METHOD AND DEVICE FOR EVALUATING PARAMETERS CHARACTERIZING ATMOSPHERIC TURBULENCE

Abstract

A method for characterizing the atmospheric turbulence, includes acquiring images of a celestial object by means of a camera coupled to a small telescope; analyzing the acquired images to determine angle of arrival fluctuations of wavefronts from positions of spots formed by the celestial object in the acquired images; determining variances of the angle of arrival fluctuations; and estimating the Fried parameter from the variances of the angle of arrival fluctuations, by setting an outer scale parameter of the atmospheric turbulence to a fixed median value.

Claims

1. A method for characterizing the atmospheric turbulence, comprising: acquiring images (SIM) of a celestial object (PS, CB) by means of a camera (CM) coupled to a telescope (T); analyzing the acquired images to determine angle of arrival fluctuations of wavefronts from positions of at least one spot (IPS, P1, P2) formed by the celestial object in the acquired images; determining variances of the angle of arrival fluctuations; and estimating a Fried parameter (r.sub.0) from the variances of the angle of arrival fluctuations, by setting an outer scale parameter (L.sub.0) of the atmospheric turbulence to a fixed median value, wherein the at least one spot is formed by the Polar star (PS) to which the telescope is rigidly pointed, or the at least one spot comprises two spots (P1, P2) spaced apart from each other in each acquired image.

2. The method of claim 1, wherein the outer scale parameter (L.sub.0) is set to 20 m plus or minus 10%.

3. The method of claim 1, wherein the telescope (T) is rigidly secured to a fixed support in a direction towards the Polar star (PS).

4. The method of claim 3, wherein the Fried parameter is estimated from the following equation:
σ.sup.2=0.18λ.sup.2r.sub.0.sup.−1/3(D.sup.−1/3−1.525L.sub.0.sup.−1/3) wherein σ.sup.2 is the variance of the angle of arrival fluctuations, λ is the wavelength of the light emitted by the observed star (PS), r.sub.0 is the Fried parameter, D is an aperture diameter of the telescope (T), and L.sub.0 is the outer scale parameter.

5. The method of claim 1, wherein the telescope (T) is fixed on a mount and oriented towards the Moon or Sun limb (SML), the mount being motorized and controlled to compensate the Earth rotation about its rotation axis, the acquired images being analyzed to determine the angle of arrival fluctuations of light from positions of two spots (P1, P2) spaced apart from each other on the Moon or Sun limb.

6. The method of claim 5, wherein the Fried parameter is estimated from the following equation: $D_{\alpha ,s}\left(\theta \overline{h}\right)=0.364{\lambda }^2{r}_0^{-5/3}{D}^{-1/3}\left[1-0.798{\left(\frac{\theta \overline{h}}{D}\right)}^{-1/3}\right]$ wherein h is the equivalent altitude of the whole atmospheric turbulence, λ is the wavelength of the light emitted by the observed limb, r.sub.0 is the Fried parameter, D is an aperture diameter of the telescope (T), θ is an angular separation between two viewing angles θ.sub.1 and θ.sub.2 of considered points (P1, P2) of the observed limb (SML), α(θ.sub.1) and α(θ.sub.2) are angle of arrival fluctuations at the two considered points, and D.sub.α,s(θh) is a spatial structure function of angle of arrival fluctuations α for the angular separation θ.

7. The method of claim 6, wherein the equivalent altitude h of the whole atmospheric turbulence is deduced from the following equation: $\frac{D_{\alpha ,s}\left({\theta }_a\overline{h}\right)}{D_{\alpha ,s}\left({\theta }_b\overline{h}\right)}\simeq \frac{D^{-1/3}-0.798{\left({\theta }_a\overline{h}\right)}^{-1/3}}{D^{-1/3}-0.798{\left({\theta }_b\overline{h}\right)}^{-1/3}}$ wherein θ.sub.a and θ.sub.b are angular separations of two considered pairs of points (P1, P2) on the observed limb (SML), and D.sub.α,s(θ.sub.ah) and D.sub.α,s(θ.sub.bh) are values of the spatial structure function (D.sub.α,s) of angle of arrival fluctuations for the two angular separations θ.sub.a and θ.sub.b.

8. The method of claim 1, wherein the acquired images are analyzed in a limited region around analyzed points (IPS, P1, P2) formed by the celestial object (PS, CB).

9. The method of claim 1, wherein the variance of the angle of arrival fluctuations is multiplied by cos(z), z being an angle between the direction of the observed star (PS) and the zenithal direction, at an observation site where the images are acquired.

10. The method claim 1, wherein the images are acquired by the camera (CM) at an image rate of 50 to 200 images/s.

11. A device configured to implement the method of claim 1, to analyze images (SIM) acquired by a camera (CM) coupled to a telescope (T) pointed at a celestial object (PS, CB).

12. The device of claim 11, comprising a camera (CM) coupled to a telescope (T), and a processing card (PRC) receiving and processing images acquired by the camera.

13. The device of claim 12, wherein the telescope (T) has at least one of the following features: it has an aperture diameter set to a value between 4 and 12 cm it comprises a Barlow lens (B) interfacing with the camera (CM) to increase the focal length (F) of the telescope, and it is of the type Cassegrain.

14. A computer program product loadable into a computer memory and comprising code portions which, when carried out by a computer, configure the computer to carry out the method of claim 1, to analyze images (SIM) acquired by a camera (CM) coupled to a telescope (T) pointed at a celestial object (PS, CB).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0024] The method and/or device may be better understood with reference to the following drawings and description. Non-limiting and non-exhaustive descriptions are described with the following drawings. In the figures, like referenced signs may refer to like parts throughout the different figures unless otherwise specified.

[0025] FIG. 1 is a perspective view of a device for estimating parameters of the atmospheric turbulence, according to an embodiment;

[0026] FIG. 2 is a sectional view of the device, according to an embodiment;

[0027] FIG. 3 is a front view of an optical aperture of the device, according to an embodiment;

[0028] FIG. 4 is a block diagram of a processing card of the device;

[0029] FIG. 5 is a simplified optical scheme of the device, according to an embodiment;

[0030] FIG. 6 is a simplified optical scheme showing how angle of arrival fluctuations are measured in the device, according to an embodiment;

[0031] FIG. 7 shows curves of variation of the variance of angle of arrival fluctuations as a function of the Fried parameter, for different values of the outer scale;

[0032] FIG. 8 shows a curve of variation of the estimated Fried parameter as a function of the outer scale;

[0033] FIG. 9 shows a curve of variation of an estimation error of Fried parameter as a function of the outer scale;

[0034] FIG. 10 shows a curve of a temporal structure function of the angle of arrival fluctuations as a function of time delay;

[0035] FIG. 11 is an optical scheme showing the Sun or Moon limb and illustrating the transition from an angular correlation to a spatial correlation used in a method for estimating the Fried parameter from the Sun or Moon limb, according to an embodiment;

[0036] FIG. 12 is a view of the observed Sun or Moon limb, illustrating the estimation method of angle of arrival fluctuations.

DETAILED DESCRIPTION

[0037] FIG. 1 illustrates a device for estimating parameters of the atmospheric turbulence, according to an embodiment. The device comprises a fixed support 2 supporting a casing 1 housing a telescope, a camera and a data processing card. The casing 1 comprises a tube 11 preferably sealed in a watertight manner by a transparent window 12. The tube 11 has for example a cylindrical shape.

[0038] According to an embodiment, the fixed support 2 comprises a base 21 intended to be rigidly secured to a fixed and rigid location (e.g. a pilar). The support 2 comprises a plate 22 rotatably mounted parallel on the base 21 around an axis perpendicular to the base 21. A mechanism 24 is provided to accurately adjust an azimuth angle of the plate 22 with respect to the base 21 and strongly maintain this angle. The support 2 further comprises a plate 23 extending perpendicularly to the plate 22 and which can be made integral with the plate 22. The tube 11 is rotatably fixed to the plate 23 around an axis perpendicular to the plate 23, by means of screws 25, 26 arranged in slotted holes formed in the plate 23. A rod coupled with an adjustment wheel 27 and having ends respectively fixed to the plate 23 and to the tube 11 is provided to accurately adjust an elevation angle of the tube 11. When the elevation angle is adjusted, the screws 25, 26 are tightened to strongly maintain the elevation angle.

[0039] FIG. 2 shows the interior of the tube 11. The tube 11 houses a telescope T coupled with a digital camera CM connected to a processing card PRC. The processing card PRC can also be integrated into the camera CM. A Barlow lens B can be inserted between the telescope T and the camera CM to extend the focal length of the telescope T in order to increase the sensitivity of the telescope T. The telescope T can be of the type refractive or reflective, with a circular aperture AP having a diameter of about 10 cm (between 8 and 12 cm) and a central obstruction CO having a diameter of about 4 cm (FIG. 3). The obstruction CO can be supported by arms R.

[0040] According to another embodiment, no processing card is housed in the tube 11, the camera CM being coupled to a connector to be connected to an external processing device.

[0041] According to an embodiment, the telescope T is of the type Cassegrain. The camera CM can be of CCD (Charge-Coupled Device) type.

[0042] The support 2 is intended to strongly maintain and point the telescope T at an almost fixed celestial object, i.e. the polar star (α UMi), so as to prevent vibrations due to the wind. In this manner, the telescope T does not need to be motorized to follow the Earth rotation about its polar axis. Thus the vibrations generated by a motorized mount are avoided. The casing is intended to protect the telescope T, the camera CM and the card PRC against the weather.

[0043] FIG. 4 illustrates components of the processing card PRC. The processing card PRC comprises at least one processor MC, and, operatively coupled to the processor, memories MEM, and communication circuits NIT. The memories MEM comprise volatile memories and non-volatile memories. The non-volatile memories store an operating system, applications and all images provided by the camera CM. The communication circuits NIT enables the processor MC to be operatively coupled to an electronic communication network NT, and is configured to transmit the images captured by the camera CM to an external computer CP. For example, when these computer readable program code components are processed by the processor MC, the program code components are configured to cause execution of the method for analysing the images and for computing parameters characterizing the atmospheric turbulence, as described below. In another embodiment, all or a part of these parameters are computed by an external computer CP to which the images captured by the camera CM are sent.

[0044] FIG. 5 illustrates the optical path from the observed star PS through the atmosphere, the telescope T, the Barlow lens B, up to an image sensor IS of the camera CM. The light emitted by the observed star PS forms wave fronts WF1, WF2 which are perturbed by different turbulent layers of the atmosphere. FIG. 5 also shows an image SIM of observed star PS formed on the image sensor IS of the camera CM in the focal plane of the telescope T associated with the Barlow lens B.

[0045] The Fried parameter can be estimated to characterize the turbulence of the atmosphere. The Fried parameter is a measure of the quality of optical transmission through the atmosphere due to random inhomogeneities in the atmosphere's refractive index. According to an embodiment, the processing card PRC is configured to analyse the images provided by the camera CM to determine angle of arrival fluctuations of the light emitted by the observed star PS, from a number of images provided by the camera CM. FIG. 6 illustrates a method implemented by the processing card PRC to determine these angle of arrival fluctuations.

[0046] The camera CM is configured to provide an image at a rate corresponding to the coherence time of the atmospheric turbulence to distinguish the spots formed by the observed star PS on the image sensor IS. The integration time or image time interval between two images provided by the camera CM can be set to a few ms, between 5 and 20 ms (corresponding to an image rate of 50 to 200 images/s), for example 5 ms (corresponding to an image rate of 200 images/s). FIG. 6 schematically shows the telescope T with its Barlow lens B, a perturbed wave front WF, propagation axes of the light emitted by the observed star, the focal plane FP of the telescope T and the Barlow lens B corresponding to the sensitive surface of the image sensor IS of the camera CM. FIG. 6 also shows two spots SP1, SP2 formed at different moments by the observed star on the image sensor IS disposed in the focal plane FP of the telescope T, the two spots being spaced by the distance dx due to atmospheric turbulence. The distance dx can be determined by measuring the distance between the positions of the spots SP1 and SP2, the position of each spot being determined by a barycentric method. The distance dx corresponds to an angle of arrival fluctuation α, with α=dx/f, f being the focal distance of the optical system (telescope T+Barlow lens B). In a direction y perpendicular to the direction x in the focal plane FP, a distance dy corresponds to an angle of arrival fluctuation β, with β=dy/f.

[0047] The processing card PRC is configured to determine a number of values of the angle of arrival fluctuations α and β from a great number of images provided by the camera CM, thus forming time series of angle of arrival fluctuations α(t) and β(t) in x and y directions, respectively, and then to compute variances σ.sub.α.sup.2 and σ.sub.β.sup.2 or standard deviations σ.sub.αand σ.sub.βof angle of arrival fluctuations.

[0048] According to an embodiment, only a limited zone in the images provided by the camera CM is analyzed around the observed star PS, if the observed star is always present in this zone. According to an example, the analyzed zone extends on 50×50 pixels. Thus the images provided at a high rate by the camera CM can be stored and/or analyzed using a conventional processor.

[0049] It was demonstrated that the variances σ.sub.α.sup.2 and σ.sub.β.sup.2 of the angle of arrival fluctuations in x and y directions can be computed according to the following equations:

σ.sub.α.sup.2=0.18λ.sup.2r.sub.0x.sup.−5/3(D.sup.−1/3−1.525L.sub.0.sup.−1/3)  (1)

σ.sub.β.sup.2=0.18λ.sup.2r.sub.0y.sup.−5/3(D.sup.−1/3−1.525L.sub.0.sup.−1/3)  (2)

wherein λ is the wavelength of the light emitted by the observed star PS, D is the diameter of the aperture of the telescope T and L.sub.0 is the outer scale and r.sub.0x and r.sub.0y are Fried parameter values in x and y directions, respectively. The Fried parameter r.sub.0 is known to vary as a function of the wavelength λ, as λ.sup.6/5. Therefore the value of the Fried parameter r.sub.0 is only meaningful in relation to a specified wavelength. Typically, the wavelength λ can be set to 0.5 82 m.

[0050] In each equation (1) and (2), there are two unknown variables r.sub.0 (r.sub.0x or r.sub.0y) and L.sub.0. It can be observed from the statistics on all observation sites around the Earth, related to the outer scale L.sub.0 that the value of L.sub.0 is relatively stable in time and space, around a median value of 20 m. In the following, it is shown that the value of the outer scale L.sub.0 has a relatively small impact on the value of the Fried parameter r.sub.0. FIG. 7 shows curves C1, C2, C3, C4 of variation of the angle of arrival variance σ.sub.α.sup.2 or σ.sub.β.sup.2 as a function of the Fried parameter r.sub.0 when the outer scale L.sub.0 is set to 10 m (curve C1), 20 m (curve C2), 30 m (curve C3) and 50 m (curve C4). FIG. 7 shows that the effect of the outer scale L.sub.0 fades when the outer scale is greater than 20 m and when the angle of arrival variance corresponds to values of the Fried parameter greater than 10 cm.

[0051] FIG. 8 shows a curve C5 of variation of the Fried parameter r.sub.0 as a function of the outer scale L.sub.0, when the Fried parameter is estimated using equation (1) or (2) and the outer scale is fixed to 20 m. Curve C5 shows that when the outer scale L.sub.0 is set to 20 m, the Fried parameter r.sub.0 is equal to about 10 cm. FIG. 9 shows a curve C6 of variation of an estimation error of the Fried parameter r.sub.0 as a function of the outer scale L.sub.0. FIG. 9 shows that the estimation error stays lower that 5.6% when the outer scale L.sub.0 varies between 10 and 50 m, and lower than 2% when the outer scale varies between 15 and 25 m. This error is low since it is lower than statistical and instrumental errors.

[0052] In fact, the main error on the estimation of the Fried parameter r.sub.0 has a statistical origin related to the finite number of analyzed images. The statistical error on the angle of arrival variance can be computed from the following equation:

[00003]$\begin{array}{cc}{E}_{\sigma }=\frac{{\mathrm{\delta \sigma }}^2}{{\sigma }^2}=\sqrt{\frac{2}{N-1}}& \left(3\right)\end{array}$

wherein σ.sup.2=σ.sub.α.sup.2 or σ.sub.β.sup.2, and N is the number of images analyzed for the estimation of the Fried parameter r.sub.0. The statistical error on the estimation of the Fried parameter can be computed using the following equation:

[00004]$\begin{array}{cc}{E}_{r_0}=\frac{\delta r_0}{r_0}=\frac{3}{5}\sqrt{\frac{2}{N-1}}& \left(4\right)\end{array}$

When N is fixed to 400 images, E.sub.σreaches 7.1% and E.sub.r.sub.0 reaches 4.25%.

[0053] In addition, the estimation of the Fried parameter r.sub.0 is subjected to a scale error. Indeed, the displacements dx and dy are estimated in pixels. Therefore, the variance σ.sub.α.sup.2 or σ.sub.β.sup.2 is multiplied by a factor k.sup.2 transforming a variance in pixels squared into an angle variance in square arc seconds or radians. The scale factor k can be measured by observing a double star like Albireo ((β Cygni), having a known angular separation of 34.6″. According to the Shannon criterion with respect to the resolution of the telescope T (=1.22λ/D), the pixel size should be equal to 0.63″. When considering the spreading of the double star on the focal plane of the telescope, the variance error and the error on the Fried parameter due to the error on scale factor k reaches 2.6% and 1.5%, respectively. All the above-considered errors on the Fried parameter r.sub.0 reach a total of 5.75%. Other error sources should be considered such as the error due to the finite exposition time in the camera (about 5 ms), and the errors due to noise sources from the sky background and the camera readout. It appears that the cumulated error on the estimation of the Fried parameter r.sub.0 is greater than the error resulting from fixing the value of the outer scale L.sub.0 to 20 m.

[0054] According to an embodiment, the Fried parameter r.sub.0 is computed for a zenithal observation. Therefore in the application of equations (1) and (2), the variances a and σ.sub.α.sup.2 and σ.sub.β.sup.2 are multiplied by cos(z) where z is the angle between the direction of the observed star PS and the zenithal direction at the observation site. The angle z depends on the latitude of the observation site.

[0055] Then, the processing card PRC is configured to estimate the seeing parameter ϵ.sub.0 , for example using the following equation:

[00005]$\begin{array}{cc}{ϵ}_0=0.98\frac{\lambda }{r_0}& \left(5\right)\end{array}$

[0056] The processing card PRC can also be configured to estimate the isoplanatic angle from the scintillation index of the observed star PS. The scintillation can be determined using the telescope T with the central obstruction CO by measuring the variations of the intensity of the observed star image IPS, and determining the total intensity of the observed star image IPS, the mean and variance of this total intensity on several hundreds of images. The scintillation index is given by the following equation:

[00006]$\begin{array}{cc}s=\frac{{\sigma }_I^2}{I^2}& \left(6\right)\end{array}$

wherein s is the scintillation index, I is the total intensity of the observed star image IPS, σ.sub.I.sup.2 is the variance and Ī is the mean of the total intensity I. Again, the scintillation index s must be defined for the zenithal direction by multiplying it by (cos z).sup.8/3, z being the angle between the zenithal direction and the direction of the observed star PS at the observation site.

[0057] Then, the processing card PRC estimates the isoplanatic angle θ.sub.0 by means of the following equation:

θ.sub.0=(14.87 s).sup.−3/5  (7)

[0058] The device 1/2 can also be used to evaluate the coherence time defined by the following equation:

[00007]$\begin{array}{cc}{\tau }_0=0.31\frac{r_0}{v}& \left(8\right)\end{array}$

wherein τ.sub.0 is the coherence time, and v is the effective speed of the wave front at the ground, the value v of the effective speed being weighted by the energy of the atmospheric turbulence in all layers of the atmosphere. The estimation of the effective speed v is based on the measure of the temporal structure function of the angle of arrival fluctuations, defined by the following equations:

D.sub.α(τ)=[α(t)−α(t+τ)].sup.2⁽⁹⁾

D.sub.β(τ)=[β(t)−β(t+τ)].sup.2⁽¹⁰⁾

wherein D.sub.αand D.sub.βare the temporal structure functions of the angle of arrival fluctuations α and β respectively, X) represents the average of X and α(t) and β(t) are the angle of arrival fluctuations measured at time t. FIG. 10 shows a curve C7 of variation of function D.sub.α(or D.sub.β) as a function of delay time τ. Function D.sub.α(or D.sub.β) quickly increases and reaches a saturated value D.sub.s at a short delay time τ, where the correlation between the angle of arrival fluctuations is lost. In the example of FIG. 10, D.sub.s=0.72 (arc second).sup.2 reached when the delay time τ=0.9 s. Curve C7 of FIG. 10 also shows that when function D.sub.60 (or D.sub.β) reaches the value D.sub.s/e (e=2.71828), corresponding to a value of the delay time τ=τ.sub.αor τ.sub.β, the angle of arrival fluctuations are coherent with each other. Thus τ.sub.αor τ.sub.βare called “coherence times” of the angle of arrival fluctuations, respectively in the x and y directions.

[0059] In [2], it is theoretically shown that the coherence times τ.sub.αor τ.sub.βof the angles of arrival fluctuations are linked to the effective speed v of the wave front according to the following equation:

[00008]$\begin{array}{cc}v=10^3{D\left[G\left(e,\frac{D}{L_0}\right)\right]}^{-3}{\left({\tau }_{\alpha }^{1/3}+{\tau }_{\beta }^{1/3}\right)}^{-3}& \left(11\right)\end{array}$

wherein D (=10 cm) is the telescope aperture diameter, e=2.71828, τ.sub.αand τ.sub.βare the coherence times of the angles of arrival fluctuations in x and y directions, determined from the value D.sub.s/e of the temporal structure function D.sub.α(or D.sub.β), and G is a function given by the following equation:

[00009]$\begin{array}{cc}G\left(e,b\right)=\frac{\left(1-e^{-1}\right)\left[3.001{\left(\pi b\right)}^{1/3}+1.286\left({\left(\pi b\right)}^{7/3}\right)\right]+e^{-1}\left[2.882+1.628{\left(\pi b\right)}^2\right]}{0.411+0.188{\left(\pi b\right)}^2}& \left(12\right)\end{array}$$\mathrm{wherein}b=\frac{D}{L_0}.$

[0060] Thus the effective speed v can be deduced by injecting the measures of the coherence times τ.sub.αand τ.sub.βof the angle of arrival fluctuations, obtained by the device 1/2, in the equation (11) using equation (12), and by setting the outer scale L.sub.0 to 20 m (+ or − 10%). Then the effective speed v and the estimated Fried parameter r.sub.0 can be used to estimate the coherence time τ.sub.0 using the equation (8).

[0061] In the above description, a method is disclosed to determine the parameters characterizing the atmospheric turbulence, including the Fried parameter r.sub.0, the seeing parameter ϵ.sub.0 , the isoplanatic angle θ.sub.0, and the coherence time τ.sub.0, from the angle of arrival fluctuations α and β using the telescope T pointed at the polar star PS.

[0062] It should be observed that the telescope T has a diameter D close to the Fried parameter value. Thanks to this choice, the part of the wave front at the telescope aperture is almost coherent and not perturbed. Thus in the focal plane of the telescope T, the observed star PS forms a spot IPS having a position determined by a barycentric method along the two axes x and y in each individual images provided by the camera CM [3].

[0063] The parameters characterizing the atmospheric turbulence can also be determined using two points of an extended object observed in the focal plane of the telescope T. In this case, the telescope T within the housing 1 is supported by a motorized mount controlled to compensate the Earth's rotation about its polar rotation axis.

[0064] FIGS. 11 and 12 illustrate a method for estimating the angle of arrival fluctuations from two distant points P1, P2 spaced apart from each other of the limb of the Sun or the Moon. FIGS. 11 and 12 show points P1 and P2 of the limb viewed by the telescope T under respective angles θ.sub.1 and θ.sub.2, and spaced by an angle separation θ=θ.sub.1−θ.sub.2. FIG. 11 further shows a perturbed wavefront PW at the output of a turbulent layer located at an altitude h. The angular separation θ is related to a spatial distance at the altitude h equal to θ h. The angular structure function (or mean squared difference) of the angle of arrival fluctuations can be computed using the following equation:

D.sub.α(θ)=[α(θ.sub.1)−α(θ.sub.2)].sup.2=2[σ.sub.α.sup.2−C.sub.60(θ)]  (13)

wherein D.sub.αand C.sub.αare respectively the angle of arrival structure function and covariance for the angular separation θ=θ.sub.1−θ.sub.2, α(θ.sub.1) and α(θ.sub.2) are the angle of arrival fluctuations at angular positions θ.sub.1 and θ.sub.2 in y direction (transverse direction to the limb in the images), X represents the average of X, and σ.sub.α.sup.2 is the variance of angle of arrival fluctuations in y direction. When passing to a spatial transverse structure function and covariance, equation (13) becomes:

[00010]$\begin{array}{cc}\begin{array}{c}{D}_{\alpha }\left(\theta \right)=.Math.{\left[\alpha \left(r,{\theta }_1\right)-\alpha \left(r,{\theta }_2\right)\right]}^2.Math.\\ =.Math.{\left[\alpha \left(r,{\theta }_1\right)-\alpha \left(r-\theta h,{\theta }_1\right)\right]}^2.Math.\\ =D_{\alpha ,s}\left(\theta h\right)\end{array}& \left(14\right)\end{array}$

wherein D.sub.α,s(θh) is the spatial structure function in the transverse direction with respect to the observed limb of the Sun or Moon. In [3], it is shown that the spatial transverse structure function D.sub.α,s(θh) can be computed using the following equation:

[00011]$\begin{array}{cc}{D}_{\alpha ,s}\left(\theta \overline{h}\right)\simeq 0.364{\lambda }^2{r}_0^{-5/3}{D}^{-1/3}\left[1-0.798{\left(\frac{\theta \overline{h}}{D}\right)}^{-1/3}\right]& \left(15\right)\end{array}$

wherein h is the equivalent altitude of the whole atmospheric turbulence given by the following equation:

[00012]$\begin{array}{cc}\overline{h}={\left[\frac{{\int }_0^{\infty }{C}_n^2\left(h\right)h^{-5/3}dh}{{\int }_0^{\infty }{C}_n^2\left(h\right)dh}\right]}^{-3/5}& \left(16\right)\end{array}$

wherein C.sub.n.sup.2(h) is refractive index structure constant which corresponds to the atmospheric turbulence strength at the altitude h.

[0065] The measurements according to FIG. 12 make it possible to estimate the spatial transverse structure functions D.sub.α,s(θ.sub.ah) and D.sub.α,s(θ.sub.bh) for two angular separations θ.sub.a=θ.sub.a1−θ.sub.a2 and θ.sub.b=θ.sub.b1−θ.sub.b2, where (θ.sub.a1, θ.sub.a2) and (θ.sub.b1, θ.sub.b2) are two pairs of angular positions on the observed Sun or Moon limb. Then h is deduced from a ratio R.sub.h of structure functions D.sub.α,s(θ.sub.ah) and D.sub.α,s(θ.sub.bh), the ratio R.sub.h being linked to h as followed:

[00013]$\begin{array}{cc}{R}_h=\frac{D_{\alpha ,s}\left({\theta }_a\overline{h}\right)}{D_{\alpha ,s}\left({\theta }_b\overline{h}\right)}\simeq \frac{D^{-1/3}-0.798{\left({\theta }_a\overline{h}\right)}^{-1/3}}{D^{-1/3}-0.798{\left({\theta }_b\overline{h}\right)}^{-1/3}}& \left(17\right)\end{array}$

Equation (17) makes it possible to determine the equivalent altitude h which can be used in equation (15) to determine the Fried parameter r.sub.0 as a function of the angle separation θ(=θ.sub.a or θ.sub.b). The use of both values of the angle separations θ.sub.a, θ.sub.b, provides two estimated values r.sub.0a, r.sub.0b of the Fried parameter r.sub.0, each corresponding to one of the two angular separations θ.sub.a and θ.sub.b. Then the Fried parameter r.sub.0 can be fixed to an average value of r.sub.0a, r.sub.0b.

[0066] According to an embodiment, other pairs of angular positions of points on the observed Sun or Moon limb can be used to compute a number of values of h, from which estimated values of the Fried parameter r.sub.0 are deduced, the Fried parameter r.sub.0 being set to a median value of all the estimated values, to exclude outliers.

[0067] The isoplanatic angle θ.sub.0 can be computed from the following equation:

[00014]$\begin{array}{cc}{\theta }_0=0.314\frac{r_0}{\overline{h}}& \left(18\right)\end{array}$

The other parameters, ϵ.sub.0 and τ.sub.0 , can be computed from the Fried parameter as explained above, from equations (5) and (8).

[0068] Therefore, the above-disclosed method using the device 1 can characterize the atmospheric turbulence conditions both during the day and night, and everywhere over the world, even from the south hemisphere.

[0069] The above description of various embodiments of the present invention is provided for purposes of description to one of ordinary skill in the related art. It is not intended to be exhaustive or to limit the invention to a single disclosed embodiment. Numerous alternatives and variations to the present invention will be apparent to those skilled in the art of the above teaching. Accordingly, while some alternative embodiments have been discussed specifically, other embodiments will be apparent or relatively easily developed by those of ordinary skill in the art.

[0070] In this respect, it is apparent to a person skilled in the art that all or a part of the operations performed by the processing card PRC can be performed by a computer CP connected to the processing card, the images acquired by the camera CM, or only the displacements dx, dy, or the angle of arrival fluctuations α0 and β or the variances σ.sub.α.sup.2 and σ.sub.β.sup.2 computed from the angle of arrival fluctuations, being transmitted to the computer CP.

[0071] In some implementations, the method for characterizing the atmospheric turbulence only estimates the Fried parameter r.sub.0 or some other of the above-disclosed parameters. When the coherence time τ.sub.0 does not need to be estimated, the image rate of the camera can be as low as 50 images/s. When the scintillation s or the isoplanatic angle θ.sub.0 does not need to be determined, the obstruction CO is not necessary. In this case, the telescope diameter can be reduced up to 4 to 6 cm.

[0072] The above description is intended to embrace all alternatives, modifications and variations of the present invention that have been discussed herein, and other embodiments that fall within the spirit and scope of the above description. Limitations in the claims should be interpreted broadly based on the language used in the claims, and such limitations should not be limited to specific examples described herein.

CITED REFERENCES

[0073] [1] “Experimental estimation of the spatial-coherence outer scale from a wavefront statistical analysis”, A. Ziad, J. Borgnino, F. Martin and A. Agabi, Astronomy & Astrophysics, 282 (1994), pp. 1021-1033.

[0074] [2] “Temporal characterization of atmospheric turbulence with the GSM instrument”, A. Ziad, J. Borgnino, W. Dali Ali, A. Berdja, J. Maire and F. Martin, Journal of Optics: Pure & Applied Optics, 14, 045705-8pp, (2012)

[0075] [3] “The Generalised Differential Image Motion Monitor”, E. Aristidi, A. Ziad, J. Chabé, Y. Fantëi-Caujolle, C. Renaud, C. Giordano, Monthly Notices of the Royal Astronomical Society, Vol. 486, Issue 1, p.915-925 (2019)