11institutetext: Instituto de Astrofísica de Canarias, C/ Vía Láctea s/n,
E-38205 La Laguna, Tenerife, Spain
11email<EMAIL_ADDRESS>22institutetext: Departamento de Astrofísica,
Universidad de La Laguna, E-38200 La Laguna, Tenerife, Spain

# Integral field spectroscopy supports atmospheric optics to reveal the finite
outer scale of the turbulence

B. García-Lorenzo  1122 D. Esparza-Arredondo  1122 J.A. Acosta-Pulido  1122
J.A. Castro-Almazán 1122

(Received October 24, 2023; accepted , 2024)

###### Abstract

Context. The spatial coherence wavefront outer scale ($\mathcal{L}_{0}$ )
characterizes the size of the largest turbulence eddies in Earth’s atmosphere,
determining low spatial frequency perturbations in the wavefront of the light
captured by ground-based telescopes. Advances in adaptive optics (AO)
techniques designed to compensate for atmospheric turbulence emphasize the
crucial role of this parameter for the next generation of large telescopes.

Aims. The motivation of this work is to introduce a novel technique for
estimating $\mathcal{L}_{0}$ from seeing-limited integral field spectroscopic
(IFS) data. This approach is based on the impact of a finite $\mathcal{L}_{0}$
on the light collected by the pupil entrance of a ground-based telescope.

Methods. We take advantage of the homogeneity of IFS observations to generate
band filter images spanning a wide wavelength range, enabling the assessment
of image quality (IQ) at the telescope’s focal plane. Comparing the measured
wavelength-dependent IQ variation with predictions from Tokovinin (2002)
analytical approach offers valuable insights into the prevailing
$\mathcal{L}_{0}$ parameter during the observations. We applied the proposed
technique to observations from the Multi Unit Spectroscopic Explorer (MUSE) in
the Wide Field Mode obtained at the Paranal Observatory.

Results. Our analysis successfully validates Tokovinin’s analytical
expression, which combines the seeing ($\epsilon_{0}$) and the
$\mathcal{L}_{0}$ parameters, to predict the IQ variations with the wavelength
in ground-based astronomical data. However, we observed some discrepancies
between the measured and predictions of the IQ that are analyzed in terms of
uncertainties in the estimated $\epsilon_{0}$ and dome-induced turbulence
contributions.

Conclusions. This work constitutes the empirical validation of the analytical
expression for estimating IQ at the focal plane of ground-based telescopes
under specific $\epsilon_{0}$ and finite $\mathcal{L}_{0}$ conditions.
Additionally, we provide a simple methodology to characterize the
$\mathcal{L}_{0}$ and dome-seeing ($\epsilon_{\mathrm{dome}}$) as by-products
of IFS observations routinely conducted at major ground-based astronomical
observatories.

###### Key Words.:

Atmospheric effects – Telescopes – Instrumentation: high angular resolution –
Techniques: imaging spectroscopy – Site testing

##  1 Introduction

Earth’s atmospheric turbulence significantly affects light propagation through
it, distorting both intensities and wavefronts. Consequently, ground-based
astronomical observations often produce blurred images compared to those
obtained from space using identical telescopes and instruments.

Atmospheric turbulence results from stochastic fluctuations in the refractive
index attributed to temperature variations. The strength of this turbulence
can be quantified using the refractive index structure parameter
(C${}_{n}^{2}$), which is a function of the position. The Kolmogorov model
provides a satisfactory description of the statistical properties of
atmospheric turbulence, assuming that it is homogeneous and isotropic (e.g.,
Roddier, 1981). This model pictures the turbulence as a cascade of energy,
following a power-law distribution, from large- to small-scale turbulence
eddies until dissipation. The Kolmogorov model applies only to the inertial
range between inner ($\it{l}_{0}$) and outer ($\it{L}_{0}$) scales determined
by the smallest and largest sizes of turbulence eddies. This model holds for
any atmospheric turbulence layer that light passes through on its way to the
pupil entrance of ground-based telescopes, each characterized by a
C${}_{n}^{2}$ and an $\it{L}_{0}$ depending on the local conditions at each
layer height (h). Hence, the wavefront of the light reaching the telescope
pupil will suffer perturbations from the distinct atmospheric turbulence
layers. We can define the spatial coherence wavefront outer scale
($\mathcal{L}_{0}$ ) as an equivalent outer scale determining the image
quality (IQ) of any observation taken with that telescope (Borgnino, 1990).
$\mathcal{L}_{0}$ is related to the $\it{L}_{0}$ of the atmospheric turbulence
layers as follows (e.g., Abahamid et al., 2004):

$\mathcal{L}_{0}=\left(\frac{\int\it{L}_{0}(h)^{-1/3}C_{N}^{2}(h)dh}{\int
C_{N}^{2}(h)dh}\right)^{-3}$ (1)

The $\mathcal{L}_{0}$ is a parameter independent of the wavelength of the
observed light, representing a size and typically measured in meters. In many
applications, equations are simplified by assuming infinite outer scales.
Under this assumption, the full width at half maximum (FWHM) of the point
spread function (PSF) in a long-exposure (LE) image, captured with a ground-
based telescope limited by atmospheric turbulence, commonly referred to as the
seeing-limited IQ ($\epsilon_{LE}$), can be expressed in terms of the strength
of atmospheric turbulence as follows:

$\mathrm{\epsilon_{LE}(\lambda)\approx\epsilon_{0}}(\lambda)=\frac{0.976\lambda}{r_{0}(\lambda)}$
(2)

where $\epsilon_{0}$ is the seeing, $\lambda$ is the wavelength, and r0 is the
Fried parameter, defined in terms of the refractive index structure constant
profile with height (C${}_{N}^{2}$(h)) as (Fried, 1965):

$\mathrm{r{{}_{0}}}(\lambda)=0.185\lambda^{6/5}(\cos{\zeta})^{3/5}\left(\int
C_{N}^{2}(h)dh\right)^{-3/5}$ (3)

with $\zeta$ being the observing zenith angle. We note that Eq. 3 implies
that:

$\mathrm{\epsilon_{LE}(\lambda)\approx\epsilon_{0}}(\lambda)\propto\lambda^{-1/5}$
(4)

However, interferometric measurements and the growing size of telescope pupils
evidence the limitations of applying the Kolmogorov model for large distances
(i.e., $\mathcal{L}_{0}$ $\approx\infty$), showing deviations from Eq. 4 that
reveal the finite nature of the $\mathcal{L}_{0}$ . Hence, with the increasing
diameter of ground-based telescopes and adaptive optics (AO) developments, the
relevance of $\mathcal{L}_{0}$ has increased in the last few years (e.g.,
Fusco et al., 2020). For example, the size of $\mathcal{L}_{0}$ strongly
impacts the low-order modes in AO applications, showing a significant tilt
attenuation even for pupil sizes much smaller than $\mathcal{L}_{0}$ . The
validity of the Kolmogorov theory beyond the inertial range can be assessed
using empirical models. These models describe the spectrum of the refractive-
index fluctuations by means of the $\mathcal{L}_{0}$ parameter based on
general physical considerations (e.g., Voitsekhovich, 1995). A widely used is
the von Kármán model, where the $\mathcal{L}_{0}$ corresponds to the distance
at which the structure-function of the phase fluctuations saturates (see e.g.,
Fig. 2 in Voitsekhovich, 1995).

Different techniques enable the determination of the magnitude of
$\mathcal{L}_{0}$ either directly (e.g., GSM, Ziad et al., 2000) or indirectly
(e.g., from AO telemetry and applying Eq. 1, Guesalaga et al., 2017), and both
provide comparable results (see Ziad, 2016, for a review). At a telescope
focal plane, the effect of a finite $\mathcal{L}_{0}$ is to smooth low spatial
frequency perturbations, resulting in a reduction of image motion compared to
the expected for an infinite $\mathcal{L}_{0}$ (e.g., Winker, 1991). As a
result, the actual IQ improves over the $\epsilon_{0}$ by a factor that
depends on both the $\mathcal{L}_{0}$ and the $\epsilon_{0}$. Tokovinin (2002)
proposed the following first-order approximation to $\epsilon_{LE}$:

$\epsilon_{LE}(\lambda)\approx\epsilon_{0}(\lambda)\sqrt{1-2.183\left(\frac{\mathrm{r_{0}}(\lambda)}{\mathcal{L}_{0}}\right)^{0.356}}$
(5)

This approach results from an analytical simplification of the turbulence
statistics using the von Kármán model, and it seems to work with an
uncertainty of about 1% for $\mathcal{L}_{0}$/r0 ¿ 20. Equation 5 was refined
later to account for the pupil size relative to $\mathcal{L}_{0}$ , as follows
(Sarazin, 2017).:

$\epsilon_{LE}(\lambda,\mathcal{D})\approx\epsilon_{0}(\lambda)\sqrt{1-2.183\left(1-\frac{1}{1+300\times\frac{\mathcal{D}}{\mathcal{L}_{0}}}\right)\left(\frac{\mathrm{r_{0}}(\lambda)}{\mathcal{L}_{0}}\right)^{0.356}}$
(6)

where $\mathcal{D}$ is the telescope pupil diameter. Previous studies have
adopted these analytical expressions to analyze the deviation of IQ from
$\epsilon_{0}$ using numerical simulations (Martinez et al., 2010) and through
simultaneous visible and infrared images taken with different instruments
(Tokovinin et al., 2007). Moreover, Floyd et al. (2010) derived an average
turbulence $\mathcal{L}_{0}$ of 25 m for Las Campanas observatory by applying
Eq. 5 to a large series of IQ measurements from band filter images of stars.
Currently, first-class astronomical observatories commonly use Eq. 5 (or 6,
depending on the telescope) to predict the IQ of their queue observations
using the $\epsilon_{0}$ provided by Differential Image Motion Monitors
(DIMMs, Sarazin & Roddier, 1990), and assuming a typical $\mathcal{L}_{0}$ for
the site. However, to our knowledge, the actual $\epsilon_{LE}$ behavior with
wavelength has not been verified empirically to follow such analytical
approaches.

In this work, we take advantage of integral field spectroscopic observations
limited by atmospheric turbulence to track the wavelength behavior of the IQ
at the telescope focal plane. We compare our findings with the analytical
model predictions proposed by Tokovinin (2002) to determine the prevailing
$\mathcal{L}_{0}$ during observations. We applied the methodology to data from
the Multi Unit Spectroscopic Explorer
(MUSE111https://www.eso.org/sci/facilities/develop/instruments/muse.html) at
the Paranal Observatory, empirically validating the analytical model
predictions with wavelength. Thanks to the spectral range covered by MUSE, our
analysis reveals the prevailing $\epsilon_{0}$ and $\mathcal{L}_{0}$
parameters best matching the IQ measurements. We explore different conditions,
including those influenced by dome-induced turbulence.

The structure of this article is as follows. In Sect. §2, we discuss the
factors influencing IQ at the focal plane of a ground-based telescope, with
special attention to dome-induced turbulence. Section §3 introduces integral
field spectroscopy (IFS) as a valuable technique for assessing the
$\mathcal{L}_{0}$ during observations. In Sect. §4, we apply the proposed
methodology to data cubes observed with MUSE, analyzing different scenarios.
Finally, Sect. §5 summarizes the main findings and conclusions.

##  2 Telescopes and turbulence contributors

The IQ at the focal plane of an instrument attached to a ground-based
telescope results from various factors, including the telescope/instrument
optics and stability, and the air turbulence conditions during observations.
Through this work, we use the FWHM of the PSF to quantify the IQ, although
other metrics might also be employed, such as the Strehl ratio in AO
applications. The design and operation of modern telescopes try to minimize
optical and stability aspects, while advanced AO systems primarily compensate
for atmospheric turbulence and small optical aberrations. However, air
turbulence will depend on environmental conditions both inside and outside of
the telescope enclosure. We identify the external turbulence to the
atmospheric turbulence, assuming Eqs. 5 and 6 (see Appendix A for a comparison
of predictions from these equations) describe its effect on IQ. The influence
of turbulence within the telescope enclosure is complex and challenging to
parameterize. We explore it further in the following subsection. Hereafter, we
assume an optimal optical design of the telescope+instrument system providing
negligible image degradation in comparison to air turbulence.

###  2.1 The dome-seeing

Telescope enclosures protect from external factors such as adverse weather
conditions, but they can also induce turbulence that degrades the quality of
the images, the so-called dome-seeing ($\epsilon_{\mathrm{dome}}$). While
$\epsilon_{\mathrm{dome}}$ is recognized as a variable source of IQ
degradation, only recent efforts actively promote its monitoring, even aiming
for real-time quantification (Lai et al., 2019; Stubbs, 2021; Osborn & Alaluf,
2023). However, there remains a limited quantitative understanding of
$\epsilon_{\mathrm{dome}}$, particularly concerning its wavelength behavior.

The non-Kolmogorov regime of the dome turbulence combines thermal inversion
and convection, and its strength will depend on factors such as the dome
design, the inside-outside temperature gradient, the temperature of the
primary mirror surface, the outside wind, the orientation of the dome window,
etc (Woolf, 1979; Henry et al., 1987; Munro et al., 2023). We can define
$\tilde{r}_{0}$ as an equivalent parameter to Fried’s parameter, which will be
a power law of the wavelength and the non-Kolmogorov refractive index
structure constant ($\tilde{C}_{N}^{2}$) for the dome turbulence:

$\mathrm{\tilde{r}{{}_{0}}}(\lambda)=\left(\frac{16.7}{\cos{\zeta}\times\lambda^{2}}\int\tilde{C}_{N}^{2}dh\right)^{1/(2-\gamma)}$
(7)

with the power law index in the range $3<\gamma<4$ (Toselli et al., 2007). We
note that Eq. 7 reduces to the Kolmogorov case (i.e., Eq. 3) when
$\gamma=\frac{11}{3}\approx 3.67$. Assuming that the
$\epsilon_{\mathrm{dome}}$ varies in proportion to the relationship between
$\epsilon_{0}$ and r0 in Eq. 2, and taking into account Eq. 7:

$\epsilon_{\mathrm{dome}}(\lambda)\propto\frac{\lambda}{\mathrm{\tilde{r}{{}_{0}}}(\lambda)}\
\stackrel{{\scriptstyle(\ref{fried_general})}}{{\implies}}\epsilon_{\mathrm{dome}}(\lambda)\propto\lambda^{(\gamma-4)/(\gamma-2)}$
(8)

Considering the dome’s contribution to turbulence as an additional atmospheric
layer, we can linearly combine the dome-induced turbulence using the
corresponding refractive index structure constants. Nevertheless, the
resulting image blur from dome-turbulence could only be added to atmospheric
$\epsilon_{0}$ using a 5/3-power law within the constraints of the Kolmogorov
regime (Woolf, 1979; Racine et al., 1991). However, the effect of
$\epsilon_{\mathrm{dome}}$ on the focal plane is typically represented as a
Gaussian blur (e.g., Bustos & Tokovinin, 2018), making a quadratic
contribution to the overall IQ:

$[\mathrm{IQ}(\lambda)]^{2}=[\epsilon_{\mathrm{LE}}(\lambda)]^{2}+[\epsilon_{\mathrm{dome}}(\lambda)]^{2}$
(9)

We note that we have considered a negligible degradation of the IQ due to
instrumental factors, adopting such a quadratic approach hereafter. Figure 1
presents examples of the quadratic contribution to IQ for the extreme values
of the power law $\gamma$-index of the non-Kolmogorov dome-induced turbulence,
assuming a $\epsilon_{\mathrm{dome}}$ of 0.2 arcsecs at 5000 Å. We also
include the $\gamma=\frac{11}{3}$ corresponding to the Kolmogorov regime. The
dome contribution remains constant with wavelength as the $\gamma$ index
approaches 4, whereas it becomes inversely proportional to wavelength as
$\gamma$ approaches 3. As shown in Fig. 1, the IQ will consistently exceed
that associated with atmospheric turbulence at any wavelength. Furthermore,
its wavelength dependence deviates from the natural atmospheric turbulence
behavior (described by Eqs. 5 and 6), which is particularly noticeable for the
smallest $\gamma$ value.

Figure 1: Estimation of the IQ at the focal plane of an 8-m telescope as a
function of the wavelength under a $\epsilon_{0}$ of 1.0 (left) and 0.7
(right) arcsecs at 5000 Å, considering four different wavelength behaviors of
the dome-turbulence (blue lines) contribution: (1)
$\epsilon_{\mathrm{dome}}$=0 (solid green line), and (2, 3, and 4)
$\epsilon_{\mathrm{dome}}$=0.2 arcsecs at 5000 Å, with a power-law wavelength
dependence of index $\gamma$=3 (2, dotted lines), $\gamma$=11/3 (3, dashed
lines), and $\gamma$=4 (4, dotted-dashed lines). We assumed a typical
$\mathcal{L}_{0}$ value of 22 m. The black solid curve follows the
$\epsilon_{0}$ wavelength dependence ($\epsilon_{0}\propto\lambda^{-1/5}$).
The upper panels depict the general spectral range covered by instruments in
ground-based optical-infrared observatories. The lower panels specifically
showcase the spectral range of the MUSE integral field spectrograph, the
selected instrument for implementing the proposed methodology on real data.

Large telescopes invest significant efforts to mitigate the occurrence of
dome-induced turbulence (i.e., $\epsilon_{\mathrm{dome}}\approx 0$), thereby
ensuring seeing-limited conditions, and consequently, IQ
$\approx\epsilon_{\mathrm{LE}}$. Then, we can reasonably assume negligible or
minimal $\epsilon_{\mathrm{dome}}$ contributions at any wavelength for large
telescopes. Nonetheless, the monitoring of $\epsilon_{\mathrm{dome}}$ at
distinct wavelengths will largely benefit the characterization of the actual
IQ at the focal plane.

##  3 Using Integral Field Spectroscopy to reveal $\mathcal{L}_{0}$

IFS is a powerful technique that allows for the simultaneous recording of
spatial and spectral information over a field. Since its first applications in
astronomy in the 1980s and 1990s, IFS rapidly grew in popularity during the
first decade of the 21st century. IFS has become a widely used technique in
both ground-based and space-borne astronomy enabling many scientific programs,
from studies in our solar system to high-redshifted objects of cosmological
interest. Some of the most advanced IFS instruments in operation at the
largest ground-based telescopes include the MUSE spectrograph on the Very
Large Telescope (VLT) in Chile, the Gemini Multi-Object Spectrograph
(GMOS222http://www.gemini.edu/instrumentation/gmos/) on the Gemini Observatory
in Chile and Hawaii, the Keck Cosmic Web Imager
(KCWI333https://www2.keck.hawaii.edu/inst/kcwi/) on the Keck Observatory in
Hawaii, and the Multi-Espectrógrafo en GTC de Alta Resolución para Astronomía
(MEGARA444http://www.gtc.iac.es/instruments/megara/megara.php) on the Gran
Telescopio Canarias on the Roque de los Muchachos Observatory in La Palma
island (Spain).

The basic instrumental concept behind IFS is to spatially split the telescope
focal plane into several sub-apertures, rebinning them to create a pseudo-slit
to feed on long-slit spectrographs. Different instrumental approaches allow
achieving IFS (see e.g., Fig. 1 in Allington-Smith, 2006), all of them
resulting in obtaining the flux density (F) over a sky area in three
dimensions (3D): right ascension ($\alpha$), declination ($\delta$), and
wavelength ($\lambda$) (i.e., F($\alpha$,$\delta$,$\lambda$)), under the same
instrumental and atmospheric conditions. For this work, the
F($\alpha$,$\delta$,$\lambda$) data cube should be viewed as a collection of
narrow-band filter images at different wavelengths obtained simultaneously,
ensuring data homogeneity (see e.g., Fig. 2.1 in Harrison, 2016). This set of
images enables us to empirically assess the variation of IQ with wavelength by
characterizing the PSF using any point-like source in the observed field of
view (for the initial exploration of this concept see García-Lorenzo et al.,
2023). Fitting the $\epsilon_{LE}$($\lambda$) to the curve described by Eq. 5
(or 6 when applicable) will yield the $\mathcal{L}_{0}$ and $\epsilon_{0}$
parameters during observations.

Hence, we recognize IFS instruments operating under atmospheric turbulence-
limited conditions as valuable tools for validating the first-order analytical
approximation to estimate $\epsilon_{LE}$($\lambda$) and for assessing the
prevailing $\mathcal{L}_{0}$ during IFS observations. For that, low-spectral
resolution IFS covering a broad spectral range will be preferable to a high-
spectral resolution IFS sampling a short wavelength range.

The F($\alpha$,$\delta$,$\lambda$) data cube will be characterized by a
certain number of spatial and spectral pixels (spaxels) of $\Delta\alpha$,
$\Delta\delta$, and $\Delta\lambda$ sizes. $\Delta\alpha$ and $\Delta\delta$
should achieve Nyquist or better sampling for the typical FWHM of the
atmospheric turbulence-limited PSF at the observing site (i.e.,
$\epsilon_{LE}$($\lambda$) ¿ 2$\Delta\alpha$ & 2$\Delta\delta$). Along the
spectral direction, the critical point will be the detection of point-like
sources with a signal-to-noise ratio (S/N) larger than a threshold to
characterize the PSF properly. To achieve a minimum S/N threshold, IFS enables
the easy recovery of band-filter images (I${}_{\mathrm{BW}_{j}}$) by summing N
consecutive slices of the data cube within a chosen wavelength range, as
follows:

$I_{{\mathrm{BW}j}}(\alpha,\delta)=\sum_{i-N}^{i+N}F(\alpha,\delta,\lambda_{i}),\hskip
4.26773pt\\\ |\lambda_{i}-\lambda_{j}|\leq\frac{N\Delta\lambda}{2}$ (10)

where $\lambda_{j}$ and BWj=N$\Delta\lambda$ are the central wavelength and
width of the desired filter, respectively. However, the final IQ of the
recovered I${}_{{\mathrm{WB}_{j}}}$ image may differ from the actual IQ of the
particular slide image F($\alpha,\delta,\lambda_{j}$) due to instrumental
behavior, the observed target characteristics, or a combination of both
factors (see Sect. §B.1 in appendix B). It is important to note that we assume
a filter with constant transmission within the selected wavelength band. In
any different case, Eq. 10 should be multiplied by the specific transmission
function. The characteristics of the point-like source will determine the
maximum number and wavelength distribution of IBW. In some cases, the number
of bands can equal the total spectral channels in the IFS data cube, such as
when a bright star allows for characterizing the PSF at each channel in the
IFS field of view. Conversely, the number of bands can be few (or even just
one), as in IFS observations of active galaxies, where the PSF can be imaged
in isolation from the host galaxy using specific spectral lines unique to the
point-like active nucleus (e.g., Husemann et al., 2016; Esparza-Arredondo et
al., 2024).

Finally, to determine the optimal $\mathcal{L}_{0}$ and $\epsilon_{0}$ values
that best fit the IQ($\lambda$) measurements, we might fit the curve described
by Eq. 5 (or 6 when applicable). Different fitting approaches may be
considered based on the intended use of the derived parameters and the
specific characteristics of the measurements, including the number of
empirical data points. It is recommended to experiment with various methods
and assess the fit quality using appropriate metrics, such as the coefficient
of determination or residual analysis.

While a global fitting algorithm considering all IQ measurements at once is
the most suitable and elegant procedure, in this study we adopt a
straightforward fitting approach due to its versatility in accommodating
different scenarios (see Sect §4), ranging from a low to a high number of
measurements, as the two cases mentioned above.

##  4 Application to real IFS observations

To apply the proposed methodology to actual data, we identified the MUSE
instrument (Bacon et al., 2010) installed on the Nasmyth B platform of the
European Southern Observatory (ESO) VLT Unit Four (UT4) at the Paranal
Observatory555https://www.eso.org/public/spain/teles-instr/paranal-
observatory/. MUSE is an integral field spectrograph providing spectral
information in the range of 4800 to 9300 Å with a mean resolution of 3000. In
the seeing-limited mode (Wide-field mode), MUSE samples a one-squared arcmin
field of view with 0.2 arcsecs spatial pixels. The VLT UT4 has a
$\sim$30-m-high dome, and the observing floor is $\sim$10 m above ground
level666https://www.eso.org/sci/facilities/paranal/telescopes/ut.html. At
about 100 m to the North of the UT4 building, a robotic DIMM monitors the
turbulence conditions at Paranal on top of a 7-m-high tower (Sarazin, 2017).
Due to the difference in altitude between the DIMM tower and the Nasmyth VLT
platform, we expect a slightly better $\epsilon_{0}$ at the MUSE focal station
($\Delta\epsilon_{\mathrm{h}}$), with a projected improvement of about
$\Delta\epsilon_{\mathrm{h}}\sim$0.05 arcsecs compared to the DIMM
measurements (see Figure 15 in Butterley et al., 2020), assuming a negligible
dome turbulence contribution.

Table 1: Selection criteria used in the ESO Science Portal for identifying processed data cubes from IFS observations. Instrument, Mode | MUSE, WFM-NOAO
---|---
Sensitivity (AB mag) |  $\geq$ 19
Date | 1 Jan 2017 - 31 May 2023
Total exposure time (s) | 100-200
Sky resolution (arcsec) |  $\leq$ 1
Table 2: Parameter for the MUSE processed data cubes downloaded from the ESO
Science Portal.

ID | PI | Field | Obs. date | UTC | IT (s) | $T_{mm}$ | $T_{in-en}$ | $T_{out-en}$ | Air mass | $\epsilon_{DIMM}$
---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
0102.C-0589(A) | Vogt, F. | HD90177a | 29 Dec 2018 | 05:58:47 | 180 | 12.38 | 14.88 | 12.70 | 1.40 | 0.47$\pm$0.10
0102.C-0589(A) | Vogt, F. | HD90177b | 29 Dec 2018 | 06:40:43 | 180 | 12.18 | 14.83 | 12.48 | 1.31 | 0.39$\pm$0.05
0102.C-0589(A) | Vogt, F. | HD90177c | 29 Dec 2018 | 07:23:27 | 180 | 12.08 | 14.75 | 12.37 | 1.25 | 0.51$\pm$0.04
0102.C-0589(A) | Vogt, F. | HD90177d | 29 Dec 2018 | 08:04:58 | 105 | 11.98 | 14.66 | 12.22 | 1.23 | 0.51$\pm$0.02
106.21GS.001 | Ivanov, V. | OGLEII | 20 Feb 2021 | 08:54:08 | 140 | 10.78 | 13.18 | 11.62 | 1.56 | 0.41$\pm$0.02

777 (1) ESO program and run identification code, (2) principal investigator,
(3) observed field/object, (4) observing date (day month year), (5) date time
(UTC), (6) integration time (s), (7) main mirror temperature (in ∘C), (8)
temperature inside the enclosure (in ∘C). This temperature corresponds to the
average of the values provided for sensors at different locations, (9)
observatory ambient temperature (in ∘C), (10) average air mass of MUSE
observations, (11) average and standard deviation of DIMM $\epsilon_{0}$
measurements in arcsecs during MUSE observations. Temperatures and air mass
for each night were retrieved from the headers of the MUSE data cubes.

We used the extensive ESO science archive888http://archive.eso.org/cms.html to
collect MUSE data. This archive offers thousands of reduced seeing-limited
MUSE data cubes, ready for analysis. We note that any point-like source within
the field of view of these data cubes can be a potential candidate for
retrieving the prevailing atmospheric parameters during observations. To apply
the method described in Sect. §3 to real data, we systematically searched on
the ESO science portal for processed data using the selection criteria
outlined in Table 1 to constrain the number of potential data cubes to
analyze. This search yielded a total of 718 data cubes, of which 389 were
identified as being obtained in seeing-limited mode (labeled as WFM-NOAO in
the MUSE data cubes header). About 95.9% of them correspond to sky exposures,
and 2.8% to unfocused calibration stars or extended objects, data cubes that
are useless for the purpose of this work. Eventually, we concluded with five
(1.3%) MUSE data cubes of two fields showing a few stars (see Figs. 8 in Sect.
§B.2 of appendix B). Table 2 presents relevant parameters for the retrieved
MUSE data cubes. Observations for the HD90177 field (four data cubes) come
from the same observing night, acquired within an interval of around two and a
half hours (see Table 2).

We used the source detection tool within the _photutils Astropy_
package999https://photutils.readthedocs.io/en/stable/detection.html to
retrieve the positions of stars in the field-of-view of the selected MUSE
data. For the subsequent analysis, we focused on non-saturated bright stars
isolated from companions within a radius of at least 5 arcsecs. Then, we used
the MUSE Python Data Analysis Framework
(MPDAF101010https://mpdaf.readthedocs.io/en/latest/, Piqueras et al., 2019) to
recover band-filter images of 8 arcsec2 field-of-view centered on each
selected star. For this, we designed a set of 100 Å wide bandpass filters
uniformly distributed along the MUSE spectral axis, starting with the bluest
at 4850 Å. The resulting 44 filter-band images for each star depict the
seeing-limited MUSE PSF in the spectral coverage of those bands. We selected
the analytical elliptical Moffat profile defined in Eq. 11 to model our band-
filter images. We ensure a zero background in the Moffat profile by
subtracting any sky background level from the filter-band images before
performing the fitting.

$EM1=\frac{AM}{\left[1+A(x-x_{0})^{2}+B(y-y_{0})^{2}+C(x-x_{0})(y-y_{0})\right]^{\beta}}$
(11)

In Eq. 11, AM and $\beta$ are the amplitude and the power index of the Moffat
model centered on the star image peak at the field-of-view coordinates (x0,
y0). Furthermore, the coefficients A, B, and C are defined as follows:

$A=\left(\frac{cos\theta}{\sigma_{1}}\right)^{2}+\left(\frac{sin\theta}{\sigma_{2}}\right)^{2}\rm{;\,}$

$B=\left(\frac{sin\theta}{\sigma_{1}}\right)^{2}+\left(\frac{cos\theta}{\sigma_{2}}\right)^{2}\rm{;and}$
$C=2sin\theta
cos\theta\left(\frac{1}{\sigma_{1}^{2}}-\frac{1}{\sigma_{2}^{2}}\right)$

where $\theta$ defines the orientation of the Moffat model with respect to the
image axes, while $\sigma_{1}$ and $\sigma_{2}$ represent the characteristic
widths of the profile along the major and minor axes, respectively, defining
the ellipticity of the Moffat model. The Moffat function serves as a general
model encompassing the Gaussian and Lorentzian functions, approaching each
depending on the values of its parameters (e.g., Fétick et al., 2019; Trujillo
et al., 2001). Specifically, for large values of $\beta$, the Moffat function
converges to a Gaussian profile, while for values close to unity, it tends
toward a Lorentzian profile.

We used a modified version of the Moffat2D model, incorporating an elliptical
Moffat instead of a circular one. This model was introduced in the _modeling_
module within the _Astropy_ package, enabling the determination of optimal
parameters for fitting the Moffat model to the band-filter images recovered
from MUSE data cubes. The FWHM of the PSF along the major and minor axes for
each band-filter image is then determined from the Moffat parameters as
follows:

$FWHM_{n}=2\times\sigma_{n}\times\sqrt{2^{(1/\beta)}-1},\,\mathrm{with}\
n=1,2$ (12)

Previous works, based on the analysis of stars, found that a circular Moffat
profile with a power index of $\beta\approx 2.5$ effectively reproduces the
MUSE PSF across wavelengths (e.g., Bacon et al., 2015; Husser et al., 2016) in
its seeing-limited mode. For other MUSE modes, including AO compensation, the
PSF core is also approximated using a Moffat function (Fétick et al., 2019;
Fusco et al., 2020).

We explored four distinct model configurations analyzing the residuals
resulting from the subtraction of Moffat models from star images: (1)
$\sigma_{1}=\sigma_{2}$ and $\beta$ as a free parameter, (2)
$\sigma_{1}=\sigma_{2}$ with a fixed $\beta$ of 2.5, (3) both $\sigma_{1}$ and
$\sigma_{2}$, as well as $\beta$, as free parameters, and (4) $\sigma_{1}$ and
$\sigma_{2}$, as free parameters with $\beta$ fixed at 2.5 (see Esparza-
Arredondo et al., 2024). Our findings show that residual levels remain
comparable whether the $\beta$ parameter is constrained to 2.5 or not.
However, we obtained smaller residuals for an elliptical Moffat profile, even
when $\sigma_{1}-\sigma_{2}$ is significantly smaller than the spatial
sampling. Subsequently, as a reference for analyzing the wavelength-dependent
variation of MUSE IQ, we adopt the average FWHM derived from the
$\sigma_{1,2}$ parameters obtained with configuration (4) of the Moffat model.
Section §B.2 of appendix B presents the Moffat model parameters obtained for
each individual star analyzed in this study.

Since the size and wavelength variation of IQ obtained from stars at distinct
positions within the MUSE field-of-view are comparable within uncertainties,
we computed the average IQ for each MUSE field
(IQ${\mathrm{MUSE}}$($\lambda$), see Fig. 2) to compare it with the behavior
predicted by Eq. 5. The standard deviation of these averaged IQ was taken as
the uncertainty for IQ${\mathrm{MUSE}}$($\lambda$). From the ESO ambient
conditions database for
Paranal111111http://archive.eso.org/wdb/wdb/asm/dimm_paranal/form, we
retrieved the DIMM measurements recorded on the corresponding nights and
computed the DIMM average value ($\epsilon_{\mathrm{DIMM}}$) during the MUSE
observations (see Table 2). We estimated the natural $\epsilon_{0}$ affecting
the selected MUSE data cubes ($\epsilon_{\mathrm{MUSE}}$, listed in Table 3)
by compensating for the expected improvement due to the DIMM-MUSE platform
difference in height and projecting the result to the zenith angle ($\zeta$)
for the observations (Table 2), that is:

$\epsilon_{\mathrm{MUSE}}=\left(\frac{[\epsilon_{\mathrm{DIMM}}]^{5/3}-[\Delta\epsilon_{\mathrm{h}}]^{5/3}}{\cos\zeta}\right)^{3/5}$
(13)

For two of the five MUSE fields (i.e., HD90177b, and OGLEII), the measured
IQsMUSE consistently exceed $\epsilon_{0}$ at any wavelength (Fig. 2), while
for the other three fields (i.e., HD90177a, HD90177c and HD90177d), IQs are
comparable or slightly smaller across regions of the analyzed spectral range.
This scenario is consistent with either an infinite $\mathcal{L}_{0}$ with
some contribution from dome-induced turbulence or a finite $\mathcal{L}_{0}$
with a significant dome-induced blur. Previous works have reported a finite
$\mathcal{L}_{0}$ at the Paranal observatory, with a typical value of about 22
m (e.g., Ziad, 2016), supporting the second as the most plausible option.
Furthermore, the measured IQMUSE($\lambda$) follows a similar curve to the
predicted $\epsilon_{LE}$ through Eq. 5, although we observe some slope
discrepancies for the analyzed fields. Such discrepancies may suggest that the
estimated $\epsilon_{\mathrm{MUSE}}$ may not accurately represent the
atmospheric turbulence affecting MUSE observations, indicating a non-
negligible difference between the reported $\epsilon_{0}$ by the DIMM and the
actual $\epsilon_{0}$ along the MUSE observation direction. Additionally, as
discussed in Sect. §2.1, a non-negligible $\epsilon_{\mathrm{dome}}$
contribution may also contribute to these slope discrepancies.

Fortunately, the empirical IQsMUSE measurements enable us to estimate the
optimal combination of $\epsilon_{0}$ and $\mathcal{L}_{0}$ values that best
align with the observed data by fitting the analytical curve derived by
Tokovinin (2002). Various fitting strategies may be employed for this purpose.
However, the choice of a fitting method for implementing the proposed method
to a specific instrument should be made after analyzing each particular case.
The selected fitting method should align with the nature of the measurements,
considering factors such as the number of empirical data points, sampling,
spectral range, uncertainties, etc. It should also align with the intended use
of the derived parameters. In this study, we adopt a straightforward numerical
approach, varying parameters to minimize residuals with the observed IQsMUSE.

In particular, we used an iterative process involving variations of the input
$\epsilon_{0}$ within the analytical approach (i.e., Eq. 5) across a range of
possible values for the $\epsilon_{0}$ along the MUSE observing direction. For
our analysis, we explored $\epsilon_{0}$ values from
$\epsilon_{\mathrm{MUSE}}-0.2$ to $\epsilon_{\mathrm{MUSE}}+0.5$ arcsecs, with
0.01 arcsecs steps. For each specific $\epsilon_{0}$ within this range, we
determined the unknown parameter $\mathcal{L}_{0}$ by computing
$\mathcal{L}_{0i}$ for each $\lambda_{i}$ to match the measured
IQMUSE($\lambda_{i}$) and the predicted $\epsilon_{LE}$($\lambda_{i}$). The
best-fit $\mathcal{L}_{0}$ value and its uncertainty are obtained as the
average and standard deviation of these $\mathcal{L}_{0i}$ values. To account
for the uncertainties in IQMUSE($\lambda_{i}$), we perform a Monte Carlo (MC)
simulation randomly varying IQMUSE($\lambda_{i}$) within their uncertainty
range many times (a thousand times for this work). The average and standard
deviation of all these MC realizations were adopted as the observed
$\mathcal{L}_{0}$ ($\mathcal{L}_{0}^{b}$) and its uncertainty for that
specific $\epsilon_{0}$. After considering all combinations of $\epsilon_{0}$
and $\mathcal{L}_{0}^{b}$ , we adopted the best-fit parameters (i.e.,
$\epsilon_{0}^{b}$ and $\mathcal{L}_{0}^{b}$) as those yielding the minimal
residual with empirical IQ measurements.

Before analyzing the empirical measurement of IQMUSE to retrieve the
combination of $\epsilon_{0}$ and $\mathcal{L}_{0}$ that best matches the
observations, we may consider a potential blur contribution from turbulence
inside the telescope dome or assume 100% atmospheric seeing-limited
conditions, implying negligible dome-induced turbulence. We explore these two
scenarios for the analyzed MUSE observed fields in the following subsections.

Figure 2: Wavelength dependence of IQ at the MUSE instrument’s focal plane
(IQMUSE), analyzed using point-like sources in the MUSE fields indicated in
each panel (see Table 2). Color dots represent IQMUSE values derived from
Moffat model fits for 44 band-filter images obtained from distinct stars
within each MUSE field (individual plots available in Sect. §B.2 of appendix
B). Black open squares with error bars denote the average and standard
deviation of measured IQMUSE across all stars at each wavelength. The black-
dashed line shows the $\lambda^{-1/5}$ behavior of the $\epsilon_{0}$ along
the MUSE observation direction ($\epsilon_{\mathrm{MUSE}}$ in arcsecs, top-
left corner), derived from average DIMM values during MUSE observations (see
Table 2) using Eq. 13. The black-solid and purple-solid lines represent the
$\epsilon_{0}$ and $\epsilon_{LE}$ best-fitting the measurements obtained for
the $\epsilon_{0}^{b}$ (in arcsecs) and $\mathcal{L}_{0}^{b}$ (in meters)
parameters in the bottom-left corner of each plot (details in text).

###  4.1 Analysis assuming negligible dome-induced turbulence

Under the assumption of a negligible dome turbulence contribution to the
observed IQ, we seek to determine the optimal combination of $\epsilon_{0}$
and $\mathcal{L}_{0}$ predicting the best matching with the IQMUSE
measurements. To achieve this, we used an iterative process involving
variations of the input $\epsilon_{0}$ within the Eq. 5 analytical approach,
following the methodology described in the preceding section. Finally, we
adopt the parameters providing the minimal residuals between the predicted and
measured IQ curves ($\epsilon_{0}^{b}$ and $\mathcal{L}_{0}^{b}$ in Fig. 2).

The derived $\epsilon_{0}^{b}$ and $\mathcal{L}_{0}^{b}$ parameters (see Fig.
2 and Table 3) for the selected MUSE fields are consistent with the
statistical distribution of these parameters at the Paranal observatory (Ziad,
2016; Otarola, 2023).We found that the obtained $\epsilon_{0}^{b}$ are on
average 0.26 arcsec larger than those estimated from DIMM measurements, with a
minimum and maximum of 0.13 and 0.40 arcsecs for HD90177d and OGLEII fields,
respectively. It is important to note that MUSE observations were obtained at
a higher airmass than typical for DIMM measurements. This, combined with the
difference in observing directions between DIMM and MUSE, may contribute to
these discrepancies. Interestingly, comparisons of $\epsilon_{0}$ measurements
from different atmospheric characterization instruments have found around 0.2
arcsecs of a discrepancy, attributed to the specific instrument locations on
the Paranal observatory plateau (e.g., Osborn et al., 2018). Consequently, the
parameters obtained using the proposed methodology for the analyzed MUSE
fields are compatible with purely seeing-limited observations, well within the
statistical behavior of atmospheric turbulence at the Paranal Observatory.

###  4.2 Analysis assuming dome turbulence contribution

As discussed in Sect. §2.1, when the contribution of dome-induced turbulence
becomes significant, it not only broadens the images but can also influence
the wavelength variation, depending on the nature of the dome’s local
turbulence. In these cases, a direct measure of $\epsilon_{\mathrm{dome}}$ at
different wavelengths is desirable to properly isolate the broadening in the
images due to the atmosphere and apply the proposed methodology in this work
to retrieve $\mathcal{L}_{0}$ .

Unfortunately, we could not find a specific analysis of the
$\epsilon_{\mathrm{dome}}$ for the VLT UT4 telescope or any direct
measurements of the dome contribution to turbulence degradation on the
selected nights. However, Racine et al. (1991) found that the principal
factors influencing the dome turbulence are the primary mirror and the
enclosure temperatures, with contributions described as follows:

$\epsilon_{\mathrm{mm}}\approx 0.4\times(T_{mm}-T_{in-en})^{6/5}\
\mathrm{,and}$ (14) $\epsilon_{\mathrm{en}}\approx 0.1\times(T_{in-en}-T_{out-
en})^{6/5}$ (15)

where $T_{mm}$ is the primary mirror temperature, and $T_{in-en}$ and $T_{out-
en}$ are the temperatures inside and outside the telescope enclosure,
respectively, assuming that $T_{in-en}$ is the same at any location inside the
dome. These contributions were empirically determined at an effective
wavelength of 7000 Å (Racine et al., 1991). Hereafter, we assume that the
Paranal VLT UT4 exhibits an equivalent behavior. Table 2 presents the recorded
temperatures for the observation nights of the analyzed MUSE data cubes. Table
3 shows the estimates for dome-induced turbulence contribution during MUSE
observations assuming Eqs. 14 and 15 apply to VLT UT4. On these nights, the
temperature of the main mirror of the telescope was lower than that inside the
dome, preventing any turbulence from it, that is $\epsilon_{\mathrm{mm}}$=0.
However, the air within the dome was warmer than the outside environment,
potentially leading to turbulence and contributing to the blur in the observed
data. Then, we adopt the estimated $\epsilon_{\mathrm{en}}$ as the total
$\epsilon_{\mathrm{dome}}$ contribution to the measured IQ at a wavelength of
7000 Å (i.e., $\epsilon_{\mathrm{dome}}(7000$ Å$)=\epsilon_{\mathrm{en}}$).

Equation 8 allows us to extend the $\epsilon_{\mathrm{dome}}$ approach from
7000 Å to any wavelength. Given the lack of a predefined power-law index
$\gamma$, we treated it as a free parameter in the fitting procedure. As
described in Sect. §4.1, we performed an iterative process involving the
variations of the input $\epsilon_{0}$ and the $\gamma$ power-law index.
Initially, we allowed $\epsilon_{0}$ to vary within a range spanning from
$\epsilon_{MUSE}-0.2$ to $\epsilon_{MUSE}+0.5$ arcsecs, with steps of 0.01
arcsecs. For each specific $\epsilon_{0}$ value within this range, we followed
these steps: (1) variations of the $\gamma$ parameter in the range of 3 to 4,
with steps of 0.01, to compute the dome turbulence contribution; (2) for each
$\gamma$ value, subtracting the resulting dome turbulence contribution from
the measured IQMUSE($\lambda_{i}$) at each wavelength to infer the
$\epsilon_{LE}$($\lambda_{i}$) using Eq. 9; (3) determining the corresponding
$\mathcal{L}_{0}$ from each $\epsilon_{LE}$($\lambda_{i}$), following the same
methodology as described in previous sections. Subsequently, the combination
of parameters ($\epsilon_{0}$, $\gamma$, $\mathcal{L}_{0}$ ) yielding minimal
residuals between the predicted (Eq. 5) and measured IQ curves was adopted.

As in the previous section, we assess the uncertainties in the derived
$\epsilon_{0}$, $\gamma$, and $\mathcal{L}_{0}$ parameters using a MC
approach. We randomly perturbed each IQMUSE($\lambda_{i}$) within its
uncertainty range for one thousand iterations, resulting in one thousand
different sets of ($\epsilon_{0}$, $\gamma$, $\mathcal{L}_{0}$ ) parameters.
Finally, we calculated the average and standard deviation of these sets to
determine the optimal combination that minimizes the residuals and best fits
the observed data (see Table 3).

Figure 3 presents the results of this procedure applied to each of the
selected MUSE data cubes. The obtained $\epsilon_{0}^{b}$ and
$\mathcal{L}_{0}^{b}$ through considering a $\epsilon_{\mathrm{dome}}$
contribution to the actual MUSE IQ are again consistent with their statistical
distribution at the site. Differences between $\epsilon_{0}^{b}$ and the
expected from DIMM measurements along the MUSE pointing are within the typical
discrepancies of $\epsilon_{0}$ measurements at different locations on the
Paranal plateau (e.g., Osborn et al., 2018).

We found some discrepancies in the estimated $\mathcal{L}_{0}^{b}$ for the
four MUSE data cubes obtained on the same night in an interval of a few hours
(see Table 2). Previous studies found significant variations in the
$\mathcal{L}_{0}$ parameter on time scales comparable to fluctuations in other
atmospheric parameters, such as $\epsilon_{0}$ (e.g., Ziad, 2016).

On the night with four data cubes of the same field (see Table 2), the
analysis of the IQMUSE reveals dome-induced turbulence that varies with
wavelength according to a power law, with an index larger than the Kolmogorov
regime for three of them (i.e., HD90177a, HD90177c, and HD90177d), while it is
within the Kolmogorov regime for the HD90177b data cube. The
$\epsilon_{\mathrm{dome}}$ parameters derived from these four MUSE data cubes
indicate a subtle evolution of dome-induced turbulence throughout the night.
This evolution primarily stems from changes in its power-law regime, while
remaining nearly constant in terms of intensity at the reference wavelength
(see Table 3). For the other field observed on a different night, the power
law index was smaller than the Kolmogorov $\gamma=\frac{11}{3}$ index.

The obtained $\epsilon_{0}^{b}$ and $\mathcal{L}_{0}^{b}$ parameters
considering the dome turbulence contribution also align with the statistical
distribution of these parameters at the Paranal observatory (see Fig. 2 and
Table 3).

Figure 3: Same as Fig. 2, but considering the dome turbulence contribution.
Black-open squares represent the average IQMUSE across each field, the same as
in Fig. 2. The black-filled squares show the IQ broadening attributed solely
to atmospheric turbulence (i.e., IQMUSE after quadratically subtracting the
estimated dome-turbulence contribution at each $\lambda$). The blue solid line
corresponds to the estimated dome-turbulence contribution, following a power-
law dependence on wavelength with the $\gamma$ index indicated in the bottom
right corner. The green line corresponds to the quadratic sum of the predicted
atmospheric (purple line) and dome (blue line) turbulence blur, fitting the
measured MUSE IQs. Table 3: Atmospheric parameters derived from the measured
IQ at different wavelengths from MUSE data cubes.

| DIMM-seeing | Negligible | Non-negligible
---|---|---|---
| along MUSE direction | dome-seeing | dome-seeing
MUSE Field | $\epsilon_{MUSE}$ | $\epsilon_{0}^{b}$ | $\mathcal{L}_{0}^{b}$ | $\epsilon_{dome}$ | $\epsilon_{0}^{b}$ | $\mathcal{L}_{0}^{b}$ | $\epsilon_{dome}$ | $\gamma$
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
HD90177a | 0.57 | 0.79 | 20.3$\pm$1.5 | 0.00 | 0.76 | 12.6$\pm$1.3 | 0.25 | 3.81$\pm$0.11
HD90177b | 0.45 | 0.76 | 20.5$\pm$1.6 | 0.00 | 0.69 | 16.1$\pm$1.9 | 0.28 | 3.61$\pm$0.06
HD90177c | 0.58 | 0.81 | 16.3$\pm$1.1 | 0.00 | 0.76 | 10.8$\pm$1.2 | 0.28 | 3.72$\pm$0.09
HD90177d | 0.57 | 0.70 | 28.1$\pm$2.6 | 0.00 | 0.69 | 9.3$\pm$0.9 | 0.29 | 3.97$\pm$0.04
OGLEII | 0.53 | 0.93 | 15.9$\pm$1.1 | 0.00 | 0.84 | 32.6$\pm$6.6 | 0.17 | 3.03$\pm$0.05

121212Columns correspond to: (1) MUSE observed field. (2) $\epsilon_{0}$ (in
arcsecs) along the MUSE observing direction estimated from the DIMM value
corrected from the Nasmyth platform altitude respecting DIMM and airmass. (3)
and (4) Combination of $\epsilon_{0}$ and $\mathcal{L}_{0}$ values minimizing
residuals when comparing the predicted $\epsilon_{LE}$ using Eq. 5 with these
values with the measured IQ at any wavelength, assuming a negligible dome
turbulence contribution (5). (6) and (7) Combination of $\epsilon_{0}$ and
$\mathcal{L}_{0}$ values minimizing residuals when comparing the predicted
$\epsilon_{LE}$ using Eq. 5 with the measured IQMUSE at any wavelength,
considering the contribution of the $\epsilon_{\mathrm{dome}}$. Uncertainties
for columns (3) and (6) are consistently 0.01 arcsecs, corresponding to the
$\epsilon_{0}$ steps in the iterative process, with units in arcsecs. Units
for columns (4) and (7) are meters. (8) Estimated turbulence contribution (in
arcsecs) of the dome enclosure at the reference wavelength of 7000 Å (Racine
et al., 1991). (9) Power-law index for the wavelength dependence of the
$\epsilon_{\mathrm{dome}}$.

##  5 Conclusions

This work introduces a novel approach using IFS to measure the spatial
coherence wavefront $\mathcal{L}_{0}$ prevailing during seeing-limited
observations. This method takes advantage of the capability of IFS techniques
to provide filter-band images over a wide wavelength range, all obtained under
the same atmospheric and instrumental conditions. The homogeneity of IFS data
enables the tracing of IQ variation with wavelength at the focal plane,
directly linked to the prevailing $\epsilon_{0}$ and $\mathcal{L}_{0}$
characterizing the atmospheric turbulence conditions. We applied the proposed
technique to MUSE observations obtained at the Paranal Observatory, exploring
two situations: one assuming negligible dome-induced turbulence and the other
considering the $\epsilon_{\mathrm{dome}}$ contribution. Through this
analysis, the methodology accurately derived essential atmospheric parameters
such as $\epsilon_{0}$ and $\mathcal{L}_{0}$ , while also revealing the
wavelength dependence of the dome-induced turbulence, providing valuable
insights into atmospheric conditions and instrument performance. The main
findings and conclusions of this work can be summarized as follows.

  * $\bullet$

IFS is a powerful tool for astronomical observations, enabling simultaneous
spatial and spectral datasets over wide fields and wavelength ranges.

  * $\bullet$

The homogeneity of data cubes resulting from IFS observations provides a
collection of filter-band images that allow for comprehensive analysis of IQ
variations with wavelength. The S/N of each filter-band image is crucial for
precise IQ assessment.

  * $\bullet$

IFS enables the empirical verification of the analytical approximation to IQ
variations with wavelength (Eqs. 5 and 6), derived from simplified models of
atmospheric turbulence theory. We accomplished this validation using IFS data
obtained with the MUSE instrument at the Paranal Observatory.

  * $\bullet$

The MUSE PSF is well-described by a Moffat Model with the power-law index set
to a value of 2.5 at any wavelength. We adopt the Moffat model FWHM as the
metric for assessing the MUSE IQ.

  * $\bullet$

We analyze the IQMUSE wavelength variation to estimate the $\mathcal{L}_{0}$
parameter under two different scenarios:

    1. 1.

Both the $\mathcal{L}_{0}$ and the $\epsilon_{0}$ as free parameters with
negligible dome-induced turbulence contribute to MUSE IQ.

       * –

An iterative approach led to determining the optimal $\epsilon_{0}$ and
$\mathcal{L}_{0}$ parameters ($\epsilon_{0}^{b}$ and $\mathcal{L}_{0}^{b}$)
matching the predicted and the measured IQMUSE($\lambda$) with minimal
residuals.

       * –

Both $\epsilon_{0}^{b}$ and $\mathcal{L}_{0}^{b}$ parameters derived for the
analyzed MUSE fields are in agreement with the statistical behavior of these
parameters at the Paranal Observatory, supporting that MUSE provides
predominantly seeing-limited observations.

       * –

The differences between the retrieved $\epsilon_{0}^{b}$ and the MUSE seeing
estimated from DIMM are within typical discrepancies of $\epsilon_{0}$
measurements at various locations on the Paranal plateau with different
instruments.

    2. 2.

$\mathcal{L}_{0}$ and $\epsilon_{0}$ as free parameters, and considering a
non-negligible dome-induced turbulence, estimated using the approach by Racine
et al. (1991), with a power-law dependence on wavelength and the power-law
index as a free parameter.

       * –

An iterative process involving variations in the $\epsilon_{0}$ and the power-
law index controlling the wavelength behavior of the
$\epsilon_{\mathrm{dome}}$ was performed to identify optimal
$\epsilon_{0}^{b}$, $\mathcal{L}_{0}^{b}$, and $\gamma$ parameters, minimizing
residuals between predicted and measured IQMUSE($\lambda$).

       * –

The derived $\epsilon_{0}^{b}$ and $\mathcal{L}_{0}^{b}$ parameters for this
scenario also align with the statistical turbulence behavior at the Paranal
Observatory.

       * –

Direct measurements of $\epsilon_{\mathrm{dome}}$ at various wavelengths are
crucial for accurately isolating atmospheric broadening and determining
$\mathcal{L}_{0}$ .

Seeing-limited integral field spectrographs commonly operate in ground-based
observatories. As a by-product of these observations, the $\mathcal{L}_{0}$
parameter could be routinely obtained and also facilitate the tracking of
dome-induced turbulence. A reassessment of observing strategies for IFS
observations, especially during the observation of calibration stars, may be
beneficial to implement the proposed methodology for site characterization
purposes.

###### Acknowledgements.

Based on data obtained from the ESO science archive facility with DOI(s):
https://doi.org/10.18727/archive/41. We thank the referee and David Mouillet
for their valuable feedback. The authors acknowledge support from the Spanish
Ministry of Science and Innovation through the Spanish State Research Agency
(AEI-MCINN/10.13039/501100011033) through grants ”Participation of the
Instituto de Astrofísica de Canarias in the development of HARMONI: D1 and
Delta-D1 phases with references PID2019-107010GB-100 and PID2022-140483NB-C21
and the Severo Ochoa Program 2020-2023 (CEX2019-000920-S).

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##  Appendix A Comparison of the two analytical functions approaching the
image quality

We have computed the predicted IQ as a function of wavelength considering
various telescope diameters and turbulence conditions (Fig. 4) using Eqs. 5
and 6, and tracking the atmospheric conditions where $\mathcal{L}_{0}$/r0 ¿
20. Figure 4 shows that both equations provide similar assessments for the IQ
at the focal plane of telescopes. Differences become more notable for
decreasing telescope diameter and increasing the size of the $\mathcal{L}_{0}$
parameter of the turbulence. As a general trend, it is important to note that
these approaches apply only to wavelengths within the range of 3700 to 9300 Å
for $\mathcal{L}_{0}$ $\leq$ 12 m and excellent $\epsilon_{0}$ conditions, to
maintain relative error within $\pm$1%. For typical $\epsilon_{0}$ (e.g.,
Skidmore et al. 2009; Vázquez Ramió et al. 2012) and $\mathcal{L}_{0}$ (e.g.,
Ziad 2016) at astronomical sites, the applicability without exceeding the
$\pm$1% error extends up to 15000 Å. In the case of poor atmospheric
turbulence conditions, these approaches are usable with $\pm$1% uncertainty up
to 3 $\mu$m. Figure 5 zooms in on the predicted IQ($\lambda$) for the spectral
range of the MUSE instrument, the data of which are used in this work to
illustrate the applicability of the proposed method for estimating the
$\mathcal{L}_{0}$ of turbulence from IFS observations.

Figure 6 shows the wavelength average ratio of $\epsilon_{LE}$($\lambda$) and
$\epsilon_{LE}$($\lambda$, $\mathcal{D}$) as a function of the
$\mathcal{L}_{0}$ for different $\epsilon_{0}$ conditions and telescope
diameters. The diameter becomes relevant for small telescopes (i.e.,
$\mathcal{D}$ ¡ 1.5 m), underestimating the image quality by more than
$\sim$2%, $\sim$4%, and $\sim$6% in the optical (i.e., 3700-9300 Å), 0.93-2.5
$\mu$m, and 2.5-5 $\mu$m spectral ranges, respectively. For middle to large
telescopes (4m to 10m-class telescopes), the image quality is underestimated
by a factor always ¡ 5% in the entire near-infrared range (i.e., 0.93-5
$\mu$m), being ¡2% at the optical for any turbulence condition. For extremely
large telescopes (i.e., $\mathcal{D}$ ¿ $\sim$20 m), the
$\epsilon_{LE}$($\lambda$)/$\epsilon_{LE}$($\lambda$, $\mathcal{D}$) is always
almost 1.

Figure 4: Predicted image quality for a long-exposure image (e.g.,
$\epsilon_{LE}$) of a point-like source obtained using ground-based telescopes
with diameters of (top) 1.5 m, (middle) 8 m, and (bottom) 39 m, as a function
of the wavelength. Panels correspond to different $\epsilon_{0}$ conditions at
5000 Å: (left panels) 0.4 arcsecs, (central panels), 0.7 arcsecs, and (right
panels) 1.5 arcsec. Each curve corresponds to the indicated value of the
$\mathcal{L}_{0}$ of the turbulence in meters: $\mathcal{L}_{0}$ =100
$\rightarrow$ red, $\mathcal{L}_{0}$ =32$\rightarrow$ orange,
$\mathcal{L}_{0}$ =22 $\rightarrow$ olive green, $\mathcal{L}_{0}$ =12
$\rightarrow$ purple, and $\mathcal{L}_{0}$ =5 $\rightarrow$ lime green

. The black curve follows the $\epsilon_{0}$ wavelength dependence
($\epsilon{0}(\lambda)\propto\lambda^{-1/5}$), corresponding to
$\mathcal{L}_{0}$ =$\infty$. Solid curves correspond to the first-order
approach to $\epsilon{LE}$ following Eq. 5, while dashed curves to the second-
order approach considering the telescope diameter (Eq. 6). The dotted line
delineates the $\mathcal{L}_{0}$/r0¿20 boundary, above which errors are
constrained within $\pm 1$%.

Figure 5: Same as Fig. 4, but for the parameters of the MUSE instrument, the
IFS installed at the 8-m VLT operating in the optical range between 4750 and
9350 Å. Figure 6: Wavelength average ratio of the two approaches for
estimating the image quality at the focal plane of a telescope (i.e.,
$\epsilon_{LE}$($\lambda$)/$\epsilon_{LE}$($\lambda$, $\mathcal{D}$ from Eqs.
5 and 6) as a function of the $\mathcal{L}_{0}$ of the turbulence for the
spectral ranges (top-panels) 3700-9300 Å; (middle-panels) 0.93-2.5 $\mu$m; and
(bottom-panels) 2.5-5.0 $\mu$m at excellent (left-panels), typical (central-
panels), and fair (right-panels) atmosheric turbulence conditions. We have
considered only those conditions with $\mathcal{L}_{0}$/r0¿20, ensuring errors
are within $\pm$1%. The shaded area marks the one-sigma of the mean. Colors
indicate different telescope diameters as indicated at the end-right.

##  Appendix B Filter-band images recovered from IFS data

In this appendix, we use a simple model to analyze the putative impact of the
filter-band width selected to recover filter-band images from IFS data cubes
on the expected IQ of the image. We also include the parameters derived for
the Moffat profiles fit to model the IQ of filter-band images recovered from
MUSE observations.

###  B.1 Impact of the filter-band width

We performed a simple simulation of seeing-limited IFS observations in the
optical range spanning from 3500 to 7500 Å for two point-like sources to
explore the effect of bandwidth on the measured IQ. We chose the spectra of
HD000319 (type A1V) and HD001326B (type M6 V) stars (see Fig. 7) from the
MILES stellar library (Sánchez-Blázquez et al. 2006). We selected two stars
with different spectral emissions to also assess the influence of the source
spectral behavior on the IQ measurements from filter-band images.

We generated a mock datacube for a $1\arcmin\times 1\arcmin$ field-of-view,
using spaxels with dimensions
($\Delta\alpha,\Delta\delta,\Delta\lambda$)=($0\aas@@fstack{\prime\prime}1,0\aas@@fstack{\prime\prime}1,1$Å).
We modeled the seeing-limited PSF using a circular Gaussian distribution, with
the FWHM varying with wavelength in accordance with the predicted
$\epsilon_{LE}$($\lambda)$ derived from equation 5 for a $\epsilon_{0}$ and
$\mathcal{L}_{0}$ of 0.83 arcsecs and 22 m, respectively (Martinez et al.
2010), typical values at the Paranal observatory. Each wavelength slice was
perturbed by normally-distributed pseudo-random noise with a mean of zero and
a standard deviation of one, ensuring a minimum S/N of 5 per slice.

From the mock data cubes, we chose four central wavelengths
($\lambda_{1}=4000$, $\lambda_{2}=5000$, $\lambda_{3}=6000$, and
$\lambda_{4}=7000$ Å) to recover five band-filter images at each wavelength,
using bandwidths of 10, 30, 100, 250, and 500 Å. Additionally, we recovered
broad-band filter images with widths of 1000 and 1500 Å, centered on
$\lambda_{2}$ and $\lambda_{3}$.

Figure 7 presents a comparison between the expected $\epsilon_{LE}$ at
$\lambda_{i}$ and the measured FWHM${\lambda_{i}}$ of recovered filter images
within the spectral bands [$\lambda_{i}-\frac{N}{2},\lambda_{i}+\frac{N}{2}$],
with i=1,2,3,4, and N=30,100,250,500, and N=1000, 1500 for i=2,3. Note that we
selected $\Delta\lambda$=1 Å in the mock data cube to simplify the assessment.
The resulting IQ varies less than $\pm 1\%$ with respect to the input
$\epsilon_{LE}$($\lambda_{i})$ of the slice at $\lambda_{i}$ corresponding to
the central wavelength of the bands in almost all the recovered images.

For 10 Å bandwidth filter images, the S/N strongly contributes to the
discrepancies between IQ and $\epsilon_{LE}$. A minimum S/N of 25 seems to be
an adequate threshold for the measurement of the image PSF widths. The
spectral energy distribution of the observed object also impacts the outcomes,
with more notable deviations occurring when the central wavelength of the band
filter coincides with a spectral region of weak emission in the source.
Therefore, in general, deviations tend to increase with wider bands, although
the S/N of recovered band-filter images and the spectrum of the observed
object could also contribute to these deviations. Henceforth, for band-filter
images obtained from seeing-limited IFS observations with S/N$\geq$25, we
assume uncertainties less than 1% of the PSF FWHM, specifically associated
with the chosen filter bandwidths for image recovery.

Figure 7: IQ of band-filter images recovered from mock IFS observations of the
stars in the inset plot, relative to the input $\epsilon_{LE}$ for the
simulation, plotted as a function of the bandwidths used for image recovery.
Turbulence conditions were set to $\epsilon_{0}$=0.83 arcsecs and
$\mathcal{L}_{0}$ =22 m to compute $\epsilon_{LE}$ from equation 5. Filled and
open circles correspond to mock IFS observations of the A1V (HD000136) and M6
V (HD001326) type stars, respectively. Colors correspond to the central
wavelength of the band filters: $\lambda_{1}=4000\AA\rightarrow$ purple,
$\lambda_{2}=5000\AA\rightarrow$ blue, $\lambda_{3}=6000\AA\rightarrow$ green,
and $\lambda_{4}=7000\AA\rightarrow$ red.

###  B.2 Parameters of the Moffat models fitted to filter-band images of
isolated stars in MUSE data cubes

We present here the images of the MUSE observed fields identifying the stars
analyzed in this study to trace image quality across wavelengths (see Fig. 8).
We also present the wavelength variation of the centroids and widths (i.e.,
x0, y0, $\sigma_{1}$, and $\sigma_{2}$ parameters in section §3) for the
Moffat fitting profiles models to the 44 narrow-band filter images recovered
for each selected star for each observed field within the MUSE retrieved data
cubes. Differences in FWHMs derived from the $\sigma_{i}$ (i=1,2) parameters
along the two axes are almost always smaller than the MUSE spatial sampling
(i.e., 0.2 arcsec). Therefore, we define the IQ($\lambda_{i}$) as the average
of the FWHM1 and FWHM2 at each $\lambda_{i}$. Following the same approach as
outlined in section §4.1, we determine a combination of $\epsilon_{0}^{b}$ and
$\mathcal{L}_{0}$ b values that best fit the IQ($\lambda$) for each of the
stars analyzed in each MUSE field. Below, we provide a case-by-case
description of the MUSE fields, Moffat fittings, and the optimal
$\epsilon_{0}^{b}$ and $\mathcal{L}_{0}$ b combinations that match the
measured IQ($\lambda$) for each star, and Table 4 summarizes the results. The
dispersion of the derived atmospheric parameters from individual stars in the
same field provides the typical error of the proposed methodology.

Figure 8: Broad-band filter images recovered from MUSE observations of the
fields: (left) HD90177, and (right) OGLEII

.

Table 4: Atmospheric parameters derived from the measured image quality at
different wavelengths from MUSE data cubes.

HD90177a | HD90177b | HD90177c | HD90177d | OGLEII
---|---|---|---|---
Star | $\epsilon_{0}^{b}$ | $\mathcal{L}_{0}^{b}$ | Star | $\epsilon_{0}^{b}$ | $\mathcal{L}_{0}^{b}$ | Star | $\epsilon_{0}^{b}$ | $\mathcal{L}_{0}^{b}$ | Star | $\epsilon_{0}^{b}$ | $\mathcal{L}_{0}^{b}$ | Star | $\epsilon_{0}^{b}$ | $\mathcal{L}_{0}^{b}$
(1) | (2) | (3) | (1) | (2) | (3) | (1) | (2) | (3) | (1) | (2) | (3) | (1) | (2) | (3)
1 | 0.79 | 18.1$\pm$1.3 | 1 | 0.78 | 14.0$\pm$0.9 | 1 | 0.81 | 15.3$\pm$1.0 | 1 | 0.71 | 21.3$\pm$1.7 | 1 | 0.99 | 15.9$\pm$1.0
2 | 0.82 | 15.5$\pm$1.0 | 2 | 0.78 | 17.6$\pm$1.3 | 2 | 0.82 | 14.8$\pm$1.0 | 2 | 0.69 | 32.9$\pm$3.2 | 2 | 0.98 | 10.3$\pm$0.6
3 | 0.79 | 28.9$\pm$2.5 | 3 | 0.76 | 31.4$\pm$3.0 | 3 | 0.83 | 19.5$\pm$1.4 | 3 | 0.77 | 21.1$\pm$1.6 | 4 | 0.96 | 10.4$\pm$0.6
11 | 0.77 | 22.0$\pm$1.8 | 11 | 0.72 | 29.8$\pm$2.6 | 11 | 0.79 | 17.5$\pm$1.3 | 11 | 0.63 | 47.9$\pm$6.2 | 7 | 0.93 | 29.1$\pm$2.5
|  |  |  |  |  |  |  |  |  |  |  | 9 | 0.87 | 21.4$\pm$1.6
|  |  |  |  |  |  |  |  |  |  |  | 10 | 0.91 | 11.2$\pm$0.6
$\overline{P}$ | 0.79 | 21.1 |  | 0.76 | 23.2 |  | 0.81 | 16.8 |  | 0.70 | 30.8 |  | 0.94 | 16.3
$\sigma_{P}$ | 0.02 | 5.8 |  | 0.03 | 8.7 |  | 0.02 | 2.2 |  | 0.06 | 12.7 |  | 0.05 | 7.6

131313Columns indicate the MUSE observed field, and each sub-column
corresponds to: (1) the number label of the star in the field (see Fig. 8),
and (2) and (3) are the best combination of $\epsilon_{0}$ and
$\mathcal{L}_{0}$ values minimizing residuals when comparing the predicted
$\epsilon_{LE}$ using Eq. 5 with these values to the measured IQ at any
wavelength, assuming a negligible dome turbulence contribution. The last two
lines present the average ($\overline{P}$) and standard deviation
($\sigma_{P}$) of the estimated parameters. These values are derived by
averaging the individual estimations from each star in the field.

####  B.2.1 HD90177 field

HD90177 is a luminous blue variable star in the Carina Nebula, one of the
largest diffuse nebulae in the southern sky, hosting numerous massive and
young stars. HD90177 is known for intense mass-loss episodes, and it is
surrounded by a unique stellar population distribution suggesting a potential
association with a moving group of B-type stars in a spiral arm (Mehner et al.
2021).

We obtained four MUSE data cubes labeled as HD90177a, HD90177b, HD90177c, and
HD90177d (see Table 2) for the HD90177 field. All were observed on the same
night within a temporal window of just under three hours. The HD90177 star,
located at the center of the MUSE data cubes, generates strong stray light
artifacts due to its brightness, affecting some stars in the observed MUSE
field, which includes many other fainter stars than HD90177 (Fig. 8-left). We
identified 14 stars in the field without detected companions within a
5-arcsec-radius circular aperture. Among them, we discarded four stars due to
contamination by stray light from the central bright star. Additionally, five
stars had low S/N in at least 50% of the defined filter bands in any of the
four MUSE data cubes, and one star was too close to the field border,
preventing a proper fitting of the Moffat model. Figures 9, 10, 11, and 12
present the parameters for the Moffat model profiles fitted to the filter-band
images of the remaining four stars in this field for the four MUSE data cubes,
plotted against the central wavelengths of the respective bands.

The centroids of the Moffat models follow wavelength-dependent variations,
which are different for each star but consistently within $\pm$0.2 arcsec,
corresponding to one MUSE spatial pixel. The $\sigma_{i}$ (i=1,2) parameters
for the Moffat models decrease for increasing wavelengths similarly for the
four stars.

The values for $\epsilon_{0}^{b}$ derived from the analysis of the IQ for each
star in the four MUSE data cubes for the HD90177 field are compatible, with an
average and standard deviation of $0.79\pm 0.02$, $0.76\pm 0.03$, $0.81\pm
0.02$, and $0.70\pm 0.06$ arcsec for HD90177a, HD90177b, HD90177c, and
HD90177d data cubes, respectively (see Table 4). The average and standard
deviation of the $\mathcal{L}_{0}^{b}$ values derived from each star are
$21.12\pm 5.83$, $23.20\pm 8.69$, $16.77\pm 2.16$, and $30.80\pm 12.66$
meters, respectively, for HD90177a, HD90177b, HD90177c, and HD90177d data
cubes.

Figure 9: Characteristic parameters of the Moffat profile model with
$\beta=2.5$ fitting the 44 narrow-band filter images of selected point-like
sources in the MUSE Field of HD90177 labeled HD90177a (see Fig. 8(to-left)).
The top panels show the wavelength-dependent variations of the centroid
(x${0}$,y${0}$) with respect to the average star peak location (indicated by
the dashed horizontal line). The bottom panels present the variation of the
width parameter $\sigma_{i}$ (i=1,2) with wavelength. Measurements along axis
i=1 (horizontal) and i=2 (vertical) are represented by filled black circles
and blue squares, respectively. The star numbers correspond to the star labels
in Fig. 8(top-left).

Figure 10: Same as Fig. 9 but for the MUSE data cube labeled HD90177b

.

Figure 11: Same as Fig. 9 but for the MUSE data cube labeled HD90177c

.

Figure 12: Same as Fig. 9 but for the MUSE data cube labeled HD90177d

.

Figure 13:  Wavelength variation of the average FWHM for the Moffat fitting
profile to the narrow-band filter images recovered for each of the selected
stars in the MUSE HD90177 field labeled HD90177a (see Fig. 8(left). The
dashed-black curve corresponds to the wavelength behavior of the
$\epsilon_{0}$ derived from the average DIMM values during the MUSE
observations (see Table 1). The combination of $\epsilon_{0}$ and
$\mathcal{L}_{0}$ best fitting the observations is indicated in the left
corner of each panel, values obtained by minimizing the residuals while
varying $\epsilon_{0}$ and $\mathcal{L}_{0}$ values. The purple line
corresponds to the wavelength $\epsilon_{LE}$ variation using
$\epsilon_{0}^{b}$ and $\mathcal{L}_{0}$ b.

Figure 14: Same as Fig. 13 but for the MUSE data cube labeled HD90177b.

Figure 15: Same as Fig. 15 but for the MUSE data cube labeled HD90177c.

Figure 16: Same as Fig. 16 but for the MUSE data cube labeled HD90177d.

####  B.2.2 OGLEII

The MUSE field labeled as ’OGLEII’ seems to correspond to an observation of
the ’OGLEII dia bul-sc01 1821’ star according to the MUSE data cube header.
This star is a Mira variable that is included in the X-Shooter spectral
library (Gonneau et al. 2020). Located in the galactic bulge region, the MUSE
data cube shows a high density of stars of different brightness (Fig.
8-right). We found 10 stars in the MUSE field without companions brighter than
a 5-sigma background within a 5 arcsec-radius region. Among them, we excluded
three stars because the surrounding background revealed the presence of faint
stars, impacting the fitting of the Moffat model. Additionally, we discarded
one star due to a poor S/N in at least 50% of the defined filter bands. In
Fig. 17, we show the parameters against the central wavelengths of the
respective filter for the Moffat model profiles fitted to the filter-band
images of the remaining six stars. We observe wavelength-dependent variations
in the centroids within the $\pm$0.2 arcsecs of the MUSE spatial pixel for
most of the measurements, with the largest deviations along the horizontal
axis. As expected, the Moffat model’s widths decrease as wavelengths increase.
For the analyzed stars in this field, Star-7 displays the most significant
deviation from a circular Moffat model on average.

The values for $\epsilon_{0}^{b}$ derived from the analysis of the IQ for each
star in the OGLEII field are compatible, with an average and standard
deviation of $0.94\pm 0.05$ arcsec (see Table 4). The average and standard
deviation of the $\mathcal{L}_{0}^{b}$ values derived from each star are
$16.3\pm 7.6$ m.

Figure 17: Same as Fig. 9 but for the MUSE data cube of the OGLEII field.

Figure 18: Same as Fig. 13 but for the MUSE data cube of the OGLEII field.