

Rayleigh

Rician

Exponential

Correction:
Rayleigh

Swerling I has exponentially distributed RCS power ∣ci,j,q∣2|c_{i,j,q}|^2, which means the amplitude ∣ci,j,q∣|c_{i,j,q}| is Rayleigh distributed. This follows from many equal-strength scatterers via the CLT.

Quick Check

Why are stochastic channel models (e.g., 3GPP spatial channel models) insufficient for RF imaging reconstruction?

They are too computationally expensive

They only characterize statistics, not the specific realization needed to build A\mathbf{A}

They cannot model multipath

Correction:
They only characterize statistics, not the specific realization needed to build A\mathbf{A}

Imaging requires the exact sensing matrix A\mathbf{A} for the specific scene. Stochastic models provide only the distribution of A\mathbf{A}, not its realization.

⚠️Engineering Note

Speckle Reduction in Practice

Speckle degrades the visual quality and detectability of targets in coherent images. Practical speckle reduction techniques include:

  1. Multi-look averaging: Average LL independent images (different look angles or frequencies). Reduces speckle variance by 1/L1/L but degrades resolution by the same factor.
  2. Spatial filtering: Lee, Frost, or Kuan filters that adapt to local statistics.
  3. Non-local means: Exploit self-similarity in the image.
  4. Deep learning denoisers: Trained on speckled/clean image pairs (Part IV).

The tradeoff is always resolution vs. speckle suppression: aggressive filtering smooths speckle but also smooths fine target detail.

Practical Constraints
  • β€’

    Multi-look averaging requires independent measurements, which costs either bandwidth or aperture

  • β€’

    All spatial filters introduce some resolution loss

Common Mistake: Using Stochastic Channel for Image Reconstruction

Mistake:

Using a stochastic channel model (e.g., Saleh-Valenzuela, 3GPP SCM) to construct A\mathbf{A} for image reconstruction.

Correction:

The sensing matrix A\mathbf{A} must be computed from a deterministic model (ray tracing, Born approximation) that reflects the actual geometry. Stochastic models recover only statistical averages, not the specific scene realization.

Why This Matters: From Swerling Models to Detection Thresholds

The Swerling models directly determine the detection performance of radar systems. Under Swerling I, the required SNR\text{SNR} for a given detection probability PdP_d and false alarm probability PfP_f is higher than for a non-fluctuating target, because the target sometimes fades to very low RCS values.

This connects to Chapters 7 (CFAR detection) and the Neyman-Pearson framework in FSI: the fluctuation model changes the likelihood ratio, which changes the optimal detector and its ROC.

Key Takeaway

Stochastic models characterize target fluctuation, not the forward model. Swerling I-IV describe how RCS varies across measurements. Speckle (Rayleigh/Rician statistics) arises from unresolved scatterers. Spatial correlation provides a Bayesian prior. But for image reconstruction, A\mathbf{A} must remain deterministic.

Swerling Target Models: RCS Fluctuation

Comparison of Swerling I (scan-to-scan fluctuation: RCS constant within each coherent processing interval but independent between CPIs) and Swerling II (pulse-to-pulse: independent draw every pulse). The animation plots normalized RCS Οƒ/ΟƒΛ‰\sigma/\bar{\sigma} over 20 pulses, showing the qualitative difference in temporal correlation structure that drives the choice of integration strategy and detection thresholds.
Deterministic Channel Models for ImagingNear-Field and Wide-Angle Effects