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122

Appendix A

Correlation function, structure function,

and power spectrum of wavefront

distortion by a thin turbulent layer

First, the derivation of the correlation function of wavefront distortion in equation (1.16) is

shown. By substituting equation (1.14) into equation (1.15),

𝐵 𝜙 (𝒙) = 𝑘

= 𝑘2

= 𝑘2

ℎ0 +Δℎ0

ℎ0 +Δℎ0

𝑑ℎ

ℎ0

ℎ0 +Δℎ0

𝑑ℎ′ < 𝑛(𝒓, ℎ)𝑛(𝒓 + 𝒙, ℎ′) >

ℎ0

ℎ0 +Δℎ0 −ℎ

𝑑𝜂 < 𝑛(𝒓, ℎ)𝑛(𝒓 + 𝒙, ℎ + 𝜂) >

𝑑ℎ

ℎ0 −ℎ

∫ ℎ0 +Δℎ0 −ℎ

ℎ0

∫ ℎ0 +Δℎ0

𝑑𝜂𝐵𝑛 (𝒙, 𝜂).

𝑑ℎ

(A.1)

ℎ0 −ℎ

ℎ0

Since Δℎ0 is large enough compared to the typical correlation length of refractive index, one

can extend the integration range of [ℎ0 − ℎ, ℎ0 + Δℎ0 − ℎ] to (−∞, ∞) and

𝐵 𝜙 (𝒙) ∼ 𝑘

ℎ0 +Δℎ0

𝑑𝜂𝐵𝑛 (𝒙, 𝜂)

−∞

ℎ0

𝑑ℎ

𝑑𝜂𝐵𝑛 (𝒙, 𝜂).

= 𝑘 Δℎ0

−∞

123

(A.2)

Correlation function, structure function, and power spectrum of wavefront distortion

by a thin turbulent layer

Then, structure function of wavefront distortion in equation (1.18) is, using equations (1.17)

and (1.10),

𝐷 𝜙 (𝒙) = 2𝑘 Δℎ0

𝑑𝜂 [𝐵𝑛 (0, 𝜂) − 𝐵𝑛 (𝒙, 𝜂)]

−∞

∫ ∞

= 𝑘 Δℎ0

𝑑𝜂 [𝐷 𝑛 (𝒙, 𝜂) − 𝐷 𝑛 (0, 𝜂)]

−∞

∫ ∞

= 𝑘 Δℎ0

𝑑𝜂𝐶𝑛2 (𝜂) (|𝒙| 2 + 𝜂2 ) 1/3 − 𝜂2/3 )

(A.3)

−∞

Because the 𝐶𝑛2 (𝜂) is valid when 𝜂 ∼ ℎ0 ,

𝐷 𝜙 (𝒙) ∼ 𝑘

𝐶𝑛2 (ℎ0 )Δℎ0

= 2.91𝑘

𝑑𝜂 (|𝒙| 2 + 𝜂2 ) 1/3 − 𝜂2/3 )

−∞

𝐶𝑛 (ℎ0 )Δℎ0 |𝒙| 5/3

(A.4)

By taking two-dimensional Fourier transform of equation (A.2), power spectrum of wavefront

distortion in equation (1.19) is written as follows,

𝑑𝜂F [𝐵𝑛 (𝒙, 𝜂)]

Φ𝜙 ( 𝒇 ) = 𝑘 Δℎ0

−∞

= 𝑘 2 Δℎ0 Φ𝑛 ( 𝒇 , ℎ0 )

= 9.7 × 10−3 𝑘 2 | 𝒇 | −11/3𝐶𝑛2 (ℎ0 )Δℎ0 ,

where F [·] means two-dimensional Fourier transform.

124

(A.5)

Appendix B

Power spectrum of amplitude fluctuation

and wavefront distortion after Fresnel

propagation

Based on equations (2.3) and (2.4), power spectrum of amplitude fluctuation Φ 𝜒 ( 𝒇 ) and that

of wavefront distortion Φ𝜓 ( 𝒇 ) are



 2 2

|𝒙|

,

Φ 𝜒 ( 𝒇 ) = Φ𝜙 ( 𝒇 ) × F

cos 𝜋

𝜆𝑧

𝜆𝑧 

 2 2

|𝒙|

,

sin 𝜋

Φ𝜓 ( 𝒇 ) = Φ𝜙 ( 𝒇 ) × F

𝜆𝑧

𝜆𝑧 (B.1)

(B.2)

where F [·] is two-dimensional Fourier transform.

The

parts in equations (B.1) and (B.2) are real part and imaginary part

h Fourier

 transform

i

|𝒙| 2

of F 𝜆𝑧 exp 𝑖𝜋 𝜆𝑧 , as







|𝒙| 2

|𝒙| 2

|𝒙| 2

exp 𝑖𝜋

=F

cos 𝜋

+ 𝑖F

sin 𝑖𝜋

𝜆𝑧

𝜆𝑧

𝜆𝑧

𝜆𝑧

𝜆𝑧

𝜆𝑧

where 𝑖 is the imaginary unit.

125

(B.3)

Power spectrum of amplitude fluctuation and wavefront distortion after Fresnel

propagation

i

And F 𝜆𝑧

exp 𝑖𝜋 |𝒙|

is calculated, as follows,

𝜆𝑧

 ∫

 2 2

|𝒙| 2

𝑥 +𝑦

exp 𝑖𝜋

𝑑𝑥 𝑑𝑦 exp 𝑖𝜋

exp −2𝜋𝑖( 𝑓𝑥 𝑥 + 𝑓 𝑦 𝑦)

𝜆𝑧

𝜆𝑧

𝜆𝑧

𝜆𝑧

∫

 2

𝑖𝜋 2

𝑑𝑥 exp

𝑥 − 2𝜋𝑖 𝑓𝑥 𝑥

𝜆𝑧

𝜆𝑧

 2

∫

𝑖𝜋

𝑑𝑥 exp

(𝑥 − 𝜆𝑧 𝑓𝑥 ) − 𝑖𝜋𝜆𝑧 𝑓𝑥

𝜆𝑧

𝜆𝑧

 2

 ∫

𝑖𝜋

exp −𝑖𝜋𝜆𝑧( 𝑓𝑥 + 𝑓 𝑦 )

𝑑𝑥 exp

(𝑥 − 𝜆𝑧 𝑓𝑥 )

𝜆𝑧

𝜆𝑧

= 𝑖 exp −𝑖𝜋𝜆𝑧( 𝑓𝑥2 + 𝑓 𝑦2 )

(B.4)

= sin 𝜋𝜆𝑧( 𝑓𝑥 + 𝑓 𝑦 ) + 𝑖 cos 𝜋𝜆𝑧( 𝑓𝑥 + 𝑓 𝑦 )

Therefore, we obtain equations (2.5) and (2.6), i.e.

Φ 𝜒 ( 𝒇 ) = Φ𝜙 ( 𝒇 ) sin2 (𝜋𝜆𝑧| 𝒇 | 2 ),

(B.5)

Φ𝜓 ( 𝒇 ) = Φ𝜙 ( 𝒇 ) cos2 (𝜋𝜆𝑧| 𝒇 | 2 ).

(B.6)

126

Appendix C

Slope auto-covariance analysis

We show the same figures as 2.15 but for all the datasets.

127

Slope auto-covariance analysis

Fig. C.1 Top: X, Y-slope auto-covariance maps for dataset taken at 23:17, Nov.12, 2022

by SH-1. Second row: fitted X, Y-slope auto-covariance maps. Third row: residual maps

between observed and fitted auto-covariance maps. Bottom: Cross-section of the autocovariance maps at 𝛿𝑥 = 0 or 𝛿𝑦 = 0

128

Fig. C.2 Same as figure C.1 but for dataset obtained at 23:17, Nov.12, 2022 by SH-2.

129

Slope auto-covariance analysis

Fig. C.3 Same as figure C.1 but for dataset obtained at 23:37, Nov.12, 2022 by SH-1.

130

Fig. C.4 Same as figure C.1 but for dataset obtained at 23:37, Nov.12, 2022 by SH-2.

131

Slope auto-covariance analysis

Fig. C.5 Same as figure C.1 but for dataset obtained at 23:42 Nov.12, 2022 by SH-1.

132

Fig. C.6 Same as figure C.1 but for dataset obtained at 23:42 Nov.12, 2022 by SH-2.

133

Slope auto-covariance analysis

Fig. C.7 Same as figure C.1 but for dataset obtained at 23:45, Nov.12, 2022 by SH-1.

134

Fig. C.8 Same as figure C.1 but for dataset obtained at 23:45, Nov.12, 2022 by SH-2.

135

Slope auto-covariance analysis

Fig. C.9 Same as figure C.1 but for dataset obtained at 23:49, Nov.12, 2022 by SH-1.

136

Fig. C.10 Same as figure C.1 but for dataset obtained at 23:49, Nov.12, 2022 by SH-2.

137

Slope auto-covariance analysis

Fig. C.11 Same as figure C.1 but for dataset obtained at 2:42, Mar.14, 2023 by SH-1.

138

Fig. C.12 Same as figure C.1 but for dataset obtained at 2:45, Mar.14, 2023 by SH-1.

139

Slope auto-covariance analysis

Fig. C.13 Same as figure C.1 but for dataset obtained at 3:30, Mar.14, 2023 by SH-2.

140

Fig. C.14 Same as figure C.1 but for dataset obtained at 3:34, Mar.14, 2023 by SH-2.

141

Slope auto-covariance analysis

Fig. C.15 Same as figure C.1 but for dataset obtained at 3:37, Mar.14, 2023 by SH-2.

142

Fig. C.16 Same as figure C.1 but for dataset obtained at 4:44, Mar.14, 2023 by SH-1.

143

Slope auto-covariance analysis

Fig. C.17 Same as figure C.1 but for dataset obtained at 4:44, Mar.14, 2023 by SH-2.

144

Fig. C.18 Same as figure C.1 but for dataset obtained at 5:07, Mar.14, 2023 by SH-1.

145

Slope auto-covariance analysis

Fig. C.19 Same as figure C.1 but for dataset obtained at 5:07, Mar.14, 2023 by SH-2.

146

Fig. C.20 Same as figure C.1 but for dataset obtained at 5:13, Mar.14, 2023 by SH-1.

147

Slope auto-covariance analysis

Fig. C.21 Fitting slope auto-covariance map obtained at 5:13, Mar.14, 2023 by SH-2.

148

Fig. C.22 Same as figure C.1 but for dataset obtained at 5:16, Mar.14, 2023 by SH-1.

149

Slope auto-covariance analysis

Fig. C.23 Same as figure C.1 but for dataset obtained at 5:16, Mar.14, 2023 by SH-2.

150

Fig. C.24 Same as figure C.1 but for dataset obtained at 5:20, Mar.14, 2023 by SH-1.

151

Slope auto-covariance analysis

Fig. C.25 Same as figure C.1 but for dataset obtained at 5:20, Mar.14, 2023 by SH-2.

152

Appendix D

SH-MASS analysis

We show the same figures as 2.16 but for all the datasets.

153

SH-MASS analysis

Fig. D.1 Scintillation indices obtained at 23:17, Nov.12, 2022. Left and right panels show

normal and differential scintillation indices, respectively.

Fig. D.2 Same as figure D.1 but for dataset obtained at 23:37, Nov.12, 2022.

Fig. D.3 Same as figure D.1 but for dataset obtained at 23:42, Nov.12, 2022.

154

Fig. D.4 Same as figure D.1 but for dataset obtained at 23:45, Nov.12, 2022.

Fig. D.5 Same as figure D.1 but for dataset obtained at 23:49, Nov.12, 2022.

Fig. D.6 Same as figure D.1 but for dataset obtained at 2:42, Mar.14, 2023.

155

SH-MASS analysis

Fig. D.7 Same as figure D.1 but for dataset obtained at 2:45, Mar.14, 2023.

Fig. D.8 Same as figure D.1 but for dataset obtained at 3:30, Mar.14, 2023.

Fig. D.9 Same as figure D.1 but for dataset obtained at 3:34, Mar.14, 2023.

156

Fig. D.10 Same as figure D.1 but for dataset obtained at 3:37, Mar.14, 2023.

Fig. D.11 Same as figure D.1 but for dataset obtained at 4:44, Mar.14, 2023.

Fig. D.12 Same as figure D.1 but for dataset obtained at 5:07, Mar.14, 2023.

157

SH-MASS analysis

Fig. D.13 Same as figure D.1 but for dataset obtained at 5:13, Mar.14, 2023.

Fig. D.14 Same as figure D.1 but for dataset obtained at 5:16, Mar.14, 2023.

Fig. D.15 Same as figure D.1 but for dataset obtained at 5:20, Mar.14, 2023.

158

Appendix E

Scintillation auto-covariance analysis

We show the same figures as figure 2.18 but for all the datasets.

159

Scintillation auto-covariance analysis

Fig. E.1 Scintillation auto-covariance map for dataset taken by SH-1 at 23:17, Nov.12, 2022.

From top-left to bottom-right, an auto-covariance map calculated with a time lag of 0-9

frames is arranged.

Fig. E.2 Same as figure E.1 but for dataset taken by SH-2 at 23:17, Nov.12, 2022.

Fig. E.3 Same as figure E.1 but for dataset taken by SH-1 at 23:37, Nov.12, 2022.

160

Fig. E.4 Same as figure E.1 but for dataset taken by SH-2 at 23:37, Nov.12, 2022.

Fig. E.5 Same as figure E.1 but for dataset taken by SH-1 at 23:42, Nov.12, 2022.

Fig. E.6 Same as figure E.1 but for dataset taken by SH-2 at 23:42, Nov.12, 2022.

161

Scintillation auto-covariance analysis

Fig. E.7 Same as figure E.1 but for dataset taken by SH-1 at 23:45, Nov.12, 2022.

Fig. E.8 Same as figure E.1 but for dataset taken by SH-2 at 23:45, Nov.12, 2022.

Fig. E.9 Same as figure E.1 but for dataset taken by SH-1 at 23:49, Nov.12, 2022.

162

Fig. E.10 Same as figure E.1 but for dataset taken by SH-2 at 23:49, Nov.12, 2022.

Fig. E.11 Same as figure E.1 but for dataset taken by SH-1 at 2:42, Mar.14, 2023.

Fig. E.12 Same as figure E.1 but for dataset taken by SH-1 at 2:45, Mar.14, 2023.

163

Scintillation auto-covariance analysis

Fig. E.13 Same as figure E.1 but for dataset taken by SH-2 at 3:30, Mar.14, 2023.

Fig. E.14 Same as figure E.1 but for dataset taken by SH-2 at 3:34, Mar.14, 2023.

Fig. E.15 Same as figure E.1 but for dataset taken by SH-2 at 3:37, Mar.14, 2023.

164

Fig. E.16 Same as figure E.1 but for dataset taken by SH-1 at 4:44, Mar.14, 2023.

Fig. E.17 Same as figure E.1 but for dataset taken by SH-2 at 4:44, Mar.14, 2023.

Fig. E.18 Same as figure E.1 but for dataset taken by SH-1 at 5:07, Mar.14, 2023.

165

Scintillation auto-covariance analysis

Fig. E.19 Same as figure E.1 but for dataset taken by SH-2 at 5:07, Mar.14, 2023.

Fig. E.20 Same as figure E.1 but for dataset taken by SH-1 at 5:13, Mar.14, 2023.

Fig. E.21 Same as figure E.1 but for dataset taken by SH-2 at 5:13, Mar.14, 2023.

166

Fig. E.22 Same as figure E.1 but for dataset taken by SH-1 at 5:16, Mar.14, 2023.

Fig. E.23 Same as figure E.1 but for dataset taken by SH-2 at 5:16, Mar.14, 2023.

Fig. E.24 Same as figure E.1 but for dataset taken by SH-1 at 5:20, Mar.14, 2023.

167

Scintillation auto-covariance analysis

Fig. E.25 Same as figure E.1 but for dataset taken by SH-2 at 5:20, Mar.14, 2023.

168

Appendix F

SLODAR analysis

We show the same figures as figure 2.23 but for all the datasets.

169

SLODAR analysis

Fig. F.1 Top-left: Observed X-slope cross-covariance map for dataset taken at 23:17, Nov.12,

2022. bottom-left: Observed Y-slope cross-covariance map. Top-center: Model X-slope

cross-covariance map after fitting. bottom-center: Model Y-slope cross-covariance map.

Top-right: Residual of the observed and model X-slope cross-covariance map. bottom-right:

Residual of the observed and model Y-slope cross-covariance map.

Fig. F.2 Same as figure F.1 but for dataset taken at 23:37, Nov.12, 2022.

170

Fig. F.3 Same as figure F.1 but for dataset taken at 23:42, Nov.12, 2022.

Fig. F.4 Same as figure F.1 but for dataset taken at 23:45, Nov.12, 2022.

171

SLODAR analysis

Fig. F.5 Same as figure F.1 but for dataset taken at 23:49, Nov.12, 2022.

Fig. F.6 Same as figure F.1 but for dataset taken at 4:44, Mar.14, 2023.

172

Fig. F.7 Same as figure F.1 but for dataset taken at 5:07, Mar.14, 2023.

Fig. F.8 Same as figure F.1 but for dataset taken at 5:13, Mar.14, 2023.

173

SLODAR analysis

Fig. F.9 Same as figure F.1 but for dataset taken at 5:16, Mar.14, 2023.

Fig. F.10 Same as figure F.1 but for dataset taken at 5:20, Mar.14, 2023.

174

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