A 2026 Optica paper by Wang, Qi, Liu, Deng and Han established a forward stochastic-partial-differential-equation (SPDE) theory in which light propagating through a medium whose refractive-index fluctuations carry temporal fractional-Brownian memory and spatial power-law correlations maps onto the hyperbolic Anderson model, yielding exact scaling laws that tie the medium's Hurst index HH and spatial power-law exponent α\alpha to measurable field statistics. The paper itself demonstrated only a single-parameter, correlation-level estimate of HH from the received optical phase. The open problem is the systematic inverse problem: jointly recovering HH, α\alpha, and the covariance amplitude from optical measurements alone, with an identifiability analysis stating when this inversion is well posed.

Background

Classical theories of optical propagation through atmospheric turbulence — Rytov smooth-perturbation theory (a perturbative expansion of the wave equation in the small refractive-index fluctuations), higher-order moment equations, and multiple phase-screen simulation (numerical propagation through a stack of thin random refractive-index screens) — rest on a Markov approximation: refractive-index fluctuations are treated as delta-correlated, i.e. statistically memoryless, along the propagation path. Wang, Qi, Liu, Deng and Han (Optica 13(7), 1407, 2026; arXiv:2601.11213) replaced this with a non-Markovian model in which the fluctuation source W(t,x)W(t,\boldsymbol{x}) is a random field with space-time covariance

E[W(t,x) W(s,y)]=CH (t2H+s2H−∣t−s∣2H) ∣x−y∣−α,\mathbb{E}[W(t,\boldsymbol{x})\,W(s,\boldsymbol{y})] = C_H\,(t^{2H}+s^{2H}-|t-s|^{2H})\,|\boldsymbol{x}-\boldsymbol{y}|^{-\alpha},

valid for separations ∣x−y∣≥l0|\boldsymbol{x}-\boldsymbol{y}| \ge l_0, where the inner scale l0l_0 prevents divergence at the origin, with s,t∈R+s,t\in\mathbb{R}_+, x,y∈R3\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^3, and α>0\alpha>0 subject to the Dalang well-posedness condition of the resulting SPDE (which for the three-dimensional Riesz kernel ∣x−y∣−α|\boldsymbol{x}-\boldsymbol{y}|^{-\alpha} restricts to α<2\alpha<2). This combines fractional Brownian motion (fBm — the Gaussian process with stationary increments whose increments remain correlated across all lag scales, so the process carries memory) in time with a spatial multifractal power law. The Hurst index H∈(0,1)H\in(0,1) sets the temporal memory: H>1/2H>1/2 gives persistent (positively correlated) fluctuations, H<1/2H<1/2 anti-persistent ones, and H=1/2H=1/2 is the memoryless Brownian limit; the exponent α\alpha is the power-law exponent of the spatial Riesz-type kernel — in three dimensions the corresponding spectral density scales as ∣ξ∣α−3|\boldsymbol{\xi}|^{\alpha-3} — so this convention differs from, and should not be identified with, the generalized spectral exponent commonly used in the non-Kolmogov turbulence literature. Driven by this noise, the propagation equation derived directly from the full wave equation (via an operator-splitting decomposition, without a paraxial or small-angle reduction) corresponds formally to the hyperbolic Anderson model, the linear stochastic wave-type equation with multiplicative (solution-proportional) noise studied in stochastic PDE theory, going beyond the standard parabolic approximation governed by the Markov approximation. The paper proves forward scaling laws linking medium parameters to observable statistics of the propagated field u(t,x)u(t,\boldsymbol{x}): a long-time moment-growth law log⁡Eup(t,x0)=p(p−12)1/(2−α)C1(α,H) t(4H−α)/(2−α)\log\mathbb{E}u^{p}(t,\boldsymbol{x}_0) = p\left(\frac{p-1}{2}\right)^{1/(2-\alpha)}C_1(\alpha,H)\,t^{(4H-\alpha)/(2-\alpha)}, exact relations among normalized intensity moments ⟨Iq⟩/⟨I⟩q\langle I^q\rangle/\langle I\rangle^q (with intensity I=∣u∣2I=|u|^2), and a bound on the symmetrized Kullback-Leibler divergence (sKL, a symmetric distance between probability densities) showing that the aperture-averaged field converges to a complex Gaussian at rate at least R−α/2R^{-\alpha/2} as the receiver radius RR grows. The constant C1(α,H)C_1(\alpha,H) is dictated by (α,H)(\alpha,H) alone, and the normalized intensity moments and the sKL convergence rate are amplitude-free; the amplitude CHC_H enters observably through the unnormalized two-point covariance of the field (source Eq. 14). An outdoor experiment using heterodyne detection — mixing the received field with a local-oscillator reference so that its complex amplitude, including the optical phase, is measurable — with receiver apertures dilated from 2 mm to 300 mm validated these laws and used rescaled-range (R/S) analysis, the classical log-slope estimator of the Hurst index of a time series, to show that the Hurst index estimated from the received optical phase tracks that of co-located medium fluctuations (Pearson correlation 0.905 at a 2 mm aperture, falling to 0.341 at 300 mm, where spatial averaging erases the memory signature). A separate, older literature treats only the spatial exponent under memoryless models: forward laws for angle-of-arrival (wavefront-tilt) statistics of waves in non-Kolmogorov turbulence with a general spectral exponent (Du et al. 2009; Toselli et al. 2007; Cui 2015), starlight-based measurements (Fu et al. 2010), and analyses of turbulence-strength parameters in non-Kolmogorov optical experiments (Golbraikh and Kopeika 2004). Fractional-Brownian memory itself was introduced into refractive-index modeling — for correlations along the propagation path, a spatial rather than temporal effect — by Perez, Zunino and Garavaglia (2004). None of these threads combines temporal memory with spatial scaling, and none inverts a forward propagation model for both parameters.

Problem Statement

Solve the inverse counterpart of the forward SPDE theory of Wang et al. (2026): given only optical measurements of a propagated field — for example time series of the detected optical phase or intensity at one or more receiver apertures, intensity-moment scaling, and aperture-dependent distributional statistics — construct and justify a procedure that recovers the parameters of the medium covariance

E[W(t,x) W(s,y)]=CH (t2H+s2H−∣t−s∣2H) ∣x−y∣−α,\mathbb{E}[W(t,\boldsymbol{x})\,W(s,\boldsymbol{y})] = C_H\,(t^{2H}+s^{2H}-|t-s|^{2H})\,|\boldsymbol{x}-\boldsymbol{y}|^{-\alpha},

posed for separations ∣x−y∣≥l0|\boldsymbol{x}-\boldsymbol{y}| \ge l_0 (inner-scale cutoff): the temporal Hurst index HH, the spatial power-law exponent α\alpha, and the amplitude scale CHC_H. The model class is that of the source's Eq. (4): x,y∈R3\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^3, s,t∈R+s,t\in\mathbb{R}_+, H∈(0,1)H\in(0,1), and α>0\alpha>0 subject to the Dalang well-posedness condition (which for the three-dimensional Riesz kernel ∣x−y∣−α|\boldsymbol{x}-\boldsymbol{y}|^{-\alpha} restricts to α<2\alpha<2), with l0l_0 the fixed inner-scale cutoff of the covariance model. The inverse problem is to specify a set of optical observables and a reproducible estimator mapping them to (H,α,CH)(H,\alpha,C_H) — the amplitude CHC_H is carried by unnormalized field/intensity covariances, while normalized-moment and sKL-rate observables depend only on (α,H)(\alpha,H) — or, if the pair (H,α)(H,\alpha) is not identifiable from those observables, to prove that non-identifiability and characterize which subset of the parameters (H,α,CH)(H,\alpha,C_H) — as opposed to which subset of the observables — remains identifiable, together with the remaining degeneracies. The inverse problem also encompasses an identifiability analysis within this model class establishing for which measurement conditions — aperture size, sampling rate, averaging window, turbulence regime, propagation geometry — the parameter-to-statistics map is injective, and where it provably fails or becomes ill-conditioned.

References

  1. Chaoran Wang, Jinquan Qi, Shuang Liu, Chenjin Deng, Shensheng Han (2026). 'Light propagation through space-time non-Markovian random media.' Optica 13(7), 1407. DOI: 10.1364/optica.601230. https://doi.org/10.1364/optica.601230 (preprint arXiv:2601.11213, https://arxiv.org/abs/2601.11213)
  2. D.G. Pérez, L. Zunino, M. Garavaglia (2004). 'A fractional Brownian motion model for the turbulent refractive index in lightwave propagation.' Optics Communications 242(1-3), 57-63. DOI: 10.1016/j.optcom.2004.08.007. https://doi.org/10.1016/j.optcom.2004.08.007
  3. Wenhe Du, Siyuan Yu, Liying Tan, Jing Ma, Yijun Jiang, Wanqing Xie (2009). 'Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence.' Optics Communications 282(5), 705-708. DOI: 10.1016/j.optcom.2008.11.006. https://doi.org/10.1016/j.optcom.2008.11.006
  4. Italo Toselli, Larry C. Andrews, Ronald L. Phillips, Valter Ferrero (2007). 'Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence.' Proceedings of SPIE 6551, 65510E. DOI: 10.1117/12.719033. https://doi.org/10.1117/12.719033
  5. Linyan Cui (2015). 'Analysis of angle of arrival fluctuations for optical waves' propagation through weak anisotropic non-Kolmogorov turbulence.' Optics Express 23(5), 6313. DOI: 10.1364/OE.23.006313. https://doi.org/10.1364/oe.23.006313
  6. Sen Fu, Liying Tan, Jing Ma, Yanping Zhou (2010). 'Effect of non-Kolmogorov turbulence on angle-of-arrival fluctuations of starlight.' Journal of Russian Laser Research 31(4), 332-337. DOI: 10.1007/s10946-010-9152-4. https://doi.org/10.1007/s10946-010-9152-4
  7. Ephim Golbraikh, Norman S. Kopeika (2004). 'Turbulence strength parameter in laboratory and natural optical experiments in non-Kolmogorov cases.' Optics Communications 242(4-6), 333-338. DOI: 10.1016/j.optcom.2004.08.045. https://doi.org/10.1016/j.optcom.2004.08.045
  8. Jinquan Qi, Shuang Liu, Chenjin Deng, Chaoran Wang, Youzhen Gui, Shensheng Han (2026). 'Multi-mode coherent detection ghost imaging lidar with an imaging bucket detector.' Optics & Laser Technology 204, 116186. DOI: 10.1016/j.optlastec.2026.116186. https://doi.org/10.1016/j.optlastec.2026.116186