Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Estimating the time-evolving refractivity of a turbulent medium using optical beam measurements: a data assimilation approach

Anjali Nair, Qin Li, Samuel N. Stechmann

Abstract

In applications such as free-space optical communication, a signal is often recovered after propagation through a turbulent medium. In this setting, it is common to assume that limited information is known about the turbulent medium, such as a space- and time-averaged statistic (e.g., root-mean-square), but without information about the state of the spatial variations. It could be helpful to gain more information if the state of the turbulent medium can be characterized with the spatial variations and evolution in time described. Here, we propose to investigate the use of data assimilation techniques for this purpose. A computational setting is used with the paraxial wave equation, and the extended Kalman filter is used to conduct data assimilation using intensity measurements. To reduce computational cost, the evolution of the turbulent medium is modeled as a stochastic process. Following some past studies, the process has only a small number of Fourier wavelengths for spatial variations. The results show that the spatial and temporal variations of the medium are recovered accurately in many cases. In some time windows in some cases, the error is larger for the recovery. Finally we discuss the potential use of the spatial variation information for aiding the recovery of the transmitted signal or beam source.

Estimating the time-evolving refractivity of a turbulent medium using optical beam measurements: a data assimilation approach

Abstract

In applications such as free-space optical communication, a signal is often recovered after propagation through a turbulent medium. In this setting, it is common to assume that limited information is known about the turbulent medium, such as a space- and time-averaged statistic (e.g., root-mean-square), but without information about the state of the spatial variations. It could be helpful to gain more information if the state of the turbulent medium can be characterized with the spatial variations and evolution in time described. Here, we propose to investigate the use of data assimilation techniques for this purpose. A computational setting is used with the paraxial wave equation, and the extended Kalman filter is used to conduct data assimilation using intensity measurements. To reduce computational cost, the evolution of the turbulent medium is modeled as a stochastic process. Following some past studies, the process has only a small number of Fourier wavelengths for spatial variations. The results show that the spatial and temporal variations of the medium are recovered accurately in many cases. In some time windows in some cases, the error is larger for the recovery. Finally we discuss the potential use of the spatial variation information for aiding the recovery of the transmitted signal or beam source.
Paper Structure (8 sections, 1 theorem, 35 equations, 6 figures)

This paper contains 8 sections, 1 theorem, 35 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Mathematical setup: forward and inverse problems for the paraxial wave equation
  3. Estimating the medium using data assimilation
  4. Time evolution of the turbulent medium
  5. Data assimilation strategy in general form
  6. Data assimilation using the PWE
  7. Results of computational experiments
  8. Concluding discussion

Key Result

Theorem 5.1

Denote by $A$ the solution to the PWE eqn:PWE with medium being the function $V$ and the source being $\phi$, and assume the receiver is presented by the window function $\psi$, then for this fixed configuration of source and receiver $(\phi,\psi)$, the Fréchet derivative of the forward operator $\m where $h$ solves the adjoint equation

Figures (6)

  • Figure 1: Schematic illustration of estimates of refractivity, with uncertainty quantification. (a) A traditional estimate of the mean and standard deviation of the refractivity (e.g., mean refractivity and $C_n^2$ coefficient, respectively), which lacks knowledge of spatial variations. (b-c) An estimate that includes the spatial variations of the mean refractivity, $V(x)$, along with spatial variations of the uncertainty, $\sigma(x)$. Furthermore, the aim is to reduce the uncertainty from $\sigma_1(x)$ to $\sigma_2(x)$ as more measurements are accumulated over time, as indicated in moving from the prior to posterior estimates.
  • Figure 2: The behavior of the cost function \ref{['eqn:cost_fn_expt']} with $V$ specified in \ref{['eqn:V_ansatz']}.
  • Figure 3: Time series of Fourier coefficients of the medium, $V_1(t)$ and $V_2(t)$, generated by simulating the OU processes Eq. \ref{['eqn:OU_process_simul']} with parameters $\gamma=1$ s$^{-1}$ and $\sigma = 0.1$. Time is in seconds.
  • Figure 4: Intensity measurements recorded at the receiver region.
  • Figure 5: Recovery of the Fourier coefficients $V_1(t)$ and $V_2(t)$ of the refractive index, from (\ref{['eqn:V_ansatz']}), and their evolution over time as measurements are received and assimilated. The Fourier coefficients $V_1$ and $V_2$ characterize the spatial variations of the refractivity. The recovery is successful when the wavenumber $k_x$ is larger ($4\pi/L, 6\pi/L, 8\pi/L$), and it is inaccurate when $k_x$ is smaller ($2\pi/L$) at the later times of the experiment.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 5.1