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Random Media at Colorado

Website for the summer school: Summer School Waves and Particles in Random Media: Theory and Applications.

The summer school was held on campus of Colorado State University in Fort Collins.

Borcea

Weak scattering regime in open environments. Stronger regimes require an in depth study of waves in random media.

Coherent imaging: Transport mean free part-quantify the decay of the mean.

CINT passive arrays. Model of random medium: \[\frac{1}{c^2(x)} = \frac{1}{c_0^2} \left[1+\sigma \mu\left(\frac{x}{l} \right) \right] \] $\mu(x)$ is mean zero, statistically homogeneous. Model applied to clear air turbulence and captures only wave front distortions. The model is assumed in the adaptive optics methodology.

Travel time is given by Fermat's principle\[\tau(x,y) = \min\limits_{\Gamma}\int_{\Gamma}\frac{ds}{c(x(s))}. \] Derivation of the random travel time model: For $\lambda\ll l\ll L$ seek solution of Helmholtz equation: $u=\alpha e^{iw\tau}$:

  • Travel time $\tau$ solves eikonal equation: $\left|\nabla\tau\right|^2 = \frac{1}{c^2(x)}$.
  • Amplitude $\alpha$ solves transport equation $2\nabla\alpha \cdot \nabla\tau +\alpha\Delta \tau =0$.

The result extends to fairly general fluctuations: stationary, mean zero, ergodic in $z$ (mixing). Diffusion limit theorem: Kohler, Papanicolaou, Varadhan.

Scales are important!

Cross-range resolution. CINT

Synthetic aperture radar imaging - Margaret Cheney

How to get image of the Earth and stars far away

SAR: developed by engineering community; scattering theory, microlocal, ...

Antenna on an aircraft sends signals to ground satellites then receives back the reflected signal. Goal: get image on the ground from above (10km $\sim$ small photo)

1951: Carl Wiley, Goodyear

mid-50: first operational systems

1960: NASA sponsorship, first digital SAR processors

78: SEASAT-A: 100 days

1981: beginning SIR series (shuttle imaging radar)

SIR-C 1994 Weddell Sea: create false color. Assign different channel to different color: create beautiful picture.

Satellite: making amazing image of the Earth
TerraSAR cooper mine in Chile

Internal waves at Gibraltar.

ERS-1, ERS-2:

Airborne systems: AriSAR, LynxSAR: Capitol, Radar projection is not the same as optical projection.

Outline: standard matheamtical model for radar data (assumes propagation through free space). Image reconstruction from standard model.

3D Math model: we should use Maxwell's equation, but instead (scalar wave) \[\left(\nabla^2-\frac{1}{c^2(x)}\partial_t^2 \right)E(t,x)=i(t,x) \;\text{(source)}\] but not good for random medium.
One thing to need to know: Fundamental solution: Green's solution: outgoing spherical wave (fourier transform of delta function)

Scattering theory: empty universe, vacuum wave speed. Lippman-Schwinger integral equation. \[E^{sc}(t,x)=\int g(t-\tau,x-z)V(z)\partial_{\tau}^2E(\tau,z)d\tau dz.\] Inverse problem: measure scatter field, and $V(z)$. Problem, $V(z)$ is attached to the nearby scatter field.

Single-scattering and Born approximation: makes inverse problem linear but not good approximation. Math techniques to remove artifacts.

Incident wave: the field from the antenna is $E^{in}$. Real-aperture imaging versus synthetic-aperture imaging. Big antenna: narrow beam (R-A imaging), small antenna: broad beam (S-A imaing): combine data from different location.

CSU-CHILL: very large antenna disk, plot from different time-delayed data.

Putting it all together: Fourier Integral Operator (microlocal analysis techniques apply). Similar to seismic inveresion problem.

Reconstruction a function from its integrals over circles or lines. Brett Borden, Naval. Rdar FIO, Beyklin (JMP '85). Radar application: Nolan & Cheney.

Random medium is needed

  • through foliage (what's in the shadow)
  • through sea ice (complicated)
  • through ionosphere (complicated, many layers)

CARABAS uses long wavelengths.

Electromagnetic waves and SEA ICE (salt in water very conductive, like metal).

Fundamentals of Radar Imaging.

Ganesh

Attenuation

Bal

Stochastic and periodic homogenization \[-\nabla\cdot a\nabla u =f;\quad u=g\; \text{on }\partial U\\ -\nabla\cdot a\nabla u+\alpha u =f \] We have $a(x)\to a(x/\varepsilon)$. Two-scales: homogenization $\Leftrightarrow$ effective media or macroscopic models.

We have $a=a_{\varepsilon}(x)=a(x/\varepsilon)$.

First setting: $a(y)$ is periodic coefficient, $(y=x/\varepsilon)$. \[-\nabla a_{\varepsilon}\nabla u_{\varepsilon}+\alpha u_{\varepsilon}=f \] We conjecture $u_{\varepsilon}[a_{\varepsilon}]\to_{\varepsilon \to 0} {\bar u}[{\bar a}]$. Two scales $y=x/\varepsilon$ (fast scale). Two scale expansion $u_{\varepsilon}(x)=u(x,y)$$\;=u(x,x/\varepsilon)$. This is an ansatz, maybe wrong but good guess. Then write equation for $u(x,y)$, then get equations for different scales.

Fredholm alternative: spectral gap; 0 is the first one.

Effective constant \[a^* = \left({\mathbb E}\frac{1}{a} \right)^{-1} \] $a(y)$ must be elliptic between $\lambda$ and $\Lambda$. We have a result basically saying sharpness \[ \lambda \le \mathbb E(a^{-1})^{-1} \le a^*\le \mathbb E a\le \Lambda\] Justification:

  • Hilbert expansion
  • Variational structure
    • Tartar oscillatory fast function (Gregoire Allaire 2010; Stefan Neukamm 2017)
    • Second corrector $\sigma$

Presented method is not optimal, but the above two are.

Limit of the product: $ a_{\varepsilon}\nabla u_{\varepsilon} \to \;?$. We have \[ \begin{aligned} a_{\varepsilon}\nabla u_{\varepsilon} &= a_{\varepsilon}(I+\nabla \theta_1) \nabla u_0+\varepsilon O(1)\\ & = a^*\nabla +\ldots \end{aligned}\] (div-curl lemma is hidden).

Periodic setting, we summerize \[ u_{\varepsilon}=u_0 + O(\varepsilon)\\ \nabla u_{\varepsilon} = (I+\nabla \theta_1)\nabla u_0+O(\varepsilon)\]

Random settings

Introduction: Book Shen: Homogenization: periodic setting, boundary layers, ... / Alloire, Neukamm: lecture notes

Random fluctuations NOT given by ansatz.

  • A. Gloria - F. Otto - S. Neukamm
  • Scott Amstrong and C. Smart - Mourrat
  • Book: Mourat - Kuusi - Armstrong

Goal: Understand the fluctuations: know what to upscale/ Noise is influenced by the physics. \[\displaystyle \int_{0}^{\infty} \int_{0}^{\infty} \dfrac{\sin x \sin y \sin (x+y)}{xy(x+y)} \mathrm{d}x\,\mathrm{d}y = \zeta(2)\] Ergodicity:

Kozlov 79, Papanicolaou-Varadhan 79, 87, Daraso rote ,86

No rate of convergence. Gloria-otto, Armstrong-Smart

Build intuition: 1D. Domain $(0,1)$: \[ -\partial_x a_{\varepsilon}\partial_x u_{\varepsilon}=f;\;\; u_{\varepsilon}(0)=u_{\varepsilon}(1)=0.\] Theorem: sum of stationary, limit exists almost surely. Ergodicity: Birkhoff's ergodic theorem, the limit is the average.

Bal, YuGu. Rosenblatt process.

$d\ge 2$ is much harder. $a\to u$ hard/not explicit representation map (Stefan Neukamm's lecture notes; book >2017).

Calculus of variation: Dal Naso, Nodia.

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