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Relationship between turbulent image variance and average image gradient

Guy Potvin

Open Access Open Access

Abstract

Optical turbulence can cause substantial distortions in imaging over long horizontal paths. For Lambertian objects, these distortions are only seen where there is a gradient in the object’s radiance. It is possible to establish a relationship between the intensity variance of a turbulent image and the average image’s gradient squared. We test the validity of a linear relationship between these quantities using turbulent imaging data. We find that it performs reasonably well for weak and intermediate optical turbulence regimes, but that some discrepancies remain to be explained.

1. INTRODUCTION

It is well known that when an electro-optical system images an object over long distances near the ground, atmospheric turbulence can create perturbations on the received image (for an historical overview, see Roggemann and Welsh [1]). However, if the surface of the object is Lambertian (incoherent), then the turbulent perturbations are only visible in regions where there is a variation in the object’s radiance. Regions where the radiance is uniform show no perturbation, no matter how strong the turbulence. This is called the “uniformity property” [2] or “energy conservation” [3,4] in the literature.

As a result, Zamek and Yitzhaky [5] developed a relationship between the variance of the image irradiance at a point and the local gradient squared of the average image. They then used this relationship to estimate the refractive index structure function parameter $C_n^2$ of the atmosphere from a video sequence of turbulent images. It is noteworthy that their work has inspired other efforts to measure turbulence parameters using image variance [6], along with deep-learning [7]. Methods have also been developed using the temporal power spectrum of the image variance [8], the variance of image displacement [9], or a combination of these [10]. Their work has also led to methods for ranging [11] and wind estimation [12] using turbulent video sequences.

Defence Research & Development Canada (DRDC) recently developed a linear perturbation model for simulating turbulent images [13] that operates along the lines of Ref. [5]. We have recently taken turbulent imaging data for the purpose of testing the validity of the linear perturbation model over various atmospheric turbulence conditions [14]. The objective of this work is to analyze the imaging data and to validate the model.

We will therefore begin by explaining the theory behind the linear model in Section 2. We then introduce and analyze the imaging data in Section 3. Finally, we assess the quality of the validation and suggest possible explanations for observed discrepancies in the conclusions, Section 4.

2. THEORY

A. Linear Perturbation Turbulent Imaging Model

For electro-optical (EO) imaging in the infra-red (IR) or visible, the wavelength $\lambda$ is much shorter that the inner scale (minimum size) ${l_0}$ of the turbulent refractive index eddies. In such a case, it was shown [15–18] that the electric field of an electromagnetic wave can be modeled as a scalar field $E$. We further assume that the propagation occurs mainly in the $z$-direction, which means that we describe the field as $E = {A_0}{e^{\textit{ikz}}}u$, where ${A_0}$ is an arbitrary amplitude and $k = 2\pi /\lambda$ is the wavenumber. The propagator function $u$ approximately obeys the paraxial equation

$$2ik{\partial _z}u + {\nabla ^2}u + 2{k^2}{n_1}u = 0 ,$$
where the operator ${\partial _z} = \partial /\partial z$ is the partial derivative with respect to $z$ and ${\boldsymbol \nabla} = ({{\partial _x},{\partial _y}})$ is the gradient along the plane perpendicular to the $z$-axis. In addition, the function ${n_1}$ is the turbulent fluctuation of the refractive index, $n = 1 + {n_1}$, which we assume is very small, ${n_1} \ll 1$.

We decompose the propagator using the Born expansion [18–22], $u = {u_0} + {u_1} + {u_2} + \ldots$, where each term ${u_k}$ being of the order of smallness $\sigma _n^k$, where ${\sigma _n}$ is the standard deviation of ${n_1}$. In what follows, we will need only the zero-order component ${u_0}$ of the expansion, which propagates through free space, and the first-order component ${u_1}$ that depends linearly on the refractive index turbulent fluctuations. The zero-order component is

$${u_0}\!\left({{{\boldsymbol \rho}_o},{z_o};{\boldsymbol \rho},z} \right) = \left({\frac{k}{{2\pi i(z - {z_o})}}} \right)\exp\! \left[{\frac{{ik}}{2}\frac{{{{\left({{\boldsymbol \rho} - {{\boldsymbol \rho}_o}} \right)}^2}}}{{z - {z_o}}}} \right],$$
which is proportional to the small-angle approximation of a spherical wave emitted from the point $({{{\boldsymbol \rho}_o},{z_o}})$ and received at the point $({{\boldsymbol \rho},z})$. The first-order component can be written as (see Potvin [22] for the derivation and physical meaning of this result)
$${u_1}\big({{{\boldsymbol \rho}_o},{z_o};{\boldsymbol \rho},z} \big) = {u_0}\big({{{\boldsymbol \rho}_o},{z_o};{\boldsymbol \rho},z} \big){\psi _1}\big({{{\boldsymbol \rho}_o},{z_o};{\boldsymbol \rho},z,t} \big),$$
where
$$\begin{split}{\psi _1}\big({{{\boldsymbol \rho}_o},{z_o};{\boldsymbol \rho},z,t} \big) & = ik(z - {z_o}) \\ & \quad\times \int_0^1 {\rm d}\eta \int {{\rm d}^2}K {N_1}({{\boldsymbol K},L\eta ,t} )\\&\quad\times\exp \!\left[{i{\boldsymbol K} \cdot {{\boldsymbol \rho}_s} - i{K^2}{\mu ^2}\eta (1 - \eta)/2} \right].\end{split}$$

Note that in Eq. (4), the variable $\eta = z^\prime /(z - {z_o})$ is the normalized $z$-coordinate, $\mu= \sqrt {(z - {z_o})/k}$ is the Fresnel zone, and

$${{\boldsymbol \rho}_s} = (1 - \eta){{\boldsymbol \rho}_o} + \eta {\boldsymbol \rho} $$
is the straight line connecting $({{{\boldsymbol \rho}_o},{z_o}})$ to $({{\boldsymbol \rho},z})$. Note also that
$${N_1}({{\boldsymbol K},z,t} ) = \frac{1}{{{{(2\pi)}^2}}}\int {{\rm d}^2}\rho {n_1}({{\boldsymbol \rho},z,t} )\exp [{- i{\boldsymbol K} \cdot {\boldsymbol \rho}} ] $$
is the two-dimensional Fourier transform of the refractive index fluctuation ${n_1}$ in the $x - y$ plane perpendicular to the line-of-sight coordinate $z$ and where ${\boldsymbol K}$ is the wavenumber vector for the turbulent refractive index fluctuations. It is of interest to note that ${\psi _1} = {\chi _1} + i{S_1}$, where ${\chi _1}$ and ${S_1}$ are the first-order Rytov expansion terms for the log-amplitude and phase fluctuations, respectively.

The top portion of Fig. 1 shows the forward view of an ideal imaging system using a thin lens with a diameter $D$ and focal length ${f_0}$ a distance $L$ from the object plane on the right and a distance $f$ from the image plane on the left. We assume that the imaging system is in focus such that $f_0^{- 1} = {f^{- 1}} + {L^{- 1}}$. The black arrows represent rays emitted from the object plane towards the lens, through the lens, and focused on the image plane. The fact that the black rays do not converge exactly on the image plane is meant to represent the diffraction from the lens. The gray arrows represent rays deviated by turbulent fluctuations, with in turn creates a random displacement and random blur at the image plane.

 figure: Fig. 1.

Fig. 1. On the top is the forward view of a simple imaging system, where the object plane emits radiance through the thin lens and towards the image plane. The black arrows represent rays with no turbulence and the gray arrows rays with turbulence. On the bottom is the reciprocal view, where the image plane emits radiance through the thin lens and towards the object plane.

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We combine the propagation through the turbulence, through the lens, and towards the image plane in a linear way using the extended Huygens-Fresnel (EHF) principle [21,23,24]. This implies that the electric field at the image plane ${E_i}({{{\boldsymbol \rho}_i}})$ is related to the electric field at the object plane ${E_o}({{{\boldsymbol \rho}_o}})$ by an integral transformation:

$${E_i}({{{\boldsymbol \rho}_i}} ) = \int {{\rm d}^2}{\rho _o} {E_o}({{{\boldsymbol \rho}_o}} )m({{{\boldsymbol \rho}_o};{{\boldsymbol \rho}_i}} ),$$
where $m$ is the transfer function:
$$\begin{split}m({{{\boldsymbol \rho}_o};{{\boldsymbol \rho}_i}} ) & = C \exp [{ik(L + f)} ] \int {{\rm d}^2}{\rho _l} u({{{\boldsymbol \rho}_o},0;{{\boldsymbol \rho}_l},L} )\\&\quad\times{\cal L}({{{\boldsymbol \rho}_l}} ){u_0}({{{\boldsymbol \rho}_l},L; - {{\boldsymbol \rho}_i},L + f} ),\end{split}$$
and where $C$ is a normalizing constant and ${\cal L}({{{\boldsymbol \rho}_l}})$ is a function representing the thin lens with a Gaussian aperture at the lens plane:
$${\cal L}({{{\boldsymbol \rho}_l}} ) = \exp \!\left[{- \frac{2}{{{D^2}}}\rho _l^2 - \frac{{ik}}{{2{f_0}}}\rho _l^2} \right] .$$

The irradiance received at the image plane is proportional to the absolute value squared of the electric field, ${I_i} \propto {E_i}E_i^*$, such that

$${I_i}({{{\boldsymbol \rho}_i}} ) \propto \int {{\rm d}^2}{\rho _o}\int {{\rm d}^2}\rho _o^\prime {\gamma _o}({{{\boldsymbol \rho}_o},{\boldsymbol \rho}_o^\prime} )m({{{\boldsymbol \rho}_o};{{\boldsymbol \rho}_i}} ){m^*}\big({{\boldsymbol \rho}_o^\prime ;{{\boldsymbol \rho}_i}} \big),$$
where
$${\gamma _o}\left({{{\boldsymbol \rho}_o},{\boldsymbol \rho}_o^\prime} \right) = {\left\langle {{E_o}\left({{{\boldsymbol \rho}_o}} \right)E_o^*\left({{\boldsymbol \rho}_o^\prime} \right)} \right\rangle _o} $$
is the mutual coherence function (MCF) at the object plane, and where the angle brackets, ${\langle \ldots \rangle _o}$, represent the average over the randomness of the object field, which is distinct from the much slower randomness of the turbulence. Since we will deal only with spatially incoherent (Lambertian) objects, the MCF may be approximated as [23]
$${\gamma _o}\left({{{\boldsymbol \rho}_o},{\boldsymbol \rho}_o^\prime} \right) \propto \delta \left({{{\boldsymbol \rho}_o} - {\boldsymbol \rho}_o^\prime} \right){I_o}\left({\frac{{{{\boldsymbol \rho}_o} + {\boldsymbol \rho}_o^\prime}}{2}} \right) ,$$
where ${I_o}$ is the radiance of the object’s surface.

The first-order Born expansion of the propagator, $u \approx {u_0} + {u_1}$, leads to a first-order Born expansion of the transfer function, $m \approx {m_0} + {m_1}$, and consequently to a first-order Born expansion of the electric field at the image plane, ${E_i} \approx {E_0} + {E_1}$. This, in turn, leads to zero- and first-order image irradiances; ${I_i} \approx {I_0} + {I_1}$, where ${I_0} \propto {E_0}E_0^*$ and ${I_1} \propto {E_0}E_1^* + {E_1}E_0^*$. It therefore follows that the zero- and first-order images are produced through zero- and first-order point spread functions (PSFs). The zero-order image is

$${I_0}\left({{{\boldsymbol \alpha}_i}} \right) = \int {{\rm d}^2}{\alpha _o} {I_o}\left({{{\boldsymbol \alpha}_o}} \right){P_0}\left({{{\boldsymbol \alpha}_i} - {{\boldsymbol \alpha}_o}} \right) ,$$
where, for convenience, we have switched to pixel coordinates, ${{\boldsymbol \alpha}_o} = {{\boldsymbol \rho}_o}/ \epsilon$ and ${{\boldsymbol \alpha}_i} = L/(f \epsilon){{\boldsymbol \rho}_i}$, where $\epsilon$ is the size of a pixel at the object plane. The zero-order PSF is (see Ref. [13] for details on the following derivations)
$${P_0}({\boldsymbol a} ) = \frac{{{\beta ^2}}}{\pi}\exp \big[{- {\beta ^2}{a^2}} \big] ,$$
where ${\boldsymbol a} = {{\boldsymbol \alpha}_i} - {{\boldsymbol \alpha}_o}$, and $\beta = \epsilon D/(2{\mu ^2})$ is a dimensionless coefficient characterizing the imager.

The first order PSF is

$$\begin{split}{P_1}\left({{{\boldsymbol \alpha}_i};{\boldsymbol a},t} \right) &= ikL\frac{{{\beta ^2}}}{\pi}\int_0^1 {\rm d}\eta \int {{\rm d}^2}K {N_1}\left({{\boldsymbol K},L\eta ,t} \right)\exp \!\left[{- {{({\eta KD/4} )}^2} + i(1 - \eta) \epsilon {\boldsymbol K} \cdot {{\boldsymbol \alpha}_i}} \right] \\ &\quad\times \left\{{\exp \!\left[{- {\beta ^2}{{\left({{\boldsymbol a} + \frac{{{\mu ^2}\eta}}{{2 \epsilon}}{\boldsymbol K}} \right)}^2} - i(1 - \eta) \epsilon {\boldsymbol K} \cdot \left({{\boldsymbol a} + \frac{{{\mu ^2}\eta}}{{2 \epsilon}}{\boldsymbol K}} \right)} \right]} \right. \\&\quad -\left. { \exp \!\left[{- {\beta ^2}{{\left({{\boldsymbol a} - \frac{{{\mu ^2}\eta}}{{2 \epsilon}}{\boldsymbol K}} \right)}^2} - i(1 - \eta) \epsilon {\boldsymbol K} \cdot \left({{\boldsymbol a} - \frac{{{\mu ^2}\eta}}{{2 \epsilon}}{\boldsymbol K}} \right)} \right]} \right\},\end{split}$$
which gives us the first-order fluctuating image:
$${I_1}\left({{{\boldsymbol \alpha}_i},t} \right) = \int {{\rm d}^2}a {I_o}\left({{{\boldsymbol \alpha}_i} - {\boldsymbol a}} \right){P_1}\left({{{\boldsymbol \alpha}_i};{\boldsymbol a},t} \right) .$$

Expression (15) for the first-order PSF is rather unwieldy, even with the simplification afforded by the Gaussian aperture. We can obtain further simplification by assuming that the object irradiance is sufficiently smooth so that its Taylor’s expansion about the point ${{\boldsymbol \alpha}_i}$ can be reasonably well approximated to second order:

$${I_o}\left({{{\boldsymbol \alpha}_i} - {\boldsymbol a}} \right) \approx {I_o}\left({{{\boldsymbol \alpha}_i}} \right) - {a_n}{\partial _n}{I_o}\left({{{\boldsymbol \alpha}_i}} \right) + \frac{1}{2}{a_n}{a_m}\partial _{\textit{nm}}^2{I_o}\left({{{\boldsymbol \alpha}_i}} \right) ,$$
where ${\partial _n} = \partial /\partial {\alpha _n}$ and repeated indexes imply a summation. It can be shown that
$$\begin{split}{I_1}\left({{{\boldsymbol \alpha}_i},t} \right) &\approx - {\partial _n}\phi \left({{{\boldsymbol \alpha}_i},t} \right){\partial _n}{I_o}\left({{{\boldsymbol \alpha}_i}} \right) \\&\quad- \frac{1}{{2{\beta ^2}}}\partial _{\textit{nm}}^2\phi \left({{{\boldsymbol \alpha}_i},t} \right)\partial _{\textit{nm}}^2{I_o}\left({{{\boldsymbol \alpha}_i}} \right) ,\end{split}$$
where $\phi$ is a scalar field such that
$$\begin{split}\phi \left({{{\boldsymbol \alpha}_i},t} \right) = - {\left({\frac{L}{\epsilon}} \right)^2}\int_0^1 {\rm d}\eta \frac{\eta}{{1 - \eta}}\int {{\rm d}^2}K {N_1}\left({{\boldsymbol K},L\eta ,t} \right) \exp \!\left[{- {{({\eta KD/4} )}^2} - {{\left({(1 - \eta)K{\mu ^2}/D} \right)}^2} + i(1 - \eta) \epsilon {\boldsymbol K} \cdot {{\boldsymbol \alpha}_i}} \right] .\end{split}$$

Rather than using the pristine image, as in Eq. (18), the linear perturbation model uses the average turbulent image instead:

$$\begin{split}\delta I\left({{{\boldsymbol \alpha}_i},t} \right) &= - {\partial _n}\phi \left({{{\boldsymbol \alpha}_i},t} \right){\partial _n}\left\langle I \right\rangle \left({{{\boldsymbol \alpha}_i}} \right) \\&\quad- \frac{1}{{2{\beta ^2}}}\partial _{\textit{nm}}^2\phi \left({{{\boldsymbol \alpha}_i},t} \right)\partial _{\textit{nm}}^2\left\langle I \right\rangle \left({{{\boldsymbol \alpha}_i}} \right) ,\end{split}$$
where the angle brackets $\langle \ldots \rangle$ signify the average over the turbulent fluctuations. This has the advantage of smoothing sharp edges, which would make the model produce finite results. It also means that as the turbulence becomes stronger, the gradient of the average image will become weaker, which may make the linear perturbation model robust in the moderate to strong optical turbulence regimes. In practice, the image $\langle I \rangle$ will also include a convolution with the zero-order PSF, ${P_0}$.

We can gain physical insight into energy conservation, Eq. (20), and the scalar field $\phi$ by considering the reciprocal view, illustrated in the bottom portion of Fig. 1. In it, each point in the image plane is emitting the same amount of radiance towards the lens. The radiance then exits the lens as a beam focused on the object plane. The beam propagates either through free space (black arrows) or through turbulence (gray arrows). Finally, the object plane absorbs the rays in proportion to its brightness in the forward view, that is, a bright region on the object plane in the forward view will absorb more radiance in the reciprocal view than a darker region. The image in the forward view corresponds to the total amount of radiance absorbed in the reciprocal view for every point in the image plane. Note that the reciprocal view is equivalent to the forward view provided the atmosphere is perfectly transparent.

Therefore, if the object plane is uniformly bright in the forward view, then the total absorption in the reciprocal view will be the same no matter where the beam lands on the object plane. This results in an image that does not fluctuate in the forward view. If the object plane brightness has a linear gradient, then the total absorption in the reciprocal view will depend in a linear way on the position of the beam’s centroid. This corresponds to the first term on the right hand side of Eq. (20). If the object plane brightness has a quadratic term, then in the reciprocal view the total absorption will depend on the beam’s width, which corresponds to the second term on the right hand side of Eq. (20).

The scalar field $\phi$, therefore, encodes both phase (via the first-order gradient) and amplitude (via the second-order derivatives) information on the focused beams in the reciprocal view. Furthermore, since the reciprocal beams should broaden as the turbulence becomes stronger, they become less capable of resolving small scale features. It therefore seems reasonable to use the average image in Eq. (20), which will also make the data analysis compatible with the model.

B. Image Variance

We evaluate the variance of the image irradiance at a given pixel by squaring Eq. (20) and then averaging:

$$\begin{split}\sigma _I^2\left({{{\boldsymbol \alpha}_i}} \right)& = \left\langle {{{\left({\delta I\left({{{\boldsymbol \alpha}_i},t} \right)} \right)}^2}} \right\rangle\\ & = {A_{\textit{nm}}}{\partial _n}\left\langle I \right\rangle \left({{{\boldsymbol \alpha}_i}} \right){\partial _m}\left\langle I \right\rangle \left({{{\boldsymbol \alpha}_i}} \right) \\ &\quad +{B_{\textit{knm}}}{\partial _k}\left\langle I \right\rangle \left({{{\boldsymbol \alpha}_i}} \right)\partial _{\textit{nm}}^2\left\langle I \right\rangle \left({{{\boldsymbol \alpha}_i}} \right) \\&\quad+ {C_{\textit{klnm}}}\partial _{\textit{kl}}^2\left\langle I \right\rangle \left({{{\boldsymbol \alpha}_i}} \right)\partial _{\textit{nm}}^2\left\langle I \right\rangle \left({{{\boldsymbol \alpha}_i}} \right) .\end{split}$$

Equation (21) is what Charnotskii [25,26] called an object to variance (O2V) function. The general expression of the O2V function for a hard aperture function, arbitrary object brightness, and any turbulence strength is quite complicated and does not lend itself to easy application. Since any image can be expressed as the sum of a low-resolution average image and a high-resolution residual image ($I = \langle I \rangle + I^\prime $), Expression (21) can be seen as the weak turbulence and low-resolution limit of the general O2V function with a Gaussian aperture. This suggests the need for a high-resolution limit of the O2V for weak turbulence to complement the low-resolution limit.

To evaluate the tensor ${A_{\textit{nm}}} = \langle {{\partial _n}\phi {\partial _m}\phi} \rangle$, we use the Markov approximation (see Strohbehn [20] and references therein) to model the covariance of ${N_1}$ as

$$\begin{split}&\left\langle {{N_1}\left({{\boldsymbol K},L\eta ,t} \right)N_1^*\left({{{\boldsymbol K}^\prime},L{\eta ^\prime},t} \right)} \right\rangle\\& = \frac{{2\pi}}{L}\delta \left({{\boldsymbol K} - {{\boldsymbol K}^\prime}} \right)\delta \left({\eta - {\eta ^\prime}} \right){\Phi _1}\left({\frac{{{\boldsymbol K} + {{\boldsymbol K}^\prime}}}{2},0} \right) ,\end{split}$$
where ${\Phi _1}({\boldsymbol K},{K_z})$ is the three-dimensional power spectrum of the turbulent fluctuations of ${n_1}$, which we model as the modified von Karman spectrum [18]:
$${\Phi _1}(K) = \frac{{0.033C_n^2}}{{{{(K_0^2 + {K^2})}^{\frac{{11}}{6}}}}}\Gamma (K{l_0}) ,$$
where ${K_0} = 2\pi /{L_0}$ is the wavenumber corresponding to the outer scale and $\Gamma$ is the inner scale function that suppresses the power spectrum at scales smaller than ${l_0}$. We use a form proposed by Churnside [27], which is designed to model a “bump” in the spectrum around the inner scale that has been observed by Williams and Paulson [28] and by Champagne et al. [29]:
$$\Gamma (x) = \exp [{- 1.28{x^2}} ] + 1.45\exp [{- 0.97{{({\ln (x) - 0.45} )}^2}} ] .$$

Note that the turbulence power spectrum Eq. (23) is isotropic. This is a questionable assumption in the surface layer, but it provides considerable simplification. The tensor ${A_{\textit{nm}}}$ becomes

$$\begin{split}{A_{\textit{nm}}} &= 2\pi \frac{{{L^3}}}{{{\epsilon ^2}}}\int_0^1 {\rm d}\eta {\eta ^2}\int {{\rm d}^2}K {K_n}{K_m}{\Phi _1}(K)\\&\quad\times\exp \!\left[{- \frac{1}{2}{{\left({\eta K\frac{D}{2}} \right)}^2} - \frac{1}{2}{{\left({(1 - \eta)K\frac{{2{\mu ^2}}}{D}} \right)}^2}} \right] .\end{split}$$

Using the power spectrum Eq. (23) and performing the substitution $K = a/s$, where $s$ is a length scale and $a$ is a dimensionless variable, we obtain

$${A_{\textit{nm}}} = G\left({\frac{s}{\mu},\frac{D}{{2\mu}},{K_0}s,\frac{{{l_0}}}{s}} \right) \sigma _R^2{\left({\frac{\mu}{\epsilon}} \right)^2}{\left({\frac{\mu}{s}} \right)^{\frac{1}{3}}}{\delta _{\textit{nm}}} ,$$
where $\sigma _R^2 = 0.124{k^{7/6}}{L^{11/6}}C_n^2$ is the Rytov parameter. Physically, the Rytov parameter is the variance of the log-amplitude of a spherical wave propagating though weak optical turbulence [19], but it can also be used to quantify the intensity of optical turbulence. That is, the weak single-scattering regime is $0 \le \sigma _R^2 \lt 0.3$, the intermediate multiple-scattering regime is $0.3 \le \sigma _R^2 \lt 10$, and beyond that is the strong saturation regime [30]. The dimensionless function $G$ is
$$\begin{split}G({c_1},{c_2},{c_3},{c_4}) &= 5.25\int_0^1 {\rm d}\eta {\eta ^2}\int_0^\infty {\rm d}a \frac{{{a^3}\Gamma (a{c_4})}}{{{{\big({c_3^2 + {a^2}} \big)}^{\frac{{11}}{6}}}}}\\&\quad\times\exp \!\left[{- \frac{{{a^2}}}{{2c_1^2}}\left\{{{{({\eta {c_2}} )}^2} + {{\left({\frac{{(1 - \eta)}}{{{c_2}}}} \right)}^2}} \right\}} \right] .\end{split}$$

Due to the isotropy of the turbulence, it is easy to see that ${B_{\textit{knm}}} = 0$. The fourth-order tensor is

$$\begin{split}{C_{\textit{klnm}}} &= H\left({\frac{s}{\mu},\frac{D}{{2\mu}},{K_0}s,\frac{{{l_0}}}{s}} \right)\sigma _R^2{\left({\frac{\mu}{\epsilon}} \right)^4}{\left({\frac{\mu}{D}} \right)^4}{\left({\frac{\mu}{s}} \right)^{\frac{7}{3}}}\\&\quad\times\left[{\frac{1}{2}{\delta _{\textit{kl}}}{\delta _{\textit{nm}}} + {\delta _{\textit{km}}}{\delta _{\textit{nl}}}} \right] ,\end{split}$$
where
$$\begin{split}H({c_1},{c_2},{c_3},{c_4})& = 10.51\int_0^1 {\rm d}\eta {\eta ^2}{(1 - \eta)^2}\int_0^\infty {\rm d}a \frac{{{a^5}\Gamma (a{c_4})}}{{{{\left({c_3^2 + {a^2}} \right)}^{\frac{{11}}{6}}}}}\\&\quad\times\exp \!\left[{- \frac{{{a^2}}}{{2c_1^2}}\left\{{{{({\eta {c_2}} )}^2} + {{\left({\frac{{(1 - \eta)}}{{{c_2}}}} \right)}^2}} \right\}} \right] .\end{split}$$

For what follows, we will set $s = D$, ${L_0} \to \infty$, and ${l_0} \to 0$. This gives us $G(D/\mu ,D/2\mu ,0,0) \approx 7.75$ over the range of possible apertures (11 to 22.5 cm; see Table 1), whereas the function $H(D/\mu ,D/2\mu ,0,0)$ varies from 28.2 for an 11 cm aperture to 33.5 for a 22.5 cm aperture.

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Table 1. Imaging Parameters and Their Values

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 figure: Fig. 2.

Fig. 2. On the left is the camera and telescope assembly. On the right are two $2 \times 2\;{\rm m}$ panels featuring sine wave patterns.

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3. TURBULENT IMAGE DATA

A. Data Acquisition Setup

Turbulent image sequences were taken on October 12–14, 2022, at the ONERA (Office national d’études et de recherches aérospatiales) laser testing facility near Toulouse, France. In all, 44 image sequences were taken, but only 35 of those were considered to be of sufficient quality for the analysis (e.g., technical problems, shadows on the panels, etc.). The times of these sequences ranged from 11:04 to 17:00 local time on the 13th, and from 11:23 to 16:30 local time on the 14th. The images were taken with a Mikrotron EoSens 1362 digital visible camera attached to a Celestron telescope, seen on the left side of Fig. 2. Note that the telescope has an aperture diameter of 22.5 cm, but for some sequences cardboard irises were placed on the aperture, creating diameters of 11 and 14 cm. Note also that the camera-telescope assembly has a pixel size at the object plane of $\epsilon = 7\;{\rm mm} $. The range was approximately 1 km, and since we assume a nominal wavelength of 550 nm, this gives us a Fresnel zone of $\mu= 9.4\;{\rm mm} $. These imaging parameters are summarized in Table 1. Finally, it is important to point out that the assembly was attached to a stable and heavy (15.2 kg) tripod, itself resting on a concrete surface. And since the wind never exceeded 1.6 m/s (5.76 km/hr), we can assume that image jitter was negligible.

In addition, a Scintec BLS900 large aperture scintillometer was operating along the same path of the imaging system, thus providing us with measured Rytov parameter values and transverse wind speeds. The camera’s acquisition rate and interval were set according to the transverse wind speed so as to avoid temporal aliasing while providing sufficiently long sequences for adequate statistical estimates. The targets were two $2 \times 2\;{\rm m}$ panels featuring sine wave patterns, seen on the right side of Fig. 2.

B. Data Analysis

The transmitter and receiver of the BLS900 scintillometer were placed approximately 130 cm above ground and operated at a wavelength of 850 nm. The scintillometer outputs scintillation index and $C_n^2$ values every minute for that wavelength. It also outputs the temperature structure function parameter $C_T^2$ by assuming an atmospheric pressure of 1000 mb and an ambient temperature of 24°C, while neglecting humidity effects. We use this $C_T^2$ value to estimate a $C_n^2$ value at our nominal wavelength of 550 nm, from which we then estimate the Rytov parameter. Finally, we smooth the resulting Rytov parameter time series by taking a 9 min moving average, which also produces a standard deviation time series. The Rytov parameter average and standard deviation time series are plotted in Fig. 3.

 figure: Fig. 3.

Fig. 3. Time plots of the 9 min moving average Rytov parameter values (black line) for October 13, 2022 (left side), and October 14, 2022 (right side). The gray zone represents the interval between plus and minus one standard deviation evaluated over the 9 min moving interval.

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Note that the conditions for the 13th and 14th of October were sunny with intermittent low-altitude fair weather cumulus clouds. This created highly variable surface layer conditions, which is evident in Fig. 3, especially for the 14th of October. In fact, there could be significant variability over the span of the 10–30 s acquisition interval. For some sequences, the illumination on the panels could vary appreciably, which could create spurious image variance. It was therefore necessary to multiply each image in a sequence with a gain value so that the average intensity over each image remains constant over that sequence.

Given an average image that is sinusoidal, $\langle I \rangle ({{{\boldsymbol \alpha}_i}}) \propto A\cos ({{\boldsymbol \kappa} \cdot {{\boldsymbol \alpha}_i}})$, where $A$ is the amplitude and ${\boldsymbol \kappa}$ the wavenumber of the average sinusoidal pattern, then the image variance from Eq. (21) becomes

$$\sigma _I^2\left({{{\boldsymbol \alpha}_i}} \right) = \sigma _R^2{\left({\frac{\mu}{\epsilon}} \right)^2}{\left({\frac{\mu}{D}} \right)^{\frac{1}{3}}}{\kappa ^2}{A^2}\left\{{G\mathop {\sin}\nolimits^2 \left({{\boldsymbol \kappa} \cdot {{\boldsymbol \alpha}_i}} \right) + \frac{3}{2}{{\left({\frac{\mu}{\epsilon}} \right)}^2}{{\left({\frac{\mu}{D}} \right)}^6}H{\kappa ^2}\mathop {\cos}\nolimits^2 \left({{\boldsymbol \kappa} \cdot {{\boldsymbol \alpha}_i}} \right)} \right\} .$$

Further given that the vertical sinusoidal pattern on the right of Fig. 2 has a wavelength of approximately 30 pixels on the average image, then it can be shown that the second term in the curly brackets is on the order of ${10^{- 7}}$ times smaller than the first term for an aperture of 11 cm (and even smaller for an aperture of 22.5 cm). We can therefore neglect the second term and write

$$\sigma _I^2\left({{{\boldsymbol \alpha}_i}} \right) \approx 7.75\sigma _R^2{\left({\frac{\mu}{\epsilon}} \right)^2}{\left({\frac{\mu}{D}} \right)^{\frac{1}{3}}}{\left[{{\boldsymbol \nabla}\left\langle I \right\rangle \left({{{\boldsymbol \alpha}_i}} \right)} \right]^2} ,$$
in accordance with Ref. [14].

Figure 4 shows the treatment for the image sequence taken at 13:24 local time on October 14, 2022. The scintillometer gave an estimated Rytov parameter of $\sigma _R^2 = 1.07 \pm 0.28$, thus placing this case in the intermediate regime, as is evident in the sample image in the upper left corner. The pair of average images in the center left is the average sub-images for each panel, where we clearly see the sinusoidal patterns along with some specs of dust on the CCD. In the lower left are the corresponding images of the gradient of the average images squared. There is a clear vertical variation, where the gradients squared at the top are stronger than at the bottom of the images.

 figure: Fig. 4.

Fig. 4. Wide image in the upper left corner is a sample image from the sequence taken at 13:24 local time on October 14, 2022. The pair of smaller images in the center left is the average images of the left and right panels. The pair in the lower left is the squares of the gradient of the averages. On the lower right are the variance images, while in the center right are the local estimate images of the Rytov parameter. In the upper right are local correlation images between the gradients squared and the variances.

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In the lower right is the pair of variance images, which have a similarly shaped pattern to the gradient squared images, but whose intensity appears roughly uniform from top to bottom. In the center right are Rytov parameter values estimated over a moving $33 \times 33$ pixel window, using the standard deviations of the gradient squared and image variance over the window, and Eq. (31). We use the standard deviation of these quantities in order to avoid contamination from constant noise in the variance and gradient squared. And in the upper right are correlation values between the gradients squared and variances over the same moving window. As we might expect for an unstable surface layer, the Rytov parameter values are strong at the bottom of the images and weaker at the top [18]. The correlation images tend to be approximately uniform, except near non-sinusoidal features, such as specs of dust and the bar between the panels.

 figure: Fig. 5.

Fig. 5. Scatter plots of the estimated Rytov parameter values versus the measured values. On the left are the results from the left panel, and on the right the results from the right panel. The black markers represent cases where the correlation between the gradient squared and the variance are 90% or greater, and the gray markers represent cases where the correlation is less than 90%.

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For each image sequence, we attribute a Rytov parameter value by averaging over the Rytov parameter images, weighted according to the correlation value. This has the advantage of emphasizing regions of strong correlation at the expense of regions of weak correlation. We also find a Rytov parameter standard deviation by the same method, which gives us a nominal error value for the Rytov parameter. Finally, we attribute a correlation value for each sequence by finding the average value over the correlation images.

These values for each of the 35 sequences are summarized in the scatter plots in Fig. 5. Error bars are included to give an idea of the variability for each point, both with respect to the measured scintillometer values and the estimated image values. We see that the measured Rytov parameter goes from about 0.07 to less than three, thereby spanning the weak to the intermediate regimes. The cases with a strong correlation (90% or more) appear in the weak and intermediate regimes, whereas the weaker correlation cases (less than 90%) are only in the intermediate regime with a Rytov parameter of approximately one. The strong correlation cases are generally consistent with the diagonal, considering the error bars. For the weaker correlation cases, the estimated Rytov values tend to overestimate the measured values, but not always, and many that do overestimate are still consistent with the diagonal if we take the error bars into account.

4. CONCLUSION

In the previous section, we have found that there is a tendency for the correlation between the variance and the squared gradient to drop and for the simple relationship in Eq. (31) to overestimate the measured value in the intermediate regime. There are numerous possible explanations for this, starting with the different sensitivities of both methods for finding the Rytov parameter. The scintillometer finds a value around its height above ground (130 cm), whereas the imaging method uses an area over most of the panels. A non-uniformity in the vertical, where there is a strong Rytov parameter near the ground as in Fig. 4, can skew the results in a way observed in Fig. 5.

In addition, the scintillometer is most sensitive to the turbulence in the middle of the path, whereas the imaging method is most sensitive to turbulence close to the aperture, as indicated in Eqs. (25) or (27). A non-uniform distribution of turbulence along the path could therefore cause a discrepancy between the two methods, but it would not explain the persistent overestimation of the imaging method over the scintillometer. Furthermore, the linear turbulent imaging model in Section 2 includes only first-order perturbations. It could be that in the intermediate regime, higher-order perturbation terms make a contribution. Unfortunately, this would not explain why these terms affect some cases but not others.

We can also question the use of isotropic Kolmogorov turbulence in the model. A real atmosphere would have a finite outer scale and a non-zero inner scale, be non-isotropic, or have intermittent coherent structures such as convective plumes. While this could explain why the overestimation affects some cases but not others, it is beyond the scope of this work.

We therefore conclude that the simple relationship between the image variance and the square of the gradient of the average image in Eq. (31) works reasonably well for weak and intermediate optical turbulence, but more work is needed to explain the discrepancies observed in the intermediate regime. We nonetheless have reasonable confidence in the linear turbulent perturbation model in Eq. (20) over those turbulence regimes.

Future work could include taking additional imaging data featuring a variety of sine wave patterns with different wavelengths, and even patterns with sharp contrasts. A comparative analysis of such data could give clues on the role that strong gradients, small scale features, and sharp edges might play in the image variance for different turbulence strengths.

Funding

The Defence & Security, Science & Technology fund of DRDC.

Acknowledgment

The author would like to gratefully acknowledge the assistance from the personnel of the DRDC - Valcartier Research Center in obtaining the turbulent imaging data. Particular thanks are due to Martin Bérubé, without whose help the acquisition of the imaging data would not have been possible. The author is also thankful to the staff at the ONERA testing facility for their assistance. The author received funding from the Defence & Security, Science & Technology fund of DRDC.

Disclosures

The author declares no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.

REFERENCES

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. On the top is the forward view of a simple imaging system, where the object plane emits radiance through the thin lens and towards the image plane. The black arrows represent rays with no turbulence and the gray arrows rays with turbulence. On the bottom is the reciprocal view, where the image plane emits radiance through the thin lens and towards the object plane.

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Fig. 2.
Fig. 2. On the left is the camera and telescope assembly. On the right are two $2 \times 2\;{\rm m}$ panels featuring sine wave patterns.

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Fig. 3.
Fig. 3. Time plots of the 9 min moving average Rytov parameter values (black line) for October 13, 2022 (left side), and October 14, 2022 (right side). The gray zone represents the interval between plus and minus one standard deviation evaluated over the 9 min moving interval.

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Fig. 4.
Fig. 4. Wide image in the upper left corner is a sample image from the sequence taken at 13:24 local time on October 14, 2022. The pair of smaller images in the center left is the average images of the left and right panels. The pair in the lower left is the squares of the gradient of the averages. On the lower right are the variance images, while in the center right are the local estimate images of the Rytov parameter. In the upper right are local correlation images between the gradients squared and the variances.

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Fig. 5.
Fig. 5. Scatter plots of the estimated Rytov parameter values versus the measured values. On the left are the results from the left panel, and on the right the results from the right panel. The black markers represent cases where the correlation between the gradient squared and the variance are 90% or greater, and the gray markers represent cases where the correlation is less than 90%.

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Tables (1)

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Table 1. Imaging Parameters and Their Values

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Equations (31)

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2 i k ∂ z u + ∇ 2 u + 2 k 2 n 1 u = 0 ,
u 0 ( ρ o , z o ; ρ , z ) = ( k 2 π i ( z − z o ) ) exp [ i k 2 ( ρ − ρ o ) 2 z − z o ] ,
u 1 ( ρ o , z o ; ρ , z ) = u 0 ( ρ o , z o ; ρ , z ) ψ 1 ( ρ o , z o ; ρ , z , t ) ,
ψ 1 ( ρ o , z o ; ρ , z , t ) = i k ( z − z o ) × ∫ 0 1 d η ∫ d 2 K N 1 ( K , L η , t ) × exp [ i K ⋅ ρ s − i K 2 μ 2 η ( 1 − η ) / 2 ] .
ρ s = ( 1 − η ) ρ o + η ρ
N 1 ( K , z , t ) = 1 ( 2 π ) 2 ∫ d 2 ρ n 1 ( ρ , z , t ) exp ⁡ [ − i K ⋅ ρ ]
E i ( ρ i ) = ∫ d 2 ρ o E o ( ρ o ) m ( ρ o ; ρ i ) ,
m ( ρ o ; ρ i ) = C exp ⁡ [ i k ( L + f ) ] ∫ d 2 ρ l u ( ρ o , 0 ; ρ l , L ) × L ( ρ l ) u 0 ( ρ l , L ; − ρ i , L + f ) ,
L ( ρ l ) = exp [ − 2 D 2 ρ l 2 − i k 2 f 0 ρ l 2 ] .
I i ( ρ i ) ∝ ∫ d 2 ρ o ∫ d 2 ρ o ′ γ o ( ρ o , ρ o ′ ) m ( ρ o ; ρ i ) m ∗ ( ρ o ′ ; ρ i ) ,
γ o ( ρ o , ρ o ′ ) = ⟨ E o ( ρ o ) E o ∗ ( ρ o ′ ) ⟩ o
γ o ( ρ o , ρ o ′ ) ∝ δ ( ρ o − ρ o ′ ) I o ( ρ o + ρ o ′ 2 ) ,
I 0 ( α i ) = ∫ d 2 α o I o ( α o ) P 0 ( α i − α o ) ,
P 0 ( a ) = β 2 π exp ⁡ [ − β 2 a 2 ] ,
P 1 ( α i ; a , t ) = i k L β 2 π ∫ 0 1 d η ∫ d 2 K N 1 ( K , L η , t ) exp [ − ( η K D / 4 ) 2 + i ( 1 − η ) ϵ K ⋅ α i ] × { exp [ − β 2 ( a + μ 2 η 2 ϵ K ) 2 − i ( 1 − η ) ϵ K ⋅ ( a + μ 2 η 2 ϵ K ) ] − exp [ − β 2 ( a − μ 2 η 2 ϵ K ) 2 − i ( 1 − η ) ϵ K ⋅ ( a − μ 2 η 2 ϵ K ) ] } ,
I 1 ( α i , t ) = ∫ d 2 a I o ( α i − a ) P 1 ( α i ; a , t ) .
I o ( α i − a ) ≈ I o ( α i ) − a n ∂ n I o ( α i ) + 1 2 a n a m ∂ nm 2 I o ( α i ) ,
I 1 ( α i , t ) ≈ − ∂ n ϕ ( α i , t ) ∂ n I o ( α i ) − 1 2 β 2 ∂ nm 2 ϕ ( α i , t ) ∂ nm 2 I o ( α i ) ,
ϕ ( α i , t ) = − ( L ϵ ) 2 ∫ 0 1 d η η 1 − η ∫ d 2 K N 1 ( K , L η , t ) exp [ − ( η K D / 4 ) 2 − ( ( 1 − η ) K μ 2 / D ) 2 + i ( 1 − η ) ϵ K ⋅ α i ] .
δ I ( α i , t ) = − ∂ n ϕ ( α i , t ) ∂ n ⟨ I ⟩ ( α i ) − 1 2 β 2 ∂ nm 2 ϕ ( α i , t ) ∂ nm 2 ⟨ I ⟩ ( α i ) ,
σ I 2 ( α i ) = ⟨ ( δ I ( α i , t ) ) 2 ⟩ = A nm ∂ n ⟨ I ⟩ ( α i ) ∂ m ⟨ I ⟩ ( α i ) + B knm ∂ k ⟨ I ⟩ ( α i ) ∂ nm 2 ⟨ I ⟩ ( α i ) + C klnm ∂ kl 2 ⟨ I ⟩ ( α i ) ∂ nm 2 ⟨ I ⟩ ( α i ) .
⟨ N 1 ( K , L η , t ) N 1 ∗ ( K ′ , L η ′ , t ) ⟩ = 2 π L δ ( K − K ′ ) δ ( η − η ′ ) Φ 1 ( K + K ′ 2 , 0 ) ,
Φ 1 ( K ) = 0.033 C n 2 ( K 0 2 + K 2 ) 11 6 Γ ( K l 0 ) ,
Γ ( x ) = exp ⁡ [ − 1.28 x 2 ] + 1.45 exp ⁡ [ − 0.97 ( ln ⁡ ( x ) − 0.45 ) 2 ] .
A nm = 2 π L 3 ϵ 2 ∫ 0 1 d η η 2 ∫ d 2 K K n K m Φ 1 ( K ) × exp [ − 1 2 ( η K D 2 ) 2 − 1 2 ( ( 1 − η ) K 2 μ 2 D ) 2 ] .
A nm = G ( s μ , D 2 μ , K 0 s , l 0 s ) σ R 2 ( μ ϵ ) 2 ( μ s ) 1 3 δ nm ,
G ( c 1 , c 2 , c 3 , c 4 ) = 5.25 ∫ 0 1 d η η 2 ∫ 0 ∞ d a a 3 Γ ( a c 4 ) ( c 3 2 + a 2 ) 11 6 × exp [ − a 2 2 c 1 2 { ( η c 2 ) 2 + ( ( 1 − η ) c 2 ) 2 } ] .
C klnm = H ( s μ , D 2 μ , K 0 s , l 0 s ) σ R 2 ( μ ϵ ) 4 ( μ D ) 4 ( μ s ) 7 3 × [ 1 2 δ kl δ nm + δ km δ nl ] ,
H ( c 1 , c 2 , c 3 , c 4 ) = 10.51 ∫ 0 1 d η η 2 ( 1 − η ) 2 ∫ 0 ∞ d a a 5 Γ ( a c 4 ) ( c 3 2 + a 2 ) 11 6 × exp [ − a 2 2 c 1 2 { ( η c 2 ) 2 + ( ( 1 − η ) c 2 ) 2 } ] .
σ I 2 ( α i ) = σ R 2 ( μ ϵ ) 2 ( μ D ) 1 3 κ 2 A 2 { G sin 2 ⁡ ( κ ⋅ α i ) + 3 2 ( μ ϵ ) 2 ( μ D ) 6 H κ 2 cos 2 ⁡ ( κ ⋅ α i ) } .
σ I 2 ( α i ) ≈ 7.75 σ R 2 ( μ ϵ ) 2 ( μ D ) 1 3 [ ∇ ⟨ I ⟩ ( α i ) ] 2 ,
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