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.