Error correction and evaluation in astronomical speckle interferometry with low-light CCD camera

Abstract

In the astronomical speckle interferometry, low-light CCD cameras such as the electron bombarded/multiplying CCDs (EB/EMCCDs) are being widely used for taking the faint speckle images. The detector noise affects the power spectral estimate of the astronomical object as well as the speckle noise does. The estimation errors of the object power spectrum are results of the propagation of the random fluctuations due to those noises. We have formulated a method for correcting and evaluating the estimation errors based on our detection model developed for EB/EMCCDs and the conventional model of the speckle noise. In our method, the error correction and evaluation are accomplished with the aid of the auxiliary data known as the dark and flat frames. The unbiased estimator of the object power spectrum and its SNR evaluator as derived above are verified using simulated data, where we have adopted the values of the detector parameters as obtained by measuring the actually used EMCCD camera in our observations. The results of the simulated experiments show that, within the diffraction cutoff frequency, our power spectral estimator is unbiased, and our SNR evaluator for that is useful except below the seeing cutoff.

Appendices

Variance of square Fourier modulus of instantaneous TF

According to Korff et. al. [42], the TF $S({\varvec{u}})$ can be considered as the sum of $m({\varvec{u}})$ independent random phasors for $|{\varvec{u}}| > u_s$, the number of which depends on the spatial frequency ${\varvec{u}}$ and is given by

$$\begin{aligned} m({\varvec{u}}) = (D/r_0)^2 T({\varvec{u}}). \end{aligned}$$

(65)

The real and imaginary parts of the random phasor sum are uncorrelated, and have the zero-mean and the same variance. If the sufficient number of the phasors contribute, the sum of them obeys the circular complex normal distribution [44]. Under this assumption, the relation $\sigma _{|S|^2}^2({\varvec{u}}) = \langle |S({\varvec{u}})|^2\rangle ^2$ can be derived as follows.

The variance $\sigma _{|S|^2}^2({\varvec{u}})$ is defined by

$$\begin{aligned} \sigma _{|S|^2}^2({\varvec{u}}) = \left\langle \{|S({\varvec{u}})|^2 - \langle |S({\varvec{u}})|^2\rangle \}^2\right\rangle = \langle |S({\varvec{u}})|^4\rangle - \langle |S({\varvec{u}})|^2\rangle ^2 . \end{aligned}$$

(66)

For four circularly normal complex variables, $S_1$, $S_2$, $S_3$, $S_4$, the moment theorem

$$\begin{aligned} \langle S_1^* S_2^* S_3 S_4 \rangle = \langle S_1^* S_3 \rangle \langle S_2^* S_4 \rangle + \langle S_1^* S_4 \rangle \langle S_2^* S_3 \rangle \end{aligned}$$

(67)

holds. Applying this theorem to the case $S_1 = S_2 = S_3 = S_4 = S({\varvec{u}})$, we have $\langle |S({\varvec{u}})|^4\rangle = 2\langle |S({\varvec{u}})|^2\rangle ^2$, and substituting it to Eq. (66) results in the relation we intend to derive.

The normality assumption of the random phasor sum is based on the central limit theorem. Equation (65) says that the number of the contributing phasors, $m({\varvec{u}})$, is proportional to the diffraction-limited TF $T({\varvec{u}})$, the value of which falls off and reaches zero at the cutoff frequency $u_c$. This signifies that the frequency range for which the above relation holds is restricted to $|{\varvec{u}}| \ll u_c$, where the number of the contributing phasors is large enough. However, this restriction may be relaxed when $D \gg r_0$. For example, in the typical case of $D = 2~\text{ m }$ and $r_0 = 10~\text{ cm }$, $m({\varvec{u}}) = 15$ at $|{\varvec{u}}| = 0.9u_c$, that is, even for the spatial frequencies close to the cutoff, there are still enough contributors to make their sum a normal variable.

Autocovariance function of frame data

The raw frame, ${i}'_n({\varvec{x}})$, is modeled by Eq. (15), and represented by the sum of the terms of statistically independent random functions. The autocovariance function of ${i}'_n({\varvec{x}})$ is defined by $\overline{\varDelta {i}'_n({\varvec{x}})\varDelta {i}'_n({\varvec{x}}')}$, where the fluctuating component $\varDelta {i}'_n({\varvec{x}}) = {i}'_n({\varvec{x}})-\overline{{i}'_n({\varvec{x}})}$ is given by

$$\begin{aligned} \varDelta {i}'_n({\varvec{x}}) = \varDelta \omega _n({\varvec{x}})\ast g({\varvec{x}}) + \varDelta {\text {d}}_n({\varvec{x}})\ast \psi ({\varvec{x}}) + e_n({\varvec{x}}), \end{aligned}$$

(68)

and is comprised of 3 statistically independent fluctuating terms. The autocovariance function is then comprised of 3 autocovariance terms with vanishing 6 cross-covariance terms, and is given by

$$\begin{aligned} \overline{\varDelta {i}'_n({\varvec{x}})\varDelta {i}'_n({\varvec{x}}')}&= \overline{\left\{ \varDelta \omega _n({\varvec{x}})\ast g({\varvec{x}})\right\} \left\{ \varDelta \omega _n({\varvec{x}}')\ast g({\varvec{x}}')\right\} }\nonumber \\&\quad + \overline{\left\{ \varDelta d_n({\varvec{x}})\ast \psi ({\varvec{x}})\right\} \left\{ \varDelta d_n({\varvec{x}}')\ast \psi ({\varvec{x}}')\right\} }\nonumber \\&\quad + \overline{e_n({\varvec{x}})e_n({\varvec{x}}')}. \end{aligned}$$

(69)

Due to the properties of the random function $\omega _n({\varvec{x}})$ that its fluctuations are uncorrelated, as given by Eq. (18), and that its variance $\sigma _{\omega _n}^2({\varvec{x}})$ is given by Eq. (14), the 1st term of the right side of Eq. (69) is reduced as follows:

$$\begin{aligned} \overline{\left\{ \varDelta \omega _n({\varvec{x}})\ast g({\varvec{x}})\right\} \left\{ \varDelta \omega _n({\varvec{x}}')\ast g({\varvec{x}}')\right\} }&= \int \int \overline{\varDelta \omega _n({\varvec{x}}'')\varDelta \omega _n({\varvec{x}}''')}g({\varvec{x}}-{\varvec{x}}'')g({\varvec{x}}'-{\varvec{x}}'''){\text {d}}{\varvec{x}}''{\text {d}}{\varvec{x}}'''\nonumber \\&= \int \int \sigma _{\omega _n}^2({\varvec{x}}'')\delta ({\varvec{x}}''-{\varvec{x}}''')g({\varvec{x}}-{\varvec{x}}'')g({\varvec{x}}'-{\varvec{x}}''')d{\varvec{x}}''{\text {d}}{\varvec{x}}'''\nonumber \\&= M^2F^2\int \overline{\xi }_n({\varvec{x}}'')g({\varvec{x}}-{\varvec{x}}'')g({\varvec{x}}'-{\varvec{x}}''){\text {d}}{\varvec{x}}''. \end{aligned}$$

(70)

Due to Eq. (16) and the uniformities of $\overline{k}$ and $\eta$, the 1st term is further reduced to

$$\begin{aligned} \overline{\left\{ \varDelta \omega _n({\varvec{x}})\ast g({\varvec{x}})\right\} \left\{ \varDelta \omega _n({\varvec{x}}')\ast g({\varvec{x}}')\right\} }&= M^2F^2\eta \int \overline{i}_n({\varvec{x}}'')g({\varvec{x}}-{\varvec{x}}'')g({\varvec{x}}'-{\varvec{x}}''){\text {d}}{\varvec{x}}''\nonumber \\&\quad + M^2F^2\overline{k}~(g\star g)({\varvec{x}}'-{\varvec{x}}). \end{aligned}$$

(71)

Similarly, due to the uniform, uncorrelating and Poissonian properties of $d_n({\varvec{x}})$, the 2nd term is reduced to

$$\begin{aligned} \overline{\left\{ \varDelta d_n({\varvec{x}})\ast \psi ({\varvec{x}})\right\} \left\{ \varDelta d_n({\varvec{x}}')\ast \psi ({\varvec{x}}')\right\} } =&\overline{d}~(\psi \star \psi )({\varvec{x}}'-{\varvec{x}}). \end{aligned}$$

(72)

The 4th term is rewritten as

$$\begin{aligned} \overline{e_n({\varvec{x}})e_n({\varvec{x}}')} = \sigma _e^2\delta ({\varvec{x}}-{\varvec{x}}'). \end{aligned}$$

(73)

Substituting Eqs. (71), (72) and (73) into Eq. (69) yields the expression for the autocovariance function of ${i}'_n({\varvec{x}})$, as given by Eq. (20).

Average power spectrum of frame data

The average power spectrum of the L data frames, $P'({\varvec{u}})$, is an average of the finite number of samples, and is a random variable due to the detector noise. The mean (expectation) of the power spectral frame average, $\overline{P'({\varvec{u}})} = \overline{\langle |I({\varvec{u}})|^2\rangle _L}$, is given by

$$\begin{aligned} \overline{P'({\varvec{u}})} = \left\langle \overline{|I({\varvec{u}})|^2}\right\rangle _L, \end{aligned}$$

(74)

where taking the expectation and averaging over the frames (samples) have been exchanged in their orders. The mean value of the power spectrum of the frame at a certain instant n, $\overline{|I_n({\varvec{u}})|^2}$, is the Fourier transform of the mean autocorrelation function of $i_n({\varvec{x}})$ as given by

$$\begin{aligned} \overline{(i_n \star i_n)({\varvec{x}})} = \int \overline{i_n({\varvec{t}})~i_n({\varvec{t}}-{\varvec{x}})}d{\varvec{t}}. \end{aligned}$$

(75)

The statistical autocorrelation function $\overline{i_n({\varvec{t}})~i_n({\varvec{t}}-{\varvec{x}})}$ is related to the autocovariance function by

$$\begin{aligned} \overline{i_n({\varvec{t}})~i_n({\varvec{t}}-{\varvec{x}})} = \overline{i_n({\varvec{t}})}~\overline{i_n({\varvec{t}}-{\varvec{x}})} + \overline{\varDelta i_n({\varvec{t}})~\varDelta i_n({\varvec{t}}-{\varvec{x}})}, \end{aligned}$$

(76)

where, since $\varDelta i_n({\varvec{x}}) = \varDelta i'_n({\varvec{x}})$, the autocovariance is given by Eq. (20). Substituting Eqs. (20) and (30), and invoking the convolution integral yield

$$\begin{aligned} \overline{i_n({\varvec{t}})~i_n({\varvec{t}}-{\varvec{x}})}&= M^2\eta ^2\int \int \overline{i}_n({\varvec{x}}')\overline{i}_n({\varvec{x}}'')g({\varvec{t}}-{\varvec{x}}')g({\varvec{t}}-{\varvec{x}}-{\varvec{x}}''){\text {d}}{\varvec{x}}'{\text {d}}{\varvec{x}}''\nonumber \\&\quad + M^2F^2\eta \int \overline{i}_n({\varvec{x}}')g({\varvec{t}}-{\varvec{x}}')g({\varvec{t}}-{\varvec{x}}-{\varvec{x}}'){\text {d}}{\varvec{x}}'\nonumber \\&\quad + M^2F^2\overline{k}~(g\star g)({\varvec{x}})\nonumber \\&\quad + \overline{d}~(\psi \star \psi )({\varvec{x}})\nonumber \\&\quad + \sigma _e^2\delta ({\varvec{x}}). \end{aligned}$$

(77)

Integrating Eq. (77) over ${\varvec{t}}$ gives the expression for Eq. (75),

$$\begin{aligned} \overline{(i_n \star i_n)({\varvec{x}})}&= M^2\eta ^2~(\overline{i}_n\star \overline{i}_n\star g\star g)({\varvec{x}})\nonumber \\&\quad + M^2F^2\eta ~(g\star g)({\varvec{x}})~\int \overline{i}_n({\varvec{x}}')d{\varvec{x}}'\nonumber \\&\quad + M^2F^2\overline{k}A~(g\star g)({\varvec{x}})\nonumber \\&\quad + \overline{d}A~(\psi \star \psi )({\varvec{x}})\nonumber \\&\quad + \sigma _e^2A\delta ({\varvec{x}}), \end{aligned}$$

(78)

where the 1st term of the right side is the cross-correlation function of the two autocorrelation functions, and A denotes the area of the frames.

Fourier transforming it, we have

$$\begin{aligned} \overline{|I_n({\varvec{u}})|^2}&= M^2\eta ^2|\overline{I}_n({\varvec{u}})|^2|G({\varvec{u}})|^2 \nonumber \\&\quad + M^2F^2\eta |G({\varvec{u}})|^2 \overline{I}_n({\varvec{0}}) + M^2F^2\overline{k}A|G({\varvec{u}})|^2 + \overline{d}A|\varPsi ({\varvec{u}})|^2 + \sigma _e^2 A . \end{aligned}$$

(79)

From $I_n({\varvec{u}}) = \overline{I_n({\varvec{u}})} + N_n({\varvec{u}})$ and Eq. (31), the 1st term in the right side of the above equation represents $\left| \overline{I_n({\varvec{u}})}\right| ^2$, and the sum of the latter four terms $\overline{|N({\varvec{u}})|^2}$. The mean power spectrum of the detector noise, $\overline{|N({\varvec{u}})|^2}$, does not depend on n, because Eq. (2) has been assumed, and thus $\overline{I}_n({\varvec{0}}) = O({\varvec{0}})$ being constant regardless of the frame number.

Averaging Eq. (79) over L frames, the expression for Eq. (74) results:

$$\begin{aligned} \overline{P'({\varvec{u}})}&= M^2\eta ^2\langle |\overline{I}({\varvec{u}})|^2\rangle _L|G({\varvec{u}})|^2 \nonumber \\&\quad + M^2F^2\eta |G({\varvec{u}})|^2\langle \overline{I}({\varvec{0}})\rangle _L + M^2F^2\overline{k}A|G({\varvec{u}})|^2 + \overline{d}A|\varPsi ({\varvec{u}})|^2 + \sigma _e^2 A . \end{aligned}$$

(80)

Recalling Eqs. (1) and (7), and the assumption Eq. (2), we have Eq. (33) with Eq. (34).

Variance of average power spectrum due to detector noise

Because of the relation $P({\varvec{u}}) = P'({\varvec{u}}) - B({\varvec{u}})$, the variance of $P({\varvec{u}})$ equals that of $P'({\varvec{u}})$, that is, $\sigma _N^2({\varvec{u}}) = \overline{\left\{ P({\varvec{u}}) - \overline{P({\varvec{u}})}\right\} ^2} = \overline{\left\{ P'({\varvec{u}}) - \overline{P'({\varvec{u}})}\right\} ^2}$, and thus we have

$$\begin{aligned} \sigma _N^2({\varvec{u}}) = \overline{P'^2({\varvec{u}})} - \left\{ \overline{P'({\varvec{u}})}\right\} ^2 , \end{aligned}$$

(81)

where $P'({\varvec{u}}) = \langle |I({\varvec{u}})|^2 \rangle _L$ is computed by

$$\begin{aligned} P'({\varvec{u}}) = L^{-1}\sum _{n=1}^L |I_n({\varvec{u}})|^2 . \end{aligned}$$

(82)

Substituting Eq. (82) into the right side of Eq. (81), the 1st term is

$$\begin{aligned} \overline{P'^2({\varvec{u}})} = L^{-2}\left\{ \sum _{n=1}^L\overline{|I_n({\varvec{u}})|^4} + \sum _{n=1}^L\sum _{n'\ne n}\overline{|I_n({\varvec{u}})|^2}~\overline{|I_{n'}({\varvec{u}})|^2}\right\} , \end{aligned}$$

(83)

where $|I_n({\varvec{u}})|^2$ and $|I_{n'}({\varvec{u}})|^2$ are independent for $n\ne n'$, and the 2nd term is

$$\begin{aligned} \left\{ \overline{P'({\varvec{u}})}\right\} ^2 = L^{-2}\sum _{n=1}^L\sum _{n'=1}^L \overline{|I_n({\varvec{u}})|^2}~\overline{|I_{n'}({\varvec{u}})|^2}, \end{aligned}$$

(84)

thus

$$\begin{aligned} \sigma _N^2({\varvec{u}}) = L^{-1}\left\langle \overline{|I({\varvec{u}})|^4} - \left( \overline{|I({\varvec{u}})|^2}\right) ^2\right\rangle _L, \end{aligned}$$

(85)

where we have used the notation for the average over L realizations, $\langle X \rangle _L = L^{-1}\sum _{n=1}^L X_n$.

The Fourier spectrum of the speckle frame, $I_n({\varvec{u}})$, is represented by $I_n({\varvec{u}}) = \overline{I_n({\varvec{u}})} + N_n({\varvec{u}})$, where $N_n({\varvec{u}})$ is given by the Fourier transform of the zero-mean detector noise term $\nu _n({\varvec{x}}) = \varDelta i'_n({\varvec{x}})$, that is,

$$\begin{aligned} N_n({\varvec{u}}) = \int \varDelta i'_n({\varvec{x}}) e^{-j2\pi {\varvec{x}}\cdot {\varvec{u}}} d{\varvec{x}} ~~~ (j=\sqrt{-1}), \end{aligned}$$

(86)

and $\overline{N({\varvec{u}})} = 0$. Then, the 2nd and the 4th moments of $I_n({\varvec{u}})$ are represented by

$$\begin{aligned} \overline{|I_n({\varvec{u}})|^2}&= \left| \overline{I_n({\varvec{u}})}\right| ^2 + \overline{|N({\varvec{u}})|^2}, \end{aligned}$$

(87)

$$\begin{aligned} \overline{|I_n({\varvec{u}})|^4}&= \left| \overline{I_n({\varvec{u}})}\right| ^4 + \overline{|N_n({\varvec{u}})|^4} + 4\left| \overline{I_n({\varvec{u}})}\right| ^2\overline{|N({\varvec{u}})|^2}\nonumber \\&\quad + \left\{ \overline{I_n({\varvec{u}})}\right\} ^2\left\{ \overline{N_n^2({\varvec{u}})}\right\} ^* + c.c.\nonumber \\&\quad + 2\overline{I_n({\varvec{u}})} ~ \overline{N_n^*({\varvec{u}})|N_n({\varvec{u}})|^2} + c.c., \end{aligned}$$

(88)

where c.c. stands for the complex conjugate of the preceding term. The 2nd moment $\overline{|N({\varvec{u}})|^2}$ is equal to the detector noise bias $B({\varvec{u}})$. The other 2nd moment $\overline{N_n^2({\varvec{u}})}$ is given by the average of the square of Eq. (86), that is,

$$\begin{aligned} \overline{N_n^2({\varvec{u}})} = \int \int \overline{\varDelta i'_n({\varvec{x}}) ~ \varDelta i'_n({\varvec{x}}')} e^{-j2\pi ({\varvec{x}}+{\varvec{x}}')\cdot {\varvec{u}}} d{\varvec{x}}d{\varvec{x}}'. \end{aligned}$$

(89)

Substituting Eq. (20) yields the expression for $\overline{N_n^2({\varvec{u}})}$,

$$\begin{aligned} \overline{N_n^2({\varvec{u}})} = M^2F^2\eta ~\overline{I}_n(2{\varvec{u}})G^2({\varvec{u}}) + M^2F^2\overline{k}|G({\varvec{u}})|^2\delta (2{\varvec{u}}) + \overline{d}|\varPsi ({\varvec{u}})|^2\delta (2{\varvec{u}}) + \sigma _e^2\delta (2{\varvec{u}}). \end{aligned}$$

(90)

Equation (86) implies that both the real and the imaginary parts of $N_n({\varvec{u}})$ are given as the weighted sums of large numbers of the zero-mean variables $\varDelta i'_n({\varvec{x}})$, and can be assumed to obey the zero-mean normal distributions because of the central limit theorem [49]. Additionally, we assume that the both parts are jointly normal. Thus, the 3rd and the 4th moments of $N_n({\varvec{u}})$ can be computed under the normality assumption, that is, $N_n({\varvec{u}})$ being assumed to be zero-mean complex normal variables. Letting $N_n({\varvec{u}}) = X + jY$ and invoking the moment theorem for the normal variables, we have

$$\begin{aligned} \overline{N({\varvec{u}})|N({\varvec{u}})|^2}&= 0 \end{aligned}$$

(91)

$$\begin{aligned} \overline{|N_n({\varvec{u}})|^4}&= 3\sigma _X^4 + 2\sigma _X^2\sigma _Y^2 + 4\sigma _{XY}^2 + 3\sigma _Y^4, \end{aligned}$$

(92)

where $\sigma _X^2$ and $\sigma _Y^2$ are the variances of X and Y, and $\sigma _{XY}$ is the covariance of them. Similarly, the 2nd moments are represented by $\overline{|N({\varvec{u}})|^2} = \sigma _X^2 + \sigma _Y^2$ and $\overline{N_n^2({\varvec{u}})} = \sigma _X^2 + 2j\sigma _{XY} - \sigma _Y^2$, and thus the 4th moment in Eq. (92) can be expressed by the two 2nd moments:

$$\begin{aligned} \overline{|N_n({\varvec{u}})|^4} = 2\left\{ \overline{|N({\varvec{u}})|^2}\right\} ^2 + \left| \overline{N_n^2({\varvec{u}})}\right| ^2. \end{aligned}$$

(93)

Substituting Eqs. (87), (88), (91) and (93), and letting $\overline{|N({\varvec{u}})|^2} = B({\varvec{u}})$, the expression for $\sigma _N^2({\varvec{u}})$ as given by Eq. (85) is rewritten by

$$\begin{aligned} \sigma _N^2({\varvec{u}}) = L^{-1} \Biggl [ B^2({\varvec{u}}) + 2\overline{P({\varvec{u}})}B({\varvec{u}}) + \left\langle \left| \overline{N^2({\varvec{u}})}\right| ^2\right\rangle _L + \left\langle \left\{ \overline{I({\varvec{u}})}\right\} ^2\left\{ \overline{N^2({\varvec{u}})}\right\} ^* + c.c.\right\rangle _L \Biggr ], \end{aligned}$$

(94)

where we have applied the relation $\left\langle \left| \overline{I({\varvec{u}})}\right| ^2\right\rangle _L = \overline{P({\varvec{u}})}$ as derived from the relations (33) and (40). From Eq. (90), $\overline{N_n^2({\varvec{u}})}$ vanishes for $|{\varvec{u}}|\ge u_c/2$, because in this frequency range, $\overline{I}_n(2{\varvec{u}}) = O(2{\varvec{u}})S_n(2{\varvec{u}}) = 0$ and $\delta (2{\varvec{u}}) = 0$. Thus, we have the expression for $\sigma _N^2({\varvec{u}})$ as given by Eq. (43).

The condition $|{\varvec{u}}|\ge u_c/2$ for the 3rd and 4th terms in Eq. (94) vanishing and Eq. (43) holding may be relaxed, because, when considering the ensemble average of Eq. (94), the 3rd and 4th terms are ignorable as long as $|{\varvec{u}}| > u_s$. To show this, for simplicity, let us assume $\eta = 1$ and $F = \sqrt{2}$, and ignore the dark charges and the readout noise ($\overline{k} = \overline{d} = \sigma _e = 0$). Then, recalling Eqs. (90), (41), (34) and (31), the ensemble average of Eq. (94) for $|{\varvec{u}}|>u_s$ is written by

$$\begin{aligned} \langle \sigma _N^2({\varvec{u}}) \rangle = 4M^4L^{-1}\left\{ \overline{I}^2({\varvec{0}}) + \left\langle \overline{I}({\varvec{0}})|\overline{I}({\varvec{u}})|^2\right\rangle + \left\langle |\overline{I}({2\varvec{u}})|^2\right\rangle + \left\langle |\overline{I}({2\varvec{u}})| \zeta ({\varvec{u}}) |\overline{I}({\varvec{u}})|^2\right\rangle \right\} |G({\varvec{u}})|^4, \end{aligned}$$

(95)

where

$$\begin{aligned} \zeta _n({\varvec{u}}) = \cos \left( 2\theta ({\varvec{u}}) - \theta (2{\varvec{u}}) + 2\phi _n({\varvec{u}}) - \phi _n(2{\varvec{u}})\right) \end{aligned}$$

(96)

with $\theta ({\varvec{u}})$ and $\phi _n({\varvec{u}})$ being the phases of $O({\varvec{u}})$ and $S_n({\varvec{u}})$, respectively.

The factor $\overline{I}({\varvec{0}})$ indicates the number of the photons from the object collected during the exposure time. In our typical case as shown in Tables 1 and 2, the count of the photons approximately amounts to 60000. In Eq. (95), the 1st and 2nd terms including $\overline{I}({\varvec{0}})$ are superior to the 3rd term. The 4th term is the weighted average of $|\overline{I}({\varvec{u}})|^2$, the weight of which includes two factors $|\overline{I}_n({2\varvec{u}})|$ and $\zeta _n({\varvec{u}})$. According to Eqs. (6) and (10), in our case, the former factor $|\overline{I}_n({2\varvec{u}})|$ for $|{\varvec{u}}|>u_s$ is reduced from $\overline{I}({\varvec{0}})$ by $r_0/D = 1/20$ or less on average. Furthermore, due to the latter factor $\zeta _n({\varvec{u}})$ being random within $[-1,1]$, the 4th term becomes almost zero. Thus, for $|{\varvec{u}}|>u_s$, the 3rd and 4th terms are ignorable.

Quotient of average power spectra

The quotient of the average power spectra, $P'_O({\varvec{u}}) = {P({\varvec{u}})}/{P_r({\varvec{u}})}$, is represented by its ensemble mean and the residual zero-mean random component, that is,

$$\begin{aligned} {P}'_O({\varvec{u}}) = \left\langle \overline{{P}'_O({\varvec{u}})}\right\rangle + \delta P'_O({\varvec{u}}), \end{aligned}$$

(97)

where

$$\begin{aligned} \left\langle \overline{{P}'_O({\varvec{u}})}\right\rangle = \left\langle \overline{P({\varvec{u}})}\right\rangle \left\langle \overline{P^{-1}_r({\varvec{u}})}\right\rangle , \end{aligned}$$

(98)

because the two average power spectra $P({\varvec{u}})$ and $P_r({\varvec{u}})$ are independent.

Applying the Taylor’s theorem to the inverse of $P_r({\varvec{u}}) = \left\langle \overline{P_r({\varvec{u}})}\right\rangle + \delta P_r({\varvec{u}})$, we have

$$\begin{aligned} P^{-1}_r({\varvec{u}}) = \left\langle \overline{P_r({\varvec{u}})}\right\rangle ^{-1} - \left\langle \overline{P_r({\varvec{u}})}\right\rangle ^{-2}\delta P_r({\varvec{u}}) + \left\langle \overline{P_r({\varvec{u}})}\right\rangle ^{-3}\{\delta P_r({\varvec{u}})\}^2, \end{aligned}$$

(99)

where the high-order terms are neglected. Taking the ensemble mean results in

$$\begin{aligned} \left\langle \overline{P^{-1}_r({\varvec{u}})}\right\rangle = \left\langle \overline{P_r({\varvec{u}})}\right\rangle ^{-1} + \left\langle \overline{P_r({\varvec{u}})}\right\rangle ^{-3}\sigma ^2_r({\varvec{u}}), \end{aligned}$$

(100)

where $\left\langle \overline{\delta P_r({\varvec{u}})}\right\rangle = 0$ and $\left\langle \overline{\{\delta P_r({\varvec{u}})\}^2}\right\rangle = \sigma ^2_r({\varvec{u}})$. Substituting Eq. (100) into Eq. (98) and recalling the definition of NSR $\mu _r({\varvec{u}})$ yield

$$\begin{aligned} \left\langle \overline{{P}'_O({\varvec{u}})}\right\rangle = \frac{\left\langle \overline{P({\varvec{u}})}\right\rangle }{\left\langle \overline{P_r({\varvec{u}})}\right\rangle }\{1 + \mu ^2_r({\varvec{u}})\}, \end{aligned}$$

(101)

and thus Eq. (49) results.

Variance of unbiased estimate of object power spectrum

By the definition, the variance of the unbiased estimate of the object power spectrum is given by

$$\begin{aligned} \sigma _O^2({\varvec{u}}) = \left\langle \overline{P_O^2({\varvec{u}})}\right\rangle - \left\langle \overline{P_O({\varvec{u}})}\right\rangle ^2, \end{aligned}$$

(102)

which is rewritten using the power spectral quotient (biased estimate) $P'_O({\varvec{u}})$ by

$$\begin{aligned} \sigma ^2_O({\varvec{u}}) = \frac{1}{\{1 + \mu ^2_r({\varvec{u}})\}^2} \left\{ \left\langle \overline{P_O^{'2}({\varvec{u}})}\right\rangle - \left\langle \overline{P'_O({\varvec{u}})}\right\rangle ^2 \right\} . \end{aligned}$$

(103)

In Eq. (103), the ensemble mean of $P_O^{'2}({\varvec{u}}) = {P^2({\varvec{u}})} / {P_r^2({\varvec{u}})}$ is computed as

$$\begin{aligned} \left\langle \overline{P_O^{'2}({\varvec{u}})}\right\rangle = \left\langle \overline{P^2({\varvec{u}})}\right\rangle \left\langle \overline{P^{-2}_r({\varvec{u}})}\right\rangle , \end{aligned}$$

(104)

where the two average power spectra $P({\varvec{u}})$ and $P_r({\varvec{u}})$ are independent. The ensemble mean of the square of $P({\varvec{u}}) = \left\langle \overline{P({\varvec{u}})}\right\rangle + \delta P({\varvec{u}})$ is computed as

$$\begin{aligned} \left\langle \overline{P^2({\varvec{u}})}\right\rangle = \left\langle \overline{P({\varvec{u}})}\right\rangle ^2 + \sigma ^2({\varvec{u}}) = \left\langle \overline{P({\varvec{u}})}\right\rangle ^2 \{1 + \mu ^2({\varvec{u}})\} . \end{aligned}$$

(105)

Applying the Taylor’s theorem to the inverse square of $P_r({\varvec{u}}) = \left\langle \overline{P_r({\varvec{u}})}\right\rangle + \delta P_r({\varvec{u}})$, we have

$$\begin{aligned} P^{-2}_r({\varvec{u}}) = \left\langle \overline{P_r({\varvec{u}})}\right\rangle ^{-2} - 2\left\langle \overline{P_r({\varvec{u}})}\right\rangle ^{-3}\delta P_r({\varvec{u}}) + 3\left\langle \overline{P_r({\varvec{u}})}\right\rangle ^{-4}\{\delta P_r({\varvec{u}})\}^2, \end{aligned}$$

(106)

where the high-order terms are neglected. Taking the ensemble mean results in

$$\begin{aligned} \left\langle \overline{P^{-2}_r({\varvec{u}})}\right\rangle = \left\langle \overline{P_r({\varvec{u}})}\right\rangle ^{-2} + 3\left\langle \overline{P_r({\varvec{u}})}\right\rangle ^{-4}\sigma ^2_r({\varvec{u}}) = \left\langle \overline{P_r({\varvec{u}})}\right\rangle ^{-2} \{1 + 3\mu _r^2({\varvec{u}})\}, \end{aligned}$$

(107)

where $\left\langle \overline{\delta P_r({\varvec{u}})}\right\rangle = 0$ and $\left\langle \overline{\{\delta P_r({\varvec{u}})\}^2}\right\rangle = \sigma ^2_r({\varvec{u}})$.

Substituting Eqs. (104) and (101), and then Eqs. (105) and (107) into Eq. (103), and using the relation (53), we have

$$\begin{aligned} \sigma ^2_O({\varvec{u}}) = \left\langle \overline{P_O({\varvec{u}})}\right\rangle ^2 ~ \frac{\mu ^2({\varvec{u}}) + \mu ^2_r({\varvec{u}}) + \mu ^2_r({\varvec{u}})\left\{ 3\mu ^2({\varvec{u}}) - \mu ^2_r({\varvec{u}})\right\} }{\left\{ 1 + \mu ^2_r({\varvec{u}})\right\} ^2}. \end{aligned}$$

(108)

For the spatial frequencies at which the NSRs are sufficiently small, say less than 0.2, their 4th powers in the numerator and the denominator of the right side can be neglected in comparison with their squares, and thus Eq. (54) is obtained.

Symbols and notation rules

Since a large number of quantities, some of which are similar but must be distinguished, appear in this paper, we summarize their symbols and notation rules in the following subsections. In Sect. G.1, we summarize for the quantities appearing in the imaging process before detecting by the camera, which are first defined in Sect. 2. In Sect. G.2, we summarize for the quantities introduced and used in Sect. 3 through Sect. 5, where the detection noise is taken into account.

1.1 Quantities related to optical images

Table 3 is the list of the symbols for the quantities arising when exclusively considering the optical imaging process. The notations for the other quantities are derived by the rules as shown in Table 4, where X represents the random variables due to the atmospheric turbulence (atmospherically random variables). Additionally, there are the following miscellaneous rules, where $f({\varvec{x}})$ represents the functions of the pixel coordinates ${\varvec{x}}$ (spatial distributions on the image plane).

• The Fourier spectrum of $f({\varvec{x}})$ is denoted by $F({\varvec{u}})$, the uppercase letter corresponding to the name of the image plane function, where ${\varvec{u}}$ denotes the pixel coordinates on the Fourier plane (spatial frequency domain).

• The power spectrum and the autocorrelation function of $f({\varvec{x}})$ are denoted by $|F({\varvec{u}})|^2$ and $(f\star f)({\varvec{x}})$, respectively.

• If the function value at any pixel position on the image plane, $f({\varvec{x}})$, is a random variable, then $F({\varvec{u}})$ and $|F({\varvec{u}})|^2$ are also similarly random.

• If the function of ${\varvec{x}}$ or ${\varvec{u}}$ is randomly variable, then the function values at different pixel positions are different random variables.

• If $f({\varvec{x}})$ is a random variable, then its n-th realization is denoted by $f_n({\varvec{x}})$, and those of $F({\varvec{u}})$ and $|F({\varvec{u}})|^2$ are by $F_n({\varvec{u}})$ and $|F_n({\varvec{u}})|^2$, respectively.

Table 3 Symbols for the quantities related to the optical images

Table 4 Notation rules for an atmospherically random variable X

1.2 Quantities related to data frames

The table of the symbols for the quantities introduced by considering the detection process is given in Table 5. The notations for the other quantities are derived by the rules as shown in Table 6, where Y represents the doubly random variables due to not only the atmospheric turbulence but also the electronic fluctuation in the detection process (detection noise), and Z represents the random variables due to only the electronic fluctuation (electronically random variables). Additionally, there are the following miscellaneous rules, where $f({\varvec{x}})$ and $F({\varvec{u}})$ represent image plane functions and Fourier plane functions, respectively.

• While $\overline{f({\varvec{x}})}$ denotes the expectation of the variable $f({\varvec{x}})$, $\overline{f}({\varvec{x}})$ denotes the value of the function with the name of $\overline{f}$. Both are usually the same, but not necessarily so. For example, the optical image (optical intensity distribution) at the specific instant, $\overline{i}_n({\varvec{x}})$, and its mean data frame $\overline{i_n({\varvec{x}})}$ are closely related, but not the same.

• The function value $f_n({\varvec{x}})$ is abbreviated to f, if it is obvious from the context that the instant n and the position ${\varvec{x}}$ are fixed to the specific ones, or if the function value is constant regardless of n and ${\varvec{x}}$.

• The variance of $f({\varvec{x}})$ is denoted by $\sigma _{f}^2({\varvec{x}})$ rather than $\sigma _{f({\varvec{x}})}^2$.

• The function having the name with the prime is sometimes offset or biased from the function having the same name but the prime. For examples, $i'({\varvec{x}})$ is offset from $i({\varvec{x}})$ ($i({\varvec{x}})$ is the de-offset version of $i'({\varvec{x}})$), $P'({\varvec{u}})$ is biased from $P({\varvec{u}})$ ($P({\varvec{u}})$ is the debiased version of $P'({\varvec{u}})$), and so on.

• The conditional variance of the value of an offset or biased function is denoted by the same notation for that of the de-offset or debiased version, because the conditional variance is not changed by de-offsetting or debiasing.

Table 5 Symbols for the quantities related to the data frames

Table 6 Notation rules for a doubly random variable Y and an electronically random variable Z