# Modeling and Estimation of Adaptive Optics-Corrected long-exposure Point Spread Function Using Gaussian Sum Approximation

Pedro Escárate, B. Osorio, C. Zamorano, M. Coronel, R. Orellana, J. C. Agüero, B. Neichel

► To cite this version:

Pedro Escárate, B. Osorio, C. Zamorano, M. Coronel, R. Orellana, et al., Modeling and Estimation of Adaptive Optics-Corrected long-exposure Point Spread Function Using Gaussian Sum Approximation. Monthly Notices of the Royal Astronomical Society, 2024, 10.1093/mnras/sta2578. insu-04846480

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# Modelling and estimation of adaptive optics-corrected long-exposure point spread function using Gaussian sum approximation

Pedro Escárate<sup>1,★</sup> B. Osorio,<sup>1</sup> C. Zamorano,<sup>1</sup> M. Coronel,<sup>2</sup> R. Orellana,<sup>3</sup> J.C. Agüero<sup>4</sup> and B. Neichel<sup>5</sup>

<sup>1</sup>Escuela de Ingeniería Eléctrica, Facultad de Ingeniería, Pontificia Universidad Católica de Valparaíso, Avenida Brasil 2147, Valparaíso, 2362804, Chile

<sup>2</sup>Departamento de Electricidad, Universidad Tecnológica Metropolitana, Avenida José Pedro Alessandri 1242, Nuñoa, Santiago, 7800002, Chile

<sup>3</sup>Departamento de Ingeniería Eléctrica, Facultad de Ingeniería, Universidad de Santiago de Chile (USACH), Avenida Víctor Jara 3519, Estación Central, Santiago, 9170124, Chile

<sup>4</sup>Departamento de Electrónica, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, 2390123, Chile

<sup>5</sup>Aix Marseille Univ., CNRS, LAM, Laboratorire d'Astrophysique de Marseille, Marseille, 13388 cedex 13, France

Accepted 2024 November 13. Received 2024 November 7; in original form 2024 October 1

## ABSTRACT

Adaptive optics (AO) enhances astronomical images by correcting perturbations in the optical system, resulting in complex point spread function (PSF) shapes, which provide critical information for optical design, calibration, and diagnostics. This paper presents a parametric model for an AO-corrected long-exposure PSF that adapts to complex shapes and various seeing conditions. A complementary estimation method is also introduced, addressing the inverse problem of estimating the atmospheric PSF or power spectral density (PSD) from observed PSF data. The model, based on Gaussian sum approximation (GSA), was tested using simulated PSF data from the OOMAO toolbox for Fried parameter ($r_0$) values ranging from 0.10 to 0.20 m and on-sky data from the Very Large Telescope's MUSE instrument. Two approaches were analysed: (i) direct PSF estimation using GSA, and (ii) PSF estimation derived from atmospheric PSD estimation. Both methods were evaluated using root mean square error (RMSE). While approach (i) provided the best performance, approach (ii) produced more accurate atmospheric PSF estimates compared to the Moffat approximation and the Fetick method. Additionally, approach (ii) enabled the estimation of key physical parameters, such as the Fried parameter ($r_0$) and residual AO variance ($\sigma_{AO}^2$). The proposed methods effectively estimate atmospheric PSF and PSD, even for complex PSF shapes influenced by spiders, central obstructions, or static aberrations.

**Key words:** instrumentation: adaptive optics – methods: analytical – methods: data analysis.

## 1 INTRODUCTION

Adaptive optics (AO) is a modern optical technique that enhances the quality of astronomical images by compensating for the effects of atmospheric disturbances, thereby partially restoring image resolution close to the diffraction limit of the telescope (Davies & Kasper 2012). Moreover, AO systems are an integral part of modern telescopes and will be an essential component in the design of future large telescopes (Roddier 1999).

The determination of the point spread function (PSF) is one of the main challenges in astronomical data processing. Accurate knowledge of the PSF is essential for estimating astronomical signals through post-processing techniques, such as deconvolution (Drummond 1998; Fusco et al. 2021). In recent years, considerable efforts have been made in PSF reconstruction (PSF-R) (Fusco et al. 2021; Grazian et al. 2022; Wagner, Niebsch & Ramlau 2023; Alonso, Zhang & Liu 2024; Millon et al. 2024), particularly in estimating the PSF from AO control loop telemetry, including AO wavefront measurements and commands (Beltramo-Martin et al. 2020a, 2020b).

For ground-based astronomical observations, atmospheric turbulence significantly distorts the PSF due to wavefront distortion of (Roddier 1981). Consequently, implementing AO correction is essential to improve astronomical images and approach the diffraction limit of the optical system. However, precise PSF estimation remains challenging, as the effectiveness of AO correction depends on several factors, including wavefront sensor noise, the number of actuators in the deformable mirror, and the parameters of the AO control loop, as discussed by Martin et al. (2017) and Rigaut, Veran & Lai (1998). As a result, AO correction produces a distinctive PSF shape characterized by a sharp peak and a wide halo. Despite efforts, even under ideal AO correction (Conan 1994), the PSF retains characteristics of the Airy function near the optical axis, along with variability that leads to perceptible image blurring (Fusco et al. 2000).

The PSF provides useful information about the optical system, which is key in preliminary design considerations, calibration procedures, testing protocols, and diagnostic evaluations (Ragland et al. 2018). Moreover, a deep understanding of the PSF is required in image post-processing, particularly in deconvolution techniques (Mugnier, Fusco & Conan 2004). Previous works of deconvolution using parametric PSFs have been demonstrated by Drummond (1998) and Fétick et al. (2019), highlighting the importance of accurate PSF knowledge in achieving accurate image correction.

<footnote>* E-mail: pedro.escarate@pucv.cl</footnote>
$r_0 = 0.10 \text{ m}, T_{exp} = 5\text{s}$

<figure><img src="image_2.png" /></figure>

$r_0 = 0.15 \text{ m}, T_{exp} = 5\text{s}$

<figure><img src="image_3.png" /></figure>

$r_0 = 0.20 \text{ m}, T_{exp} = 5\text{s}$

<figure><img src="image_4.png" /><figcaption>**Figure 1.** OOMAO generated PSF for $r_0 = \{0.10, 0.15, 0.20\}$ m with an exposure time of 5 s. Top row: (a) 2D PSF for $r_0 = 0.10$ m and (b) Central cut of the 2D PSF for $r_0 = 0.10$ m. Middle row: (c) 2D PSF for $r_0 = 0.15$ m and (d) central cut of the 2D PSF for $r_0 = 0.15$ m. Bottom row: (e) 2D PSF for $r_0 = 0.20$ m and (f) central cut of the 2D PSF for $r_0 = 0.20$ m.</figcaption></figure>

using GSA to estimate the atmospheric PSD ($\hat{W}_{\phi/AO}$), as described in equation (11), which is then used to calculate the atmospheric PSF.

To address this, the proposed method approximates the unknown PSF or PSD using GSA, respectively, as outlined below:

$$ \hat{h}_{\text{atm/AO}}(\vec{\rho}|\theta) = \sum_{i=1}^{N_g} \gamma_i \mathcal{N}(\vec{\rho}, \vec{\mu}_i, \Lambda_i), \quad (10) $$

$$ \hat{W}_{\phi/AO}(f|\theta) = \left[ \sum_{i=1}^{N_g} \gamma_i \mathcal{N}(f, \vec{\mu}_i, \Lambda_i) \right] + \left[ 0.023 r_0^{-5/3} f^{-11/3} \right]_{f<f_{AO}}, \quad (11) $$

where $N_g \in \mathbb{N}$ represents the number of Gaussian functions used to approximate the PSF, as described in equation (10), or the PSD, as described in equation (11). The Fried parameter is denoted by $r_0$, and $\gamma_i \in \mathbb{R}$ represents the weight of the $i$th Gaussian

function. The parameter vector $\theta$ for equation (10) is defined as $\theta = [\gamma_1, \vec{\mu}_1, \Lambda_1, \dots, \gamma_{N_g}, \vec{\mu}_{N_g}, \Lambda_{N_g}]$, and for equation (11), it is given by $\theta = [\gamma_1, \vec{\mu}_1, \Lambda_1, \dots, \gamma_{N_g}, \vec{\mu}_{N_g}, \Lambda_{N_g}, r_0]$. It should be noted that the $\gamma_i$ weights for equation (10) are not necessarily the same as the $\gamma_i$ weights in equation (11), as they are used to estimate different curves. Additionally, $\mathcal{N}(\vec{x}, \vec{\mu}_i, \Lambda_i)$ represents a multivariate Gaussian function with the unit area, defined as:

$$ \mathcal{N}(\vec{x}, \vec{\mu}_i, \Lambda_i) = \frac{1}{\sqrt{(2\pi)^2 |\Lambda_i|}} e^{-\frac{1}{2}(\vec{x}-\vec{\mu}_i)^T \Lambda_i^{-1} (\vec{x}-\vec{\mu}_i)}, \quad (12) $$

where $\vec{\mu}_i = [\mu_{x_i}, \mu_{y_i}] \in \mathbb{R}^2$ represent the center of $i$th Gaussian function and $\Lambda_i \in \mathbb{R}^{2 \times 2}$ represent the variance-covariance positive semidefinite matrix of the $i$th Gaussian function, given by:

$$ \Lambda_i = \begin{bmatrix} \sigma_{x_i}^2 & \rho_i \sigma_{x_i} \sigma_{y_i} \\ \rho_i \sigma_{x_i} \sigma_{y_i} & \sigma_{y_i}^2 \end{bmatrix}, \quad (13) $$

where $\sigma_{x_i} > 0$, $\sigma_{y_i} > 0$, and $-1 < \rho_i < 1$ for $i = 1, \dots, N_g$. Since the PSF and PSD are positive, all weights must also be positive, $\gamma_i \ge 0$ for $i = 1, \dots, N_g$, in order to preserve this physical property.

The AO-corrected PSD described in equation (11) shares the advantage of the Fetick PSD in correlating PSD parameters with physical parameters such as the Fried parameter $r_0$ and the residual AO variance $\sigma_{AO}^2$. This correlation allows these parameters to be compared with external measurements, thereby providing a deeper understanding of the AO system response to observing conditions. Furthermore, as indicated by equation (11), the residual AO variance can be determined as the area within the circular domain below the AO cutoff frequency, as shown by:

$$ \sigma_{AO}^2 = \iint_{f \le f_{AO}} \sum_{i=1}^{N_g} \gamma_i \mathcal{N}(f, \vec{\mu}_i, \Lambda_i) \, df_x \, df_y. \quad (14) $$

Additionally, the variance of the Kolmogorov spectrum above the AO cutoff frequency can be expressed as follows:

$$ \sigma_{\text{atm}}^2 = \iint_{f > f_{AO}} 0.023 r_0^{-5/3} f^{-11/3} \, df_x \, df_y, \quad (15) $$

$$ = 0.023 \frac{6\pi}{5} (r_0 * f_{AO})^{-5/3}. \quad (16) $$

### 3 PSF ESTIMATION METHOD

Considering equation (1) and the prior knowledge of $h_{tel}$, the inverse problem of determining the atmospheric PSF, denoted as $h_{atm/AO}$, from the observed PSF, denoted as $h_{obs}$ is described as follows:

$$ \hat{h}_{\text{obs}}(\vec{\rho}|\theta) = h_{\text{tel}}(\vec{\rho}) * \hat{h}_{\text{atm/AO}}(\vec{\rho}|\theta). \quad (17) $$

Therefore, the proposed PSF estimation method for addressing the inverse problem involves determining the parameters of the GSA PSF or GSA PSD model.

Defining the cost function $L(\theta)$:

$$ L(\theta) = \sum_{i,j}^{N_{\text{pix}}} (h_{\text{obs}}(i, j) - \hat{h}_{\text{obs}}(i, j|\theta))^2, \quad (18) $$

where $i, j$ represent the position of the pixel and $N_{pix}$ represents the total number of pixels in the PSF image.

To determine the parameter vector, the following optimization problem is solved:

$$ \hat{\theta} = \arg \min_{\vec{\theta}} L(\theta), \quad \text{s.t.} \quad \begin{cases} -1 + \epsilon \le \rho_i \le 1 - \epsilon \\ \sigma_{x_i} \ge \epsilon \\ \sigma_{y_i} \ge \epsilon \\ \gamma_i \ge 0 \end{cases}, \quad i = 0, \dots, N_g. \quad (19) $$
<figure><img src="image_5.png" /><figcaption>**Figure 2.** PSF estimation for $r_0 = \{0.10, 0.15, 0.20\}$ m and $T_{exp} = 5$ s. Top row (a): The solid blue line represents the OOMAO PSF, the dotted red line represents the best estimated PSF, and black marks represent the best estimated Moffat PSF. Bottom row (b): The solid blue line represents the GSA of atmospheric PSF, and the dotted red line represents the estimation of atmospheric PSF using the Moffat function.</figcaption></figure>

To address the minimization problem outlined in equation (18), an initial value of $\theta$ ($\theta^0$) is required. To ensure closed constraints for the optimization process using software, a small number $\epsilon = 1 \times 10^{-40}$ was incorporated into the constraints. Additionally, to prevent convergence to a local minimum, the minimization problem is computed $n_{MC}$ times, each with a different random initial value. Subsequently, the optimal $\theta^i$, corresponding to the minimum $L(\theta^i)$ value, is selected. This value is then used to compute either $\hat{h}_{atm/AO}(\bar{\rho}|\theta^i)$ for the direct PSF estimation approach (i) or $\hat{W}_{\phi/AO}(f|\theta^i)$ for the PSD estimation approach (ii).

The proposed algorithm is summarized in Table 1, where step 2a is employed for approach (i) atmospheric PSF estimation using GSA, and step 2b is utilized for approach (ii) atmospheric PSF calculation derived from the estimation of atmospheric PSD using GSA.

The minimization problem in step 4 is addressed using the MATLAB function *fmincon* to incorporate the constraints described in equation (19). The *sqp*<fnref n="1" /> algorithm is employed within *fmincon*, with the *Max Function Evaluations* parameter set to 20,000 and the *Max Iterations* parameter set to 1000 000.

Although a Gaussian function is proposed in this work, the method can be used with other basis functions to fit complex PSF shapes (see e.g. Orellana et al. 2019).

# 4 VALIDATION

## 4.1 Object-oriented Matlab adaptive optics simulations

To evaluate the performance of the proposed method described in the preceding section, we used the Object-Oriented Matlab Adaptive Optics (OOMAO) toolbox, presented by Conan & Correia (2014), to generate a set of PSFs, using the parameters of Table 2.

The set of OOMAO-generated PSFs, based on the parameters in Table 2, includes PSFs with Shannon–Nyquist sampling. These PSFs correspond to five different values of $r_0$, ranging from 0.10 to 0.20 m, covering the typical range of $r_0$ measured at astronomical sites (Sarazin et al. (2008), Osborn et al. (2018)). Additionally, three exposure times ($T_{exp}$) of 1, 2, and 5 s were considered.

Fig. 1 shows the PSFs generated using OOMAO for Fried parameter ($r_0$) values of (0.10, 0.15, and 0.20) m, and for a $T_{exp} = 5$ s.

In this work, we used $N_g = 10$ Gaussian functions for all analyses, as this value achieved a lower RMSE than the reference method.

<footnote n="1">¹sqp refers to sequential quadratic programming methods (see e.g. Nocedal & Wright 2006).</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 3.** PSD estimation for $r_0 = \{0.10, 0.15, 0.20\}$ m and $T_{exp} = 5$ s. Top row(a): The solid blue line represents the OOMAO PSF, the dotted red line represents the best estimated PSF of 20 MC simulations, and black marks represent the best estimated PSF of 20 MC simulations using the Fetick Method. Middle row (b): The solid blue line represents the estimated atmospheric PSF, and the dotted red line represents the estimation of atmospheric PSF using the Fetick method. Bottom row (c): The solid blue line represents the GSA of the atmospheric PSD, and the dotted red line represents the estimation of atmospheric PSD using the Fetick method.</figcaption></figure>

Although increasing $N_g$ slightly improves RMSE, each additional Gaussian function significantly raises computational demands. For example, in our simulations, increasing $N_g$ to 12 led to an exponential increase in computation time by a factor of 1.2984, while reducing the RMSE by only a factor of 0.9824.

## 4.2 Estimation of atmospheric PSF by Gaussian Sum Approximation

To assess the performance of this method, the set of PSFs generated with OOMAO was used to estimate the observed PSF, and its accuracy was evaluated in terms of the root mean square error
<figure><img src="image_7.png" /><figcaption>**Figure 4.** The Fried parameter $r_0$ estimated versus the $r_0$ used in OOMAO to generate the PSF. The black marks represent the estimated $r_0$ at each simulation versus the real value from OOMAO. The red line represents the linear fit between the $r_0$ estimation and the OOMAO $r_0$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 5.** On-sky MUSE PSF. (a) 2D image of MUSE PSF (b) Central cut of the 2D MUSE PSF.</figcaption></figure>

(RMSE), defined as:

$$RMSE = \sqrt{\frac{1}{N_{\text{pix}}} \sum_{i,j}^{N_{\text{pix}}} (h(i, j) - \hat{h}(i, j|\theta))^2} \quad (20)$$

Using this approach, the best RMSE between the observed PSF and the estimated PSF was obtained for $r_0 = 0.20$ m. Among the exposure times described in Table 2, the best RMSE result was achieved for $T_{\text{exp}} = 5$ s. Therefore, the analysis in the following sections was conducted using $T_{\text{exp}} = 5$ s.

Fig. 2(a) illustrates the OOMAO PSF for $r_0 = \{0.10, 0.15, 0.20\}$ m, along with the corresponding estimations using the GSA PSF approach and Moffat function. The solid blue line represents the PSF generated by OOMAO, while the dotted red line depicts the best estimation from $n_{\text{MC}} = 20$ Monte Carlo (MC) simulations of the OOMAO PSF using the proposed method. The black marks indicate the best estimation from $n_{\text{MC}} = 20$ MC simulations of the OOMAO PSF using the Moffat function (equation 6).

Fig. 2(b) displays the estimation of the atmospheric PSF for $r_0 = \{0.10, 0.15, 0.20\}$ m using the GSA PSF approach and the Moffat function. The solid blue line represents the best GSA estimation of the atmospheric PSF, while the dotted red line represents the best estimation using the Moffat function.

The results presented in Fig. 2(a) indicate that the observed PSF obtained from the atmospheric GSA PSF (dotted red line), with RMSE values of $\{1.45 \times 10^{-6}, 6.34 \times 10^{-6}, 5.05 \times 10^{-6}\}$, outperforms the observed PSF obtained from the atmospheric PSF approximated using the Moffat function (black marks), for $r_0 = \{0.10, 0.15, 0.20\}$ m, which have RMSE values of $\{2.35 \times 10^{-5}, 1.33 \times 10^{-6}, 9.08 \times 10^{-6}\}$, respectively, as expected.

Importantly, the proposed method is independent of specific $r_0$ values, meaning that it is not limited to the $r_0$ values used in this work within the OOMAO simulations (0.10 to 0.20 metres). This flexibility ensures that the proposed method can be applied to a broad range of atmospheric conditions suitable for astronomical observations, enhancing its applicability across different sites and observing scenarios.

## 4.3 Estimation of the atmospheric PSD using Gaussian sum approximation

In this section, the accuracy of the second method is evaluated in terms of RMSE using a set of PSFs generated with OOMAO, as described in the previous sections. This method was applied to the set of PSFs to obtain results, which were then compared with those obtained using the Fetick method (Fétick et al. 2019).

Fig. 3 presents the results of the proposed method and the Fetick method for PSFs with $r_0 = \{0.10, 0.15, 0.20\}$ m and $T_{\text{exp}} = 5$ m. Fig. 3(a) illustrates the PSD estimation for $r_0 = \{0.10, 0.15, 0.20\}$ m and $T_{\text{exp}} = 5$ s. The solid blue line represents the OOMAO PSF, the dotted red line represents the best estimated PSF from $n_{\text{MC}} = 20$ Monte Carlo (MC) simulations, and the black marks represent the best estimated PSF from $n_{\text{MC}} = 20$ MC simulations using the Fetick Method. In Fig. 3(b), the solid blue line represents the estimated atmospheric PSF, while the dotted red line represents the atmospheric PSF estimation using the Fetick Method. Lastly, in Fig. 3(c), the solid blue line represents the generalized spectrum analysis (GSA) of the atmospheric PSD, and the dotted red line represents the atmospheric PSD estimation using the Fetick Method.

The results presented in Fig. 3(a) indicate that for each case ($r_0 = \{0.10, 0.15, 0.20\}$ m), both methods achieve a high level of agreement between the observed PSF (solid blue line) and the PSF estimated by the proposed method (dotted red line), even in the presence of read-out noise. Although both methods are accurate, the proposed method demonstrates a slight improvement in RMSE performance, with RMSE values of $\{8.63 \times 10^{-5}, 7.12 \times 10^{-5}, 5.84 \times 10^{-5}\}$, compared to the Fetick method (black marks), which has RMSE values of $\{8.87 \times 10^{-5}, 7.37 \times 10^{-5}, 5.86 \times 10^{-5}\}$. This suggests that the proposed method is robust and effective in estimating the PSF for $r_0 = \{0.10, 0.15, 0.20\}$ m.

Furthermore, as shown in Fig. 3(c) demonstrates that the proposed method can generate complex PSD shapes that the Moffat function is unable to replicate. These findings suggest that the proposed method is a reliable tool for estimating the PSF. Additionally, similar to the Fetick method, the estimation of the atmospheric PSD using equation (5) facilitates the estimation of the Fried parameter $r_0$. As illustrated in Fig. 4, there is a linear relationship between the
<figure><img src="image_9.png" /><figcaption>**Figure 6.** PSF estimation for MUSE PSF. (a) The solid blue line represents the observed MUSE PSF, and the dotted red line represents the best estimated PSF of $n = n_{MC} = 20$ MC simulations using the approach (i). (b) The solid blue line represents the observed MUSE PSF, and the dotted red line represents the best estimated PSF of $n = n_{MC} = 20$ MC simulations using the approach (ii). (c) The solid blue line represents the observed MUSE PSF, and the dotted red line represents the estimated PSF using the Fetick method. (d) The solid blue line shows the difference between the observed MUSE PSF and the best estimated PSF from $n = n_{MC} = 20$ MC simulations using approach (i). (e) The solid blue line shows the difference between the observed MUSE PSF and the best estimated PSF from $n = n_{MC} = 20$ MC simulations using approach (ii). (f) The solid blue line shows the difference between the observed MUSE PSF and the estimated PSF obtained with the Fetick method.</figcaption></figure>

OOMAO value of $r_{0,OOMAO}$ and the estimation of $r_0$, as given by:

$$r_0 = 1.548r_{0,OOMAO} + 0.002794. \qquad (21)$$

Fig. 4 illustrates that the $r_0$ estimation achieved with the proposed method is consistent with the OOMAO value of $r_0$ used to generate the PSF. The high correlation coefficient $R^2 = 0.992$ demonstrates that the proposed method could be reliable for estimating the atmospheric Fried parameter from a long-exposure PSF.

## 4.4 Estimating a real PSF

In this section, the accuracy of the proposed methods is evaluated using an on-sky point spread function (PSF) obtained from the MUSE instrument on the Very Large Telescope (VLT) (Bacon et al. 2006, 2010). This PSF was acquired as part of the MUSE calibration programme, which observes standard stars for photometric calibration at the beginning of each night. The specific observation date for the PSF used in this work for the object HD146233 was 2019 July 8th. The

results obtained from the proposed methods were compared with those derived from the Fetick method, focusing on the root mean square error (RMSE).

Fig. 5 shows the MUSE PSF recorded on 2019 August 8, at a wavelength of 500 nm with an exposure time of 1 s.

Fig. 6(a) illustrates the observed MUSE PSF estimation using GSA described in approach (i). The solid blue line represents the MUSE PSF, and the dotted red line represents the best estimated PSF of $n = n_{MC} = 20$ MC simulations. (b) illustrates the observed PSF estimation using the approach (ii). The solid blue line represents the observed MUSE PSF, and the dotted red line represents the best estimated PSF of $n = n_{MC} = 20$ MC simulations. (c) illustrates the observed PSF estimation using the Fetick method. The solid blue line represents the observed MUSE PSF, and the dotted red line represents the estimated PSF using the Fetick method.

The results presented in Fig. 6 demonstrate that the estimation of the observed MUSE PSF using the proposed methods, with RMSE values of {6.50 × 10⁻⁶, 1.839 × 10⁻⁵}, exhibits better performance.
formance compared to the Fetick method, which has an RMSE value of {1.84 × 10⁻⁵}. The results also indicate that estimating the observed MUSE PSF from the atmospheric PSD using approach (ii) and the Fetick method yields comparable performance, with the proposed method achieving slightly better RMSE results. The best performance was achieved using approach (i), which employs a GSA to estimate the MUSE PSF. It is important to note that the proposed method does not require prior knowledge of the atmospheric conditions for real data sets. These findings demonstrate that the proposed methods provide accurate PSF estimations, even in the presence of read-out noise, for telescope optics with central obstructions and spiders, such as those found in the Very Large Telescope (VLT). Furthermore, the proposed methods are effective in fitting complex PSF shapes, including those influenced by spiders, static aberrations, or diffraction patterns.

## 5 CONCLUSIONS

This work presents a novel parametric model for an AO-corrected PSF using a Gaussian sum approximation (GSA) and introduces a new estimation method. To evaluate the performance of the proposed model and method, two approaches were analyzed for estimating the atmospheric PSF from long-exposure observed PSF data: (i) atmospheric PSF estimation using GSA and (ii) atmospheric PSF estimation based on the estimation of the atmospheric PSD using GSA.

Both approaches incorporate prior knowledge of the telescope, including the AO cutoff frequency and the theoretical telescope PSF, which accounts for factors such as the obstruction ratio, spider shapes, and static aberrations. The accuracy of these approaches was assessed in terms of RMSE using a set of OOMAO-generated PSFs and an on-sky PSF from the MUSE instrument on the VLT. The results demonstrate that the proposed model and estimation method are effective in accurately estimating the atmospheric PSF, even for telescope optics with a central obstruction and spiders, as exemplified by the VLT.

The RMSE analysis indicates that the first approach delivers the best overall performance. However, the second approach, which can also estimate the atmospheric Fried parameter $r_0$, shows slightly better performance than the Fetick method. These findings suggest that the proposed model and estimation method could be valuable tools for image post-processing techniques, such as the deconvolution of long-exposure images. Future work will focus on validating the model and estimation method for complex PSF shapes, including those with diffraction pattern effects, as seen in telescopes like the Large Binocular Telescope (LBT), the Giant Magellan Telescope (GMT), and the Extremely Large Telescope (ELT).

## ACKNOWLEDGEMENTS

This work is supported by ANID / Fondo QUIMAL 2024 / 240012, ANID-Basal Project AFB240002 (AC3E), ANID-FONDECYT1181158, 11201187, 1211630, 3220403, and 3230398. RO's work was partially supported by DICYT Regular 0624130P, Universidad de Santiago de Chile.

## DATA AVAILABILITY

The data underlying this article will be shared on reasonable request to the corresponding author.

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