11institutetext: Institute for Experimental Particle Physics, Karlsruhe
Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-
Leopoldshafen, Germany 22institutetext: Instituto de Tecnologías en Detección
y Astropartículas, Universidad, Universidad Nacional de San Martín, Av.
General Paz 1555 (B1630KNA), San Martín, Buenos Aires, Argentina
22email<EMAIL_ADDRESS>33institutetext: Department of
Astrophysics/IMAPP, Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The
Netherlands
33email<EMAIL_ADDRESS>44institutetext: Institute for Astroparticle
Physics, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1,
76344 Eggenstein-Leopoldshafen, Germany 55institutetext: Physics Department,
Vrije Universiteit Brussels, Pleinlaan 2, 1050 Brussels, Belgium
55email<EMAIL_ADDRESS>
# Uncertainties of the 30-408 MHz Galactic emission as a calibration source
for radio detectors in astroparticle physics

M. Büsken 1122 T. Fodran 33 T. Huege 4455

(Received / Accepted)

###### Abstract

Context. Arrays of radio antennas have proven to be successful in
astroparticle physics with the observation of extensive air showers initiated
by high-energy cosmic rays in the Earth’s atmosphere. Accurate determination
of the energy scale of the primary particles’ energies requires an absolute
calibration of the radio antennas for which, in recent years, the utilization
of the Galactic emission as a reference source has emerged as a potential
standard.

Aims. To apply the ”Galactic Calibration”, a proper estimation of the
systematic uncertainties on the prediction of the Galactic emission from sky
models is necessary, which we aim to determine on a global level as well as
for the specific cases of selected radio arrays. We further aim to quantify
the influence of the quiet Sun on the Galactic Calibration.

Methods. We look at four different sky models that predict the full-sky
Galactic emission in the frequency range from $30$ to
$408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ and compare them. We make an
inventory of the reference maps on which they rely and use the output of the
models to determine their global level of agreement. Next, we take the sky
exposures and frequency bands of selected radio arrays into account and repeat
the comparison for each of them. Finally, we study the relative influence of
the Sun in its quiet state by projecting it onto the sky with brightness data
from recent measurements.

Results. We find systematic uncertainty of $12\text{\,}\%$ on the predicted
power from the Galactic emission, which scales to approximately half of that
value as the uncertainty on the determination of the energy of cosmic
particles. When looking at the selected radio arrays, the uncertainty on the
predicted power varies between $10\text{\,}\%$ and $19\text{\,}\%$. The
influence of the quiet Sun turns out to be insignificant at the lowest
frequencies but increases to a relative contribution of $\sim$20\text{\,}\%$$
around $400\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$.

###### Key Words.:

Astroparticle physics – Methods: miscellaneous – Radio continuum: general –
Sun: radio radiation

##  1 Introduction

Already in the 1940s, surveys of the sky at radio frequencies were conducted.
Strong emission from the constellation of Sagittarius was found as well as
smaller maxima originating in extragalactic sources (e.g. Cygnus A, Cassiopeia
A) and at this point also a signal coming from the Sun was seen (Reber, 1944).
In measurement campaigns, the entire sky was mapped at specific frequencies.
These maps show strong point-like sources as well as the diffuse radio
emission from the Galaxy. The original motivation for this was to get a better
understanding of the Galaxy and also to use the information from the maps in
studies in the fields of astronomy and astrophysics, e.g. related to the
cosmic microwave background. To acquire accurate descriptions of the Galactic
emission not only from a handful of maps at some specific frequencies but
across a broad frequency range for the whole sky, efforts were made to build
models of the radio sky based on these reference maps.

These sky models recently became relevant in the field of astroparticle
physics for the absolute calibration of radio detection arrays. Over the past
years, the radio detection technique has gained increasing importance for the
observation of highly energetic cosmic rays and neutrinos detected through the
radio emission from particle cascades (Huege, 2016; Schröder, 2017). A number
of promising radio detection arrays for the measurement of cosmic particles in
the frequency range from a few tens to hundreds of MHz are in the phase of
development or under construction (e.g. the AugerPrime Radio Detector (Pont,
2019), the Square Kilometer Array low-frequency site (SKA-low) (Buitink et
al., 2021), the Giant Radio Array for Neutrino Detection (GRAND) (Álvarez-
Muñiz et al., 2020)) while some are already taking data (e.g. the Auger
Engineering Radio Array (AERA) (Huege, T., Aab, A., Abreu, P., et al., 2019),
the LOw Frequency ARray (LOFAR) (Schellart, P. et al., 2013)). For extracting
physics it is very important to accurately determine the absolute energy scale
of the detected particle cascades – and therefore an accurate calibration of
the radio detectors is required. Calibrating with a reference antenna emitting
a defined signal (e.g. mounted on a drone (Aab et al., 2017)) has the
disadvantage of uncertainties on the emitted signal strength which are
difficult to assess (Mulrey et al., 2019). Also, dedicated calibration
campaigns require significant effort and are almost impossible to perform on a
regular basis for large arrays. A different calibration approach uses the
diffuse Galactic radio emission – which poses a natural background to the
detection of radio emission from particle showers – as the reference signal.
It typically poses the dominant background from natural sources at these
frequencies (Union, 2019). This so-called Galactic Calibration also offers the
opportunity to directly compare the calibration of different radio arrays as
they at least partially see the same sky. Moreover, the method can be applied
as regularly as background data is available and does not require special
field campaigns. The Galactic Calibration was already applied to the LOFAR
low-band antennas (Mulrey et al., 2019) and to the engineering array of the
new radio detector of the Pierre Auger Observatory (Fodran et al., 2021).

In the Galactic Calibration, the measured background signal from the Galaxy is
compared to predictions made with the aforementioned sky models. Knowing the
uncertainty on these predictions is thus crucial for a useful absolute
calibration. In this work, we therefore conduct a comparison of four publicly
available radio sky interpolation models for the frequency range from $30$ to
$408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. We do this by generating outputs
with the models, calculating the average sky temperature from the outputs and
determining the level of agreement between the models. From this comparison we
get an estimate for the systematic uncertainties of the modeled background
predictions for the Galactic Calibration.

First we present these models (Sect. 2) and summarize the reference maps they
are based on (Sect. 3). Afterwards, we perform a global comparison while also
showing other, less detailed descriptions of the radio background (Sect. 4).
After that we adjust the comparison for the sky seen by an observer at a
specific location on Earth. We further conduct the comparison tailored to a
set of selected radio arrays from the present and future, namely the Radio
Neutrino Observatory Greenland (RNO-G) (Aguilar et al., 2021), LOFAR
(Schellart, P. et al., 2013), GRAND (Álvarez-Muñiz et al., 2020), SKA-low
(Buitink et al., 2021), the radio detectors of the Pierre Auger Observatory
(Huege, T., Aab, A., Abreu, P., et al., 2019; Pont, 2019) and IceCube (Abbasi
et al., 2021). Additionally, we study the influence of the quiet Sun as
another source in the radio sky on the background (Sect. 6). Finally, we
discuss implications of the results for the application of the Galactic
Calibration (Sect. 7) to radio detectors in astroparticle physics.

##  2 Radio sky interpolation models

In the past years, a couple of models for predicting the diffuse foreground
emission of the sky in the radio and microwave regimes were developed with the
purpose of calibrating radio arrays or removing this foreground emission from
measurements of the cosmic microwave background.

These models interpolate between reference sky surveys at various frequencies
conducted with telescopes at different locations. In the surveys, the sky
brightness at given coordinates is mapped in terms of the brightness
temperature $T_{\text{B}}$. This is the temperature of a thermal radiator,
i.e. a black body, that would show the same brightness as the one measured. In
the classical limit $h\nu\ll k_{\text{B}}T$ of Planck’s law for black body
radiation, where $h$ is the Planck constant, $\nu$ is the considered frequency
and $k_{\text{B}}$ is the Boltzmann constant, the Rayleigh-Jeans law is
applicable. The brightness temperature is then directly proportional to the
observed brightness
$I_{\nu}\>[\mathrm{Wm}^{-2}\mathrm{Hz}^{-1}\mathrm{sr}^{-1}]$:(Wilson et al.,
2009)

$T_{\text{B}}=\frac{c^{2}}{2k_{\text{B}}\cdot\nu^{2}}I_{\nu},$ (1)

where $c$ is the speed of light in vacuum. In the frequency range from a few
$10\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ to a few
$100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$, most of the electromagnetic
background in the sky is presumably synchrotron radiation from electrons
gyrating in the magnetic field of the Galaxy (Rybicki & Lightman, 1985).
Although this is non-thermal emission, the description of the radio sky by a
brightness temperature is still practical because of its proportionality to
the brightness $I_{\nu}$.

The frequency dependence of the brightness temperature can be described at
lower radio frequencies by a power-law

$T_{\text{B}}\propto\nu^{\beta}$ (2)

with a spectral index $\beta$. Recent studies of the spectral index at
frequencies from $50\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ to
$200\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ lie in a range of
$$-2.62$<\beta<$-2.46$$, depending on the region in the sky (Mozdzen et al.,
2016, 2018; Rogers & Bowman, 2008). Around
$200\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ and again above
$400\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$, changes of the spectral index
are observed with a steepening towards higher frequencies (Purton, 1966;
Bridle & Baldwin, 1967; Webster, 1974).

This relation can be used to scale a single full-sky reference map from a
survey conducted at a low frequency to any other frequency. One model, called
LFmap, uses this approach of spectral scaling with frequency- and region-
specific spectral indices for interpolation. LFmap is included in the model
comparison of this work. Other popular interpolation models use the approach
of a principal component analysis (PCA) of multiple reference maps to produce
frequency-dependent descriptions of the radio sky. In this inventory, three
such models will be considered and introduced in the following Sects. namely
the Global Sky Model (GSM) in its first (de Oliveira-Costa et al., 2008) and
improved (Zheng et al., 2016) version, as well as the Low Frequency Sky Model
(LFSM) (Dowell et al., 2017).

###  2.1 LFmap

The LFmap (Polisensky, 2007) software is a radio sky interpolation software
based on the simple power-law model for the sky brightness temperature (Eq.
2). The $408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map by Haslam et al.
(1982) in the revisited version by Platania et al. (2003) is scaled down with
spectral indices from a spectral index map from the same reference to
frequencies down to a minimum frequency of
$180\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. To take spectral bending into
account, new spectral indices are used for frequencies below
$180\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$, which are calculated such that
each pixel of the downscaled map at
$180\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ scales onto the same pixel of a
$22\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map by Roger, R. S. et al. (1999).
Part of the region around the Galactic center is affected by HII absorption,
which becomes relevant only below $45\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$.
Therefore, an intermediate step at this frequency is introduced, where the
affected region of the map is replaced by the average of the surrounding area,
that is unaffected by HII absorption. Below
$45\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ the procedure of calculating
spectral indices is done with the original
$22\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map. In a similar manner, regions
not covered in the $22\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map are
replaced with averages from surrounding areas. Most point-like sources are not
treated individually, except for the two brightest ones, for which adapted
spectral indices are used to scale their brightness temperatures. In this
study, we use the default settings of LFmap.

###  2.2 GSM (2008)

The other three considered radio sky interpolation models use a principal
component analysis (PCA) to generate all-sky maps at any intermediate
frequency. Of these models, GSM (de Oliveira-Costa et al., 2008) is the
oldest, which uses the fewest reference maps. The advantage of a PCA is that
also maps of limited sky coverage can be used. However, different angular
resolutions and levels of quality between the used maps pose challenges for
this approach. In GSM, also maps at GHz frequencies are used, which can have a
significant influence on the low-frequency results in regions where there is
otherwise only sparse map coverage.

The PCA in GSM is performed for the region of the sky that is covered by all
of the included reference maps. In a fit, the principal components are modeled
to best match in the remaining regions. At minimum, there is information from
six maps per pixel. However, for a large part of the southern hemisphere, sub-
GHz information is only available from the all-sky maps at
$45\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ and
$408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$, possibly reducing the model
accuracy in this region at these frequencies, which is relevant for
experiments located in the southern hemisphere. The resulting sky description
with principal components at the reference frequencies is finally spline-
interpolated to any frequency in between. No special treatment for point-like
sources is mentioned although they were removed in some of the used reference
maps.

The accuracy of the model is determined by taking one of the reference maps
out of the calculation of the principal components and calculating the
difference of this map to the output of the model at that frequency without
the information from the map. This is done for all frequencies at which a
reference map is available and gives an estimate for the accuracy of around
$12\text{\,}\%$ for the sub-GHz maps. However, the model does not account for
individual uncertainties associated with the reference maps or systematic
temperature shifts of a map over the whole sky, that could be introduced by
inaccurate calibration.

###  2.3 GSM (2016)

The original GSM received an update in which further reference maps were
included and the model algorithm around the PCA was improved (Zheng et al.,
2016). The frequency range of the model was extended to
$5\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ by including recently published
surveys in the microwave regime. At lower radio frequencies, maps at
$85\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ and
$150\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ were added. With this enhanced
set of surveys, there is no common region covered by all of them, that can be
used for conducting the PCA as in the original GSM. Instead, an iterative
algorithm is implemented, which fits the principal components to the reference
maps to the desired accuracy. For the zeroth iteration, a subset of maps was
used that overlap by at least $5\text{\,}\%$ in sky coverage. Then the
remaining maps were fed into the model. Finally, six principal components are
used, compared to three components in the original GSM. Within the PCA
algorithm, point-like sources are removed.

A level of model accuracy is determined in the same way as for the original
GSM and found to be around $8\text{\,}\%$ for the sub-GHz frequencies. When
compared to the original GSM, the accuracy of the improved model is better for
all sub-GHz frequencies by a factor of up to 2. However, since the original
GSM is still widely used and since most of the newly added reference maps are
at higher frequencies than we consider here, we include both versions of the
GSM in the comparison of this study.

###  2.4 LFSM

Based on the sky surveys at the Long Wavelength Array Station 1 (LWA1)
(Ellingson et al., 2013) in the USA at frequencies from
$40\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ to
$80\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$, a low-frequency sky model was
constructed (Dowell et al., 2017). Additional surveys from the literature were
included. The model itself is based on the PCA approach as performed in the
original GSM. Similar to the improved GSM, an iterative procedure is used to
tackle the problem of not having a sky region that is commonly covered by all
maps. The final number of principal components in the model is three and maps
can be generated within a total range from $10$ to
$408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. Same as for GSM no individual
treatment of point-like sources is mentioned.

As a measure of accuracy, the relative difference between the LWA1 survey at
$74\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ and the model prediction at the
same frequency was calculated. Deviations were found to be at an overall level
of $10\text{\,}\%$.

The LWA1 surveys nicely cover the low-radio regime
$<$100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$. However, the Long Wavelength
Array is stationed in the northern hemisphere. This could lead to additional
uncertainties in the interpolated maps at Galactic latitudes below
$-40\text{\,}\degr$, which would become relevant for experiments in the
southern hemisphere. Furthermore, the temperature scale of the LFSM output
strongly depends on the calibration of the LWA1 surveys, as these represent a
large fraction of the total ensemble of input maps. Potential systematic
errors in the calibration would thus directly bias the interpolation model.

###  2.5 Other parametrizations of the Galactic background brightness

Besides the interpolation models for generating all-sky maps, there are also
some parametrizations for the spectrum of the average brightness of the
Galactic background. We do not include them in the comparison of the
interpolation models but present and show them here for completeness.

One of these parametrizations was introduced by Cane (1979) and assumes a
superposition of Galactic and extragalactic contributions to the brightness
including absorption by the Galactic disk and was fitted to a multitude of
measurements of the polar regions of the Galaxy. This yields lower brightness
temperatures than expected for an average of the whole sky because the bright
Galactic center is not considered here. To accommodate for this, a correction
factor of ${\sim}1.3$ to the Galactic brightness contribution was found by
Dulk et al. (2001) and Durc et al. (2003). In this work, we use the such-
corrected parametrization and refer to it as ”Cane”.

The parametrization by Cane was revisited again by Tokarsky, Konovalenko and
Yerin (Tokarsky et al., 2017), where the corrections were summarized and
another expression for the average brightness temperature as a function of
frequency was given based on results from Krymkin (1971). We refer to this
parametrization here as ”TKY”.

Both descriptions of the Galactic background brightness are applicable only
for the low-frequency radio regime
($<$100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$), where the Galaxy is the
dominant contribution to the background.

Table 1: Summary of all presented reference maps with their sky coverage and
quoted uncertainties.

Map No. | Frequency $\nu$ | Covered Declination | $\sigma_{k}$ | $\sigma_{T_{0}}$ |  $\sigma_{T_{0}}$ (Normalized) | Used In | References
---|---|---|---|---|---|---|---
| (MHz) |  | (%) | (K) | (%) |  |
1 | 10 | $$-6\text{\,}\degr$<\delta<$74\text{\,}\degr$$ | 9* | $2\cdot 10^{4}$ | 7.0 | 1,2,3 | 1
2 | 22 | $$-28\text{\,}\degr$<\delta<$80\text{\,}\degr$$ | 16 | $5\cdot 10^{3}$ | 11.5 | 1,2,3,4 | 2
3 | 40 | $$-40\text{\,}\degr$<\delta<$90\text{\,}\degr$$ | 20 | 10 | 0.1 | 3 | 3
4 | 45 | $$-90\text{\,}\degr$<\delta<$65\text{\,}\degr$$ | 10/15 | 125† | 1.5 | 1,2,3,(4) | 4, 5
5 | 50 | $$-40\text{\,}\degr$<\delta<$90\text{\,}\degr$$ | 20 | 10 | 0.2 | 3 | 3
6 | 60 | $$-40\text{\,}\degr$<\delta<$90\text{\,}\degr$$ | 20 | 10 | 0.2 | 3 | 3
7 | 70 | $$-40\text{\,}\degr$<\delta<$90\text{\,}\degr$$ | 20 | 10 | 0.3 | 3 | 3
8 | 80 | $$-40\text{\,}\degr$<\delta<$90\text{\,}\degr$$ | 20 | 10 | 0.5 | 3 | 3
9 | 85 | $$-25\text{\,}\degr$<\delta<$25\text{\,}\degr$$ | 7 | 120 | 6.7 | 2 | 6
10 | 150 | $$-25\text{\,}\degr$<\delta<$25\text{\,}\degr$$ | 5 | 40 | 9.2 | 2 | 6
11 | 178 | $$-5\text{\,}\degr$<\delta<$88\text{\,}\degr$$ | 10 | 15 | 5.3 | (1,2,3) | 7
12.a | 408 | $$-90\text{\,}\degr$<\delta<$90\text{\,}\degr$$ | 10/5 | 3 | 8.8 | 1 | 8
12.b | 408 | $$-90\text{\,}\degr$<\delta<$90\text{\,}\degr$$ | 10/5 | 3 | 8.8 | 4 | 9
12.c | 408 | $$-90\text{\,}\degr$<\delta<$90\text{\,}\degr$$ | 10/5 | 3 | 8.8 | 2,3 | 10

111The two values for the relative scale uncertainty of map No. 4 refer to the
two different estimations for this in the publications of the northern and
southern part of the survey, respectively. For scale uncertainties with a (*)
no explicit value was given by the authors. Instead, we estimate it by taking
half of the smallest contour interval of that map and dividing it by its
minimum brightness temperature. In the same way, zero-level errors with a
$(^{\dagger})$ are estimated by taking half of the smallest contour interval.
To give the zero-level errors as a fraction of the average sky temperature, we
calculate the latter from Eq. 4, where we take the average from using all four
considered sky models. For indicating in which models the maps are used, we
use the following notation: 1 = GSM, 2 = GSM16, 3 = LFSM, 4 = LFmap. Numbers
in parentheses mark that the map was only used indirectly in the respective
model.

(1) Caswell (1976); (2) Roger, R. S. et al. (1999); (3) Dowell et al. (2017);
(4) Alvarez, H. et al. (1997); (5) Maeda, K. et al. (1999); (6) Landecker &
Wielebinski (1970); (7) Turtle & Baldwin (1962); (8) Haslam et al. (1982); (9)
Platania et al. (2003); (10) Remazeilles et al. (2015).

Furthermore for comparison purposes, we show the simple approach of scaling a
full-sky map with just one single spectral index. As the reference map, we use
the $408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map by Haslam et al. (1982)
as improved by Remazeilles et al. (2015), which we introduce in Sect. 3. Here,
we refer to this description as ”Haslam”, but also do not include it in the
comparison of the interpolation models because of its drawbacks. Using a
single spectral index does not represent reality, as the spectral index varies
for different regions of the sky and different frequency regimes (Mozdzen et
al., 2018; Dickinson et al., 2019). Although often used in studies of fast
radio bursts, the Haslam description should not be seen as equivalent to the
more sophisticated interpolation models (Price, 2021). We show the Haslam
results for a spectral index between $-2.62$ and $-2.46$.

##  3 Reference maps

The presented radio sky interpolation models rely on a number of reference
maps. Consequently, the accuracy of the models is dependent on the accuracy of
these maps and their inherited uncertainties. Therefore, in the following, we
give an overview of the reference maps used in the considered interpolation
models.

(a)

(b)

(c)

(d)

Figure 1: Sky coverages of the used reference maps for each sky model laid on
top of each other and plotted in galactic coordinates. The models are numbered
(a) LFmap, (b) GSM, (c) GSM16 and (d) LFSM. For regions of lighter color fewer
measurements from reference maps are available than for regions of darker
color.

####  $10\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ (Caswell, 1976):

This map by Caswell was obtained using the 10-MHz array at the Dominion Radio
Astrophysical Radio Observatory (DRAO) in Canada. The observations were done
during winter nights to minimize ionospheric influence. Thus the sky coverage
is limited in right ascension from $0^{\mathrm{h}}$ to $16^{\mathrm{h}}$.
Relative calibration of the received signal strength to sky brightness
temperature was performed by comparing the observations to a published
$178\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map from Turtle & Baldwin (1962),
that was scaled using a power-law relation as in Eq. 2 and spectral indices of
$-2.40$ and $-2.75$ for the Galactic and extragalactic isotropic emission,
respectively. The sky brightness is mapped in contours indicating specific
temperature levels. The contour interval between these levels is
$2\text{\cdot}{10}^{4}\text{\,}\mathrm{K}$.

####  $22\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ (Roger, R. S. et al., 1999):

The sky map at $22\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ was also produced
from measurements done at the DRAO. The applied antenna gain is based on an
assumed value for the flux density of Cygnus A (Roger et al., 1969). A cross-
check of the map with the $408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map by
Haslam et al. (1982) was performed to compare the temperature scales by
producing scatter plots of the brightness temperatures ($T$-$T$ plots) from
both maps for different declinations from the zenith. Then, the temperature
scale of the $22\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map was tuned to
overall match the ratio at the zenith, for which the authors believed their
instrument to be understood better. The $T$-$T$ plot for the zenith direction
showed no discrepancy with the zero-level of the temperature scale. Also, the
differential spectral index derived from the $T$-$T$ plot between the
$22\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map and the
$408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map is in agreement with other
measurements.

#### LFSS $40\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$,
$50\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$,
$60\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$,
$70\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$,
$80\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ (Dowell et al., 2017):

Within the LFSS at the LWA1, sky maps were produced at a variety of
frequencies. The gain of the LWA1 antennas was derived from an electromagnetic
antenna model in combination with a correction using bright pulsars at
different elevations as reference sources. Previous observations of the flux
density of Cygnus A by Baars et al. (1977) were used to convert the
measurements into sky temperature. A temperature calibration system was used
to achieve absolute accuracy of $10\text{\,}\mathrm{K}$ or better.

####  $45\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ (Alvarez, H. et al., 1997;
Maeda, K. et al., 1999):

At $45\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$, there are two separate sky
surveys available, one covering the northern hemisphere and one covering the
southern hemisphere conducted at the Middle and Upper Atmosphere Radar in
Japan and the Maipu Radio Astronomy Observatory in Chile, respectively.
Calibration of the temperature scale of the latter survey was performed by
using a spectral interpolation from data in a well-observed reference region
and comparison with the measurements. The map was later used to calibrate the
map of the northern hemisphere as both cover the region from $5\text{\,}\degr$
to $19\text{\,}\degr$ in declination.

####  $85\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$,
$150\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ (Landecker & Wielebinski, 1970):

Surveys at $85\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ and
$150\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ were made using the Parkes
$64\text{\,}\mathrm{m}$ telescope in Australia. Calibration of the temperature
scale was done after every scan with well-matched noise diodes that could be
connected to the receivers of the telescope.

####  $408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ (Haslam et al., 1982):

Multiple sites (Jodrell Bank MkI(A) telescope in England, Effelsberg
$100\text{\,}\mathrm{m}$ telescope in Germany and Parkes
$64\text{\,}\mathrm{m}$ telescope in Australia) were used to observe the sky
at $408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. The originally assembled map
by Haslam et al. was calibrated with reference to a sky survey at
$404\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ from Pauliny-Toth & Shakeshaft
(1962). (Remazeilles et al., 2015) believe that the brightness temperature
scale is more accurate (${\sim}$5\text{\,}\%$$) than originally quoted
(${\sim}$10\text{\,}\%$$). Because of its importance in the field of radio
astronomy, the map and its raw data were restudied multiple times, including
destriping it and removing point-like sources (Platania et al., 2003;
Remazeilles et al., 2015).

#### Corrections to the maps at $45\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$,
$150\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ and
$408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$:

The sky surveys at $45\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ that are used
in the interpolation models considered here were done separately for the
southern (Alvarez, H. et al., 1997) and northern hemisphere (Maeda, K. et al.,
1999). In a later study, the maps were combined into an all-sky survey with a
correction of the zero-level of $-544\text{\,}\mathrm{K}$ (Guzmán et al.,
2011) to the combined map. This corresponds to ${\sim}$6.5\text{\,}\%$$ of the
average sky temperature at that frequency and is more than three times larger
than the originally quoted error on the zero-level. In the same study also a
zero-level correction for the original
$408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map of $-3.46\text{\,}\mathrm{K}$
was determined, which corresponds to ${\sim}$10\text{\,}\%$$ of the average
sky temperature.

Another recalibration was performed for the combined
$45\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map (although without the
mentioned zero-level correction) and for the
$150\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ map (Monsalve et al., 2021).
There the maps were corrected for temperature scale and zero-level to best
match data taken with the Experiment to Detect the Global EoR Signature
(EDGES) (Bowman et al., 2018). Scale correction factors are $1.076\pm
0.017\text{\,}$ and $1.112\pm 0.012\text{\,}$ for the sky temperature of the
$45\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ and
$150\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ maps, respectively. Zero-level
corrections for the surveys are $-160\pm 78\text{\,}\mathrm{K}$ and $0.7\pm
6.0\text{\,}\mathrm{K}$, which correspond to $1.9\pm 0.9\text{\,}\%$ and
$0.2\pm 1.4\text{\,}\%$ of the average sky temperature at the respective
frequencies.

Corrections to the temperature scale and zero-level by matching to other,
sometimes newer data can be bigger than originally quoted uncertainties. This
hints towards an underestimation of the latter and poses a challenge when
trying to place trust on individual surveys.

#### Summary:

The reference maps are listed in Table 1 together with their quoted
uncertainties on the temperature scale, which are typically described by a
linear relation. The true temperature $T_{\mathrm{true}}$ relates to the
observed temperature $T_{\mathrm{obs}}$ as

$T_{\mathrm{true}}=k\cdot T_{\mathrm{obs}}+T_{0}.$ (3)

(a)

(b)

(c)

(d)

Figure 2: Sky maps showing the temperature ratio of each model to the average
from all four models at $50\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. The
models are denoted as (a) LFmap, (b) GSM, (c) GSM16 and (d) LFSM.

The scaling factor $k$ ideally has a value of 1, but it inherits the relative
scale uncertainty $\sigma_{k}$, which goes up to $20\text{\,}\%$ for the
reference maps presented here, while the largest uncertainties are quoted for
the results of the Low Frequency Sky Survey (LFSS) (map No. 3, $58$). The
zero-level $T_{0}$ represents an absolute offset of the temperature scale and
deviations from the ideal case ($T_{0}=$0\text{\,}\mathrm{K}$$) are reflected
in the zero-level error $\sigma_{T_{0}}$. The zero-level errors of the
reference maps are given in Table 1 as absolute values and also relative to
the average brightness temperature of the whole sky at the respective
frequency, where the average sky brightness temperature is calculated as in
Eq. 4. The relative zero-level errors amount up to ${\sim}$12\text{\,}\%$$,
while they are negligible for the LFSS results.

Furthermore, Table 1 gives the information, in which models each reference map
is used. This information is also visualized in Fig. 1, where for each model
the covered sky regions of all used reference maps are superimposed. LFmap
relies on only two reference maps with the approach of spectral scaling. LFSM
uses the most reference maps with a maximum of nine maps covering the region
between $-6\text{\,}\degr$ and $65\text{\,}\degr$ of declination. However,
there is a lack of available reference maps for all models around the south
celestial pole.

Further maps at higher frequencies are used in the interpolation models as
well. Many of them were generated from data taken with the space-based
instruments Wilkinson Microwave Anisotropy Probe (WMAP) and Planck (Hinshaw et
al., 2009; Planck Collaboration et al., 2016). These maps are important for
modeling sky regions where fewer low-frequency surveys were performed, but
they will not be discussed in more detail here.

##  4 Comparison of the sky model predictions

Besides the quoted accuracies of the reference maps, which propagate into the
interpolation models, we compare the output of the models directly. We study
the deviations between them and determine their level of agreement, which we
then use as an estimator for the systematic uncertainty in predicting the
diffuse Galactic radio emission on an absolute scale.

The PyGDSM (Price, 2016) package is used as an interface to the GSM, GSM16 and
LFSM models. It employs the healpy (Zonca et al., 2019) package to provide the
temperature map output of the models in the HEALPix (Górski et al., 2005)
format. LFmap is a C program and is available at the author’s
website222https://www.astro.umd.edu/~emilp/LFmap/. After generating the LFmap
output maps we convert them as well into the HEALPix format. Exemplarily, sky
maps are shown in Fig. 2, which display the temperature ratio between the
output of each model and the average at
$50\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. The average is calculated
pixelwise from all four models at the same frequency. At
$50\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ LFmap and GSM predict a hotter
Galactic center than the other two models, while away from the center their
predictions are colder. By this depiction, also spatial structures of the
differences between the models are visible, e.g. some stripe-like features for
GSM.

In this study, the comparison is conducted for the frequency range from $30$
to $408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. Typically radio arrays for
the detection of cosmic particles are not operated below
$30\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ because of the presence of strong
atmospheric noise (Huege, 2016; Union, 2019). The upper bound arises from
limitations within the LFmap and LFSM models.

###  4.1 Comparison of the total sky

Figure 3: Average sky temperature as a function of the frequency is plotted
for the four interpolation models as solid lines and for the parameterizations
Cane and TKY as dashed lines. The grey band shows the results for the
$408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ Haslam map when scaled down with
a spectral index $\beta$ between $-2.62$ and $-2.46$.

The average temperature of the sky at a given frequency $\nu$ is calculated as

$T_{\text{sky,
average}}(\nu)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}l\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\mathrm{d}b\cos{(b)}\;T(\nu;l,b),$
(4)

where $l$ is the Galactic longitude and $b$ is the Galactic latitude. The
brightness temperature at a specific location in the sky $T(\nu;l,b)$ is taken
from maps in Galactic coordinates produced with the sky models. The observable
$T_{\text{sky, average}}(\nu)$ is used for the first part of the comparison.
It gives compressed information of a sky map while ignoring spatial
structures. For the Galactic Calibration of an antenna without a narrow beam,
taking into account fine spatial structures is not as relevant, because the
measured quantity of received power is a folding of the whole visible sky
through the antenna pattern.

Figure 3 shows the average sky temperature as a function of the frequency for
the interpolation models from $30\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ to
$408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. The grey band is obtained from
the Haslam description of the sky brightness by using a spectral index $\beta$
between $-2.62$ and $-2.46$ and is shown for comparison. The range for the
spectral index is deduced from recent measurements (Mozdzen et al., 2016,
2018) around the relevant frequencies.

Furthermore, the previously introduced parametrizations of the average sky
brightness below $100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$, Cane and TKY
are shown for comparison as solid and dashed black lines, respectively.

Figure 4: Average sky temperature is plotted against the frequency as in Fig.
3 with the same line style, but normalized to the results for the scaled
Haslam map and a spectral index $\beta=-2.46$.

The comparison shows that the interpolation models agree well in their shapes
over the whole frequency range, while they are systematically shifted relative
to each other. This can be seen more quantitatively in Fig. 4, in which the
data are normalized to the Haslam results with a spectral index $\beta=-2.46$,
to better show spectral behaviour of the models.

Next, we calculate the relative difference of the average sky temperatures at
a given frequency for any combination of models m1 and m2. From that, we
evaluate the maximum relative difference

$\mathrm{r_{\mathrm{max}}}(\nu)=\max|2\cdot\frac{T_{\mathrm{{sky,\;average,}m_{1}}}(\nu)-T_{\mathrm{{sky,\;average,}m_{2}}}(\nu)}{T_{\mathrm{{sky,\;average,}m_{1}}}(\nu)+T_{\mathrm{{sky,\;average,}m_{2}}}(\nu)}|\mathrm{,}$
(5)

which we plot as a function of the frequency, shown in Fig. 5. The maximum
relative difference noticeably rises for frequencies larger than
${\sim}$50\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ to values around
$18\text{\,}\%$ and drops again for frequencies larger than
${\sim}$200\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ to values around
$8\text{\,}\%$.

Figure 5: Maximum relative difference of the average sky temperature between
any two of the considered models as a function of frequency. The maximum
relative difference is calculated as in Eq. 5.

To gauge the level of agreement between the models more concisely, for each of
them we integrate the average sky temperature over the frequency range from 30
to 408 MHz, and calculate a relative difference
$r_{\text{m}_{1},\text{m}_{2}}$ pairwise as

$r_{\text{m}_{1},\text{m}_{2}}=2\cdot\frac{\int_{30\text{MHz}}^{408\text{MHz}}T_{\text{sky,
average; m}_{1}}(\nu)-T_{\text{sky, average;
m}_{2}}(\nu)\;\mathrm{d}\nu}{\int_{30\text{MHz}}^{408\text{MHz}}T_{\text{sky,
average; m}_{1}}(\nu)+T_{\text{sky, average;
m}_{2}}(\nu)\;\mathrm{d}\nu}\mathrm{,}$ (6)

again with any combination of models m1 and m2. The results are listed in
Table 2. By this comparison, all models agree with each other at a level of
around $12\%$ or better. Here, the LFSM prediction shows the largest deviation
relative to the other models. The level of agreement is $5\%$ when neglecting
the LFSM. This could be a consequence of the latter containing the five LWA1
maps of the northern hemisphere between $30$ and
$80\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$, which have a significant
influence on the principal component analysis. The other interpolation models
have fewer reference maps in that frequency range and this regime might thus
be influenced more by the included GHz maps.

Table 2: Value of $r_{\text{m}_{1},\text{m}_{2}}$ listed for each combination
of the interpolation models.

$r_{\text{m}_{1},\text{m}_{2}}$ (%) | LFmap | GSM | GSM16 | LFSM
---|---|---|---|---
LFmap | - | $4.6\text{\,}$ | $0.7\text{\,}$ | $-7.3\text{\,}$
GSM | $-4.6\text{\,}$ | - | $-3.8\text{\,}$ | $-11.9\text{\,}$
GSM16 | $-0.7\text{\,}$ | $3.8\text{\,}$ | - | $-8.1\text{\,}$
LFSM | $7.3\text{\,}$ | $11.9\text{\,}$ | $8.1\text{\,}$ | -
333The header row gives the model m1 and the leftmost column gives the model
m2. Positive values show that the model m1 produces maps with higher average
temperatures than the model m2.

###  4.2 Comparison of the local sky

The differences between the models are not just reflected in general
deviations of the temperature scales. They also show structural variations
that become noticeable by comparing specific sky regions, e.g. on/off the
Galactic plane. These variations influence the model comparison when confining
their output maps to the sky coverage of a specific radio-detection experiment
on Earth. The local sky of an observer changes with the local sidereal time
(LST) and solely depends on the observer’s geographic latitude if the
experiment is operated both day and night.

The average temperature of a local sky is obtained by converting a map into
the horizontal coordinate system with the two angles azimuth $\alpha$ and
altitude a (the elevation angle above horizontal), thus limiting it to the
visible half of the sky above the horizon and integrating over this region.
The results are then averaged for varying LST from $0\text{\,}\mathrm{h}$ to
$24\text{\,}\mathrm{h}$ as

$\begin{split}&T_{\text{local sky, average}}(\nu,l)=\\\
&\frac{1}{2\pi}\int_{$0\text{\,}\mathrm{h}$}^{$24\text{\,}\mathrm{h}$}\mathrm{d}t_{\text{LST}}\int_{0}^{\pi}\mathrm{d}a\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\mathrm{d}\alpha\cos{(a)}\;T(\nu,l,t_{\text{LST}};a,\alpha)\end{split}$
(7)

for a given latitude $l$ of the observer on Earth. We further integrate the
average temperature of the local sky over frequency:

$\Sigma T(l)=\int_{30\mathrm{MHz}}^{408\mathrm{MHz}}T_{\text{local sky,
average}}(\nu,l)\;\mathrm{d}\nu$ (8)

The result of this integral is plotted for each interpolation model as a
function of the observer’s latitude in Fig. 6 a). It is clear that $\Sigma T$
changes depending on the exposure to the radio-bright center of the Galaxy. At
the northern celestial pole ($90\text{\,}\degr$ latitude) this exposure is the
smallest and thus $\Sigma T$ is the lowest over all frequencies, while the
maximum exposure takes place at around $-60\text{\,}\degr$ of latitude.

Furthermore, differences between the interpolation models are recognizable.
Apart from a general shift in the temperature scale there are latitude
dependent variations, which are probably due to different ratios in the models
between the coldest and hottest regions of the sky.

Figure 6 b) shows the same results as in Fig. 6 a), but normalized to those of
the GSM2008 model, which on average gives the coldest sky maps. Here the level
of agreement is found to be $15\%$ in the worst case, when comparing to the
LFSM results. The latter model shows the largest deviation with respect to the
other ones in the same way as evident from the comparison in Table 2. The
level of agreement between the three remaining models for all observer
latitudes is better than $7\text{\,}\%$.

Figure 6: Values of $\Sigma T(l)$ for each sky interpolation models as a
function of latitude.

Figure 7: Values of $\Sigma T(l)$ for each sky interpolation models, same as
in Fig. 6 but normalized to the GSM curve.

##  5 Comparison for selected radio experiments

The comparison of the sky models based on the local sky is further carried out
for selected radio arrays for the detection of cosmic particles, namely RNO-G
(surface antennas) (Aguilar et al., 2021), LOFAR (Schellart, P. et al., 2013),
GRAND (Álvarez-Muñiz et al., 2020), SKA-low (Buitink et al., 2021), the Pierre
Auger Observatory (AERA (Huege, T., Aab, A., Abreu, P., et al., 2019) and the
AugerPrime Radio Detector (Pont, 2019)) and the radio antennas of the IceCube
surface array (Abbasi et al., 2021). The sky models compare differently for
each of the experiments. This is due to them being situated at different
geographical latitudes and using different frequency bands. For this study, we
make the simplifying assumption of a constant antenna sensitivity over the
whole frequency band for all arrays. Also, we further restrict the local sky
seen by the antennas to altitude angles between $15\text{\,}\degr$ and
$90\text{\,}\degr$ above the horizon to simulate a uniform gain pattern in
that range.

Subsequently, the average temperature of the local sky integrated over LST is
calculated analogous to Eq. 8 as

$\Sigma T(l_{\text{Exp}})=\int_{\nu_{\text{Exp, lower}}}^{\nu_{\text{Exp,
upper}}}T_{\text{local sky, $15\text{\,}\degr 90\text{\,}\degr$,
average}}(\nu,l_{\text{Exp}})\;\mathrm{d}\nu,$ (9)

in which the frequency band ($\nu_{\text{Exp, lower}}$, $\nu_{\text{Exp,
upper}}$) and the latitude ($l_{\text{Exp}}$) are adapted to the respective
experiment. Further, any two sky models $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$
are compared by their relative difference

$r_{\text{Exp; m}_{1}\text{, m}_{2}}=2\cdot\frac{\Sigma
T_{\text{m}_{1}}(l_{\text{Exp}})-\Sigma
T_{\text{m}_{2}}(l_{\text{Exp}})}{\Sigma
T_{\text{m}_{1}}(l_{\text{Exp}})+\Sigma T_{\text{m}_{2}}(l_{\text{Exp}})}.$
(10)

The largest relative differences obtained for each experiment are listed in
Table 3 together with the properties of the experiment and the combination of
sky models between which the largest relative difference occurs. These
differences range from ${\sim}$10\text{\,}\%$$ to ${\sim}$19\text{\,}\%$$.
When excluding LFSM, the differences are smaller than $10\text{\,}\%$ for all
experiments except for IceCube and the high band of LOFAR with a difference of
${\sim}$13\text{\,}\%$$ between the LFmap and GSM16 models.

An extended overview of the comparison results for each of the experiments is
given in appendix A.

Table 3: For each of the selected radio experiments the largest relative
deviation $\max(r_{\text{Exp; m}_{1}\text{, m}_{2}})$ between any two sky
models m1 and m2 is tabulated, while either including or excluding LFSM.

Experiment | $l_{\text{Exp}}$ | Frequency Band | $\max(r_{\text{Exp; m}_{1}\text{, m}_{2}})$ | Corresponding |  $\max(r_{\text{Exp; m}_{1}\text{, m}_{2}})$ (Excl. LFSM) | Corresponding
---|---|---|---|---|---|---
| (°) | (MHz) | (%) | Sky Models | (%) | Sky Models
RNO-G (1) | 72.58 |  $100\text{\,}\hfil\lx@intercol&408\text{\,}$ 17.1 | LFSM / GSM16 | 9.6 | LFmap / GSM16 |
LOFAR low (2) | 52.91 |  $30\text{\,}\hfil\lx@intercol&80\text{\,}$ 13.3 | LFSM / GSM | 8.9 | GSM / GSM16 |
LOFAR high (3) | 52.91 |  $110\text{\,}\hfil\lx@intercol&190\text{\,}$ 18.4 | LFSM / GSM16 | 12.8 | LFmap / GSM16 |
GRAND (4) | 42.93 |  $50\text{\,}\hfil\lx@intercol&200\text{\,}$ 16.3 | LFSM / GSM | 2.0 | LFmap / GSM |
SKA-low (5) | -26.70 |  $50\text{\,}\hfil\lx@intercol&350\text{\,}$ 13.6 | LFSM / GSM16 | 7.5 | LFmap / GSM16 |
Auger (6) | -35.21 |  $30\text{\,}\hfil\lx@intercol&80\text{\,}$ 10.5 | LFSM / GSM | 4.8 | LFmap / GSM |
IceCube (7) | -90.0 |  $70\text{\,}\hfil\lx@intercol&350\text{\,}$ 18.6 | LFSM / GSM16 | 13.6 | LFmap / GSM16 |

444Site locations and frequency bands of the experiments are quoted from
latest design plans. For RNO-G the LPDA surface antennas are considered, which
are sensitive up to ${\sim}$1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, but
the comparison is done only up to $408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$
because of the limitations of the LFmap and LFSM sky models. The complete
table can be found in appendix A.

(1) Aguilar et al. (2021); (2) Schellart, P. et al. (2013); (3) Nelles et al.
(2015); (4) Álvarez-Muñiz et al. (2020); (5) Buitink et al. (2021); (6) De
Jong (2021); (7) Abbasi et al. (2021).

##  6 Influence of radio emission from the quiet Sun

Another source of radio background in the sky is the Sun. Here we study the
influence of the quiet Sun on the average sky brightness, which describes the
continuous thermal emission of solar radiation. There is also a concept of the
active Sun, which includes enhanced emission during sunspot activity as well
as emission in the context of solar flares. These contributions take place on
limited timescales from seconds to hours and may significantly increase the
received power in an antenna (Kraus, 1986), but we do not cover them in this
study.

Radio emission of the quiet Sun at frequencies from tens of MHz to tens of GHz
is larger than expected from a $6000\text{\,}\mathrm{K}$ black body spectrum
and reaches brightness temperatures up to ${10}^{6}\text{\,}\mathrm{K}$ around
$50\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ (Kraus, 1986). To investigate how
large the influence of the Sun is, we generate sky maps with a disc of
constant brightness projected onto it. The disc matches the apparent size of
the Sun. Data on the brightness temperature of the quiet Sun as a function of
the frequency are taken from a summary of recent measurements given in (Zhang
et al., 2022).

With the superimposed quiet Sun the average sky temperature of the map is
calculated as before and its relative difference to the average sky
temperature of the unmodified map is determined. This is shown for each of the
four sky interpolation models in Fig. 8. While the influence is negligible at
the lowest frequencies it grows to a level of $\sim$8\text{\,}\%$$ at
$400\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$.

Furthermore, we adapt this procedure to the local skies of the selected radio
arrays, again restricting them to elevations from $15\text{\,}\degr$ to
$90\text{\,}\degr$ above the horizon and taking the mean of the average sky
temperatures over the course of $24\text{\,}\mathrm{h}$ of LST. The relative
difference caused by the Sun is shown for all arrays over their frequency
bands in Fig. 9. The arrays at the lowest frequencies are shown on the left
and the ones reaching to larger frequencies are shown on the right.

Analogue to Fig. 8 the relative differences increase with frequency, while
they are scaled up here because only a portion of the total sky is considered.
The differences are on the sub-percent level for Auger and the low band of
LOFAR and go up to $20\text{\,}\%$ for RNO-G at around
$400\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. Deviations between the arrays at
the same frequencies are attributable to the different exposure to the
Galactic center due to their geographical positions.

Figure 8: Influence of the Sun on the average sky temperature for each of the
models. The plot shows the relative difference of the average sky temperature
as a function of the frequency for when a Sun sized circle of the
corresponding brightness temperature (quiet Sun) is added to the maps. Figure
9: Relative difference of the average temperature of the local sky induced by
the quiet Sun for the selected radio arrays. The lines represent the average
results from using the four sky models to produce the maps, while the maximum
and minimum contribution from any model is shown by the colored bands. Arrays
at lower frequencies are shown in the left part, while arrays at higher
frequencies are shown on the right.

##  7 Discussion

This study gives estimates for the systematic uncertainty in predicting the
diffuse Galactic radio emission in the $30$ to
$408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ range. Moreover, it provides an
overview of the available models that are used for making these predictions
and their corresponding reference measurements. A considerable realization
from this overview is that probably a significant level of correlation exists
between the systematic uncertainties of the sky models and the reference maps.
Some reference maps served as calibrators for other maps and have thus an
increased weight in the sky interpolation models. In general, for the sky
models considered here, there is no substantial attention paid to weighting
the individual reference maps during the modeling process, which can lead to
biases on the absolute temperature scale. These biases are almost impossible
to untangle.

The generally small number of reference maps in the studied frequency range
further complicates the situation. While many surveys include a careful
treatment of their uncertainties, others lack this kind of thoroughness, which
in some cases may just be a consequence of the age of these surveys and the
unavailability of some tools and techniques at the time of their execution.

In the future, these difficulties can be coped with by including more and
newer reference maps into the sky interpolation models and weighting them
based on their inherited uncertainties. New sky surveys at MHz frequencies are
on the horizon or were recently published, including direct comparisons to
other maps or sky models (Eastwood et al., 2018; Kriele et al., 2022).

Regarding the application of the Galactic Calibration for radio arrays in
astroparticle physics, we argue that the viability of this method is given if
systematic uncertainties are carefully taken care of, which means that the
parameters of the considered radio array are important. From our studies, we
see that the Galactic Calibration exhibits larger uncertainties at higher
frequencies. There, the Galaxy’s contribution to the radio background
diminishes and sources like the Sun become stronger. In the case of an array
with a large frequency band, it might be a solution to conduct the Galactic
Calibration only with night-time measurements to minimize systematic
uncertainties.

##  8 Conclusion

We compare four interpolation models that produce full-sky maps of the
Galactic radio emission at frequencies between 30 and
$408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. The models partially rely on the
same reference measurements and differ to a greater or lesser extent by their
interpolation approach. A summary of the used reference maps shows relative
uncertainties on the temperature scales of up to $20\text{\,}\%$ and zero-
level errors that can be as large as up to ${\sim}$12\text{\,}\%$$ of the
average temperature of the sky at that frequency. In a global comparison, we
find that the predictions of the interpolation models all agree within
${\sim}$12\text{\,}\%$$, which we suggest as an estimate for the systematic
uncertainty of the model predictions. However, we would like to stress that
LFSM stands out in this comparison and the other three models agree
significantly better. This estimate for the systematic uncertainty of the
model prediction incorporates both the individual temperature scale
uncertainties of the reference maps and uncertainties because of different
modeling methods.

Furthermore, we compare the models based on the local sky at a given
experiment’s latitude on Earth. This step reveals further differences between
the models and in combination with specific frequency bands of selected radio-
detection arrays deviations vary considerably. Depending on the experiment,
the level of agreement lies between $10\text{\,}\%$ and $19\text{\,}\%$.
Again, these numbers decrease notably when excluding LFSM.

Additionally, we study the influence of the quiet Sun on the radio background
of the selected arrays and find it to be negligible at lower frequencies.
However, the contribution increases to ${\sim}$20\text{\,}\%$$ at around
$400\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$.

The relative uncertainties in sky temperature quoted above determine the
signal power measured in radio detection arrays for cosmic particles – whereas
the energy scale for the energy of the detected particles scales with the
electric field amplitudes, i.e., the square root of the signal power. The
relevant systematic uncertainty on the energy scale of particle detection thus
corresponds to approximately half of the values quoted here. These
uncertainties are typically competitive with those achieved with external
calibration sources – if not better – and thus confirm the value of the
Galactic Calibration approach. If models with even higher accuracy for the
Galactic emission in the frequency band from 30 to
$408\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ become available in the future,
radio detection arrays for cosmic particles will profit from these
retroactively.

###### Acknowledgements.

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##  Appendix A Complete model comparison for selected radio experiments

The results given in Table 3 are condensed to show the maximum deviation
between any to models for all selected radio arrays. The full comparison for
all models is given in Table 4 in which the values of
$r_{\text{m}_{1},\text{m}_{2}}$ are listed for all experiments and all
combination of models m1, m2.

Table 4: Values of $r_{\text{m}_{1},\text{m}_{2}}$ for each combination of the
interpolation models and for all selected radio arrays. LOFAR low
---
$r_{\text{m}_{1},\text{m}_{2}}$ (%) | LFmap | GSM | GSM16 | LFSM
LFmap | - | $3.5\text{\,}$ | $-5.5\text{\,}$ | $-9.8\text{\,}$
GSM | $-3.5\text{\,}$ | - | $-8.9\text{\,}$ | $-13.3\text{\,}$
GSM16 | $5.5\text{\,}$ | $8.9\text{\,}$ | - | $-4.3\text{\,}$
LFSM | $9.8\text{\,}$ | $13.3\text{\,}$ | $4.3\text{\,}$ | -
LOFAR high
$r_{\text{m}_{1},\text{m}_{2}}$ (%) | LFmap | GSM | GSM16 | LFSM
LFmap | - | $7.6\text{\,}$ | $12.8\text{\,}$ | $-5.6\text{\,}$
GSM | $-7.6\text{\,}$ | - | $5.2\text{\,}$ | $-13.2\text{\,}$
GSM16 | $-12.8\text{\,}$ | $-5.2\text{\,}$ | - | $-18.4\text{\,}$
LFSM | $5.6\text{\,}$ | $13.2\text{\,}$ | $18.4\text{\,}$ | -
GRAND
$r_{\text{m}_{1},\text{m}_{2}}$ (%) | LFmap | GSM | GSM16 | LFSM
LFmap | - | $2.0\text{\,}$ | $1.5\text{\,}$ | $-14.3\text{\,}$
GSM | $-2.0\text{\,}$ | - | $-0.5\text{\,}$ | $-16.3\text{\,}$
GSM16 | $-1.5\text{\,}$ | $0.5\text{\,}$ | - | $-15.8\text{\,}$
LFSM | $14.3\text{\,}$ | $16.3\text{\,}$ | $15.8\text{\,}$ | -
SKA-low
$r_{\text{m}_{1},\text{m}_{2}}$ (%) | LFmap | GSM | GSM16 | LFSM
LFmap | - | $5.2\text{\,}$ | $7.5\text{\,}$ | $-6.1\text{\,}$
LFmap | - | $5.2\text{\,}$ | $7.5\text{\,}$ | $-6.1\text{\,}$
GSM16 | $-7.5\text{\,}$ | $-2.3\text{\,}$ | - | $-13.6\text{\,}$
LFSM | $6.1\text{\,}$ | $11.3\text{\,}$ | $13.6\text{\,}$ | -
Auger
$r_{\text{m}_{1},\text{m}_{2}}$ (%) | LFmap | GSM | GSM16 | LFSM
LFmap | - | $4.8\text{\,}$ | $0.7\text{\,}$ | $-5.6\text{\,}$
GSM | $-4.8\text{\,}$ | - | $-4.2\text{\,}$ | $-10.5\text{\,}$
GSM16 | $-0.7\text{\,}$ | $4.2\text{\,}$ | - | $-6.3\text{\,}$
LFSM | $5.6\text{\,}$ | $10.5\text{\,}$ | $6.3\text{\,}$ | -
IceCube
$r_{\text{m}_{1},\text{m}_{2}}$ (%) | LFmap | GSM | GSM16 | LFSM
LFmap | - | $9.6\text{\,}$ | $13.6\text{\,}$ | $-5.0\text{\,}$
GSM | $-9.6\text{\,}$ | - | $4.0\text{\,}$ | $-14.6\text{\,}$
GSM16 | $-13.6\text{\,}$ | $-4.0\text{\,}$ | - | $-18.6\text{\,}$
LFSM | $5.0\text{\,}$ | $14.6\text{\,}$ | $18.6\text{\,}$ | -
RNO-G
$r_{\text{m}_{1},\text{m}_{2}}$ (%) | LFmap | GSM | GSM16 | LFSM
LFmap | - | $6.5\text{\,}$ | $9.6\text{\,}$ | $-7.6\text{\,}$
GSM | $-6.5\text{\,}$ | - | $3.1\text{\,}$ | $-14.0\text{\,}$
GSM16 | $-9.6\text{\,}$ | $-3.1\text{\,}$ | - | $-17.1\text{\,}$
LFSM | $7.6\text{\,}$ | $14.0\text{\,}$ | $17.1\text{\,}$ | -


# In-Context Prompt Editing for Conditional Audio Generation

###### Abstract

_Distributional shift_ is a central challenge in the deployment of machine
learning models as they can be ill-equipped for real-world data. This is
particularly evident in text-to-audio generation where the encoded
representations are easily undermined by unseen prompts, which leads to the
degradation of generated audio — the limited set of the text-audio pairs
remains inadequate for conditional audio generation in the wild as user
prompts are under-specified. In particular, we observe a consistent audio
quality degradation in generated audio samples with user prompts, as opposed
to training set prompts. To this end, we present a _retrieval-based in-context
prompt editing_ framework that leverages the training captions as
demonstrative exemplars to revisit the user prompts. We show that the
framework enhanced the audio quality across the set of collected user prompts,
which were edited with reference to the training captions as exemplars.

Index Terms—  text-to-audio generation, prompt engineering, distributional
drift

††∗ Equal contribution.

##  1 Introduction

Recently, there has been notable progress in the task of conditional text-to-
audio (TTA) generation, where audio signals can be synthesized from textual
descriptions [1, 2]. In most setups, text encoders model text prompts as
priors for audio decoders to condition upon, and rely heavily on the amount of
parallel text-audio data for generalizability. Consequently, TTA models’
adaptability is constrained to the training prompt distributions which were
accessible during training, and collecting data from all possible prompt
distributions is impractical.

Thus, one major limitation remains as the limited ability to generalize across
the distribution shift. This shift in text distribution in the wild diverges
from the training captions that the model has been trained on, resulting in an
inadequately equipped text encoder and leading to inaccuracies in representing
unseen textual inputs. These inaccuracies further cascade into errors during
the subsequent decoding inference steps, hindering the overall synthesis
quality. Empirically, we observe a marked audio quality degradation (see
Figure 1) when there is a distributional shift from the training prompt
distribution $P_{t}(x)$ to the user prompt distribution $P_{u}(x)$. The reason
is that the learned text representation $P(z|x;\theta)$ remains constant while
the acquired prior $\theta$ is unable to adapt to unseen distribution. Thus,
this tendency for models to have better audio quality within the training
prompt distribution hinders the ability of the model to be deployed in real-
world settings, as it is impossible to train the model on all possible data
distributions that it may encounter in the real world.

In this paper, we first discuss the distributional shift in deployed
conditional audio generation systems (Section 2). We observed that the shift
in prompts leads to lower audio quality measured in terms of FAD [3] and CLAP
[4]. To handle this shift, we propose to edit the user prompts with
instruction-tuned large language models (LLMs) (i.e., LLaMA 2 [5]). However,
using LLMs as-is results in ill-formulated prompting, which can lead to sub-
par performance [6, 7]. The use of _demonstrative exemplars_ for large
language models [8, 5] has recently been shown to bridge the gap between seen
and unseen prompts. To this end, we introduce a framework for LLM-based prompt
editing with demonstrative exemplars. To validate our approach, we conducted
extensive experiments on collected user prompts consisting of a range of free-
form entered texts.

We summarize our contributions as follows:

  * •

First, we put forward a way to quantify the distributional shift in prompts
with feature-based KL divergence reduction. We compute this distributional
“prompt divergence” and establish its correlation to the audio quality in
terms of FAD scores.

  * •

We adopted in-context learning from text-only usages to text-conditioned audio
generation, and show that in-context prompt editing enhances the audio quality
across a range of evaluation metrics, including CLAP, Fréchet audio distance
(FAD), and human evaluation.

  * •

We improved upon the computationally expensive of prompt retrieval from large-
scale datasets. This is achieved via de-duplication of the training prompts
with the minHash algorithm, then using the K-means clustering technique to
split prompts into groups for fast retrieval of relevant exemplars.

##  2 Preliminaries on Prompt Distribution

To model two text prompt distributions, we need to first project them into
features where additional metrics can be computed. Here we first denote the
training and user prompt distribution as $\\{P_{t},P_{u}\\}$ respectively, and
formalize the following feature extraction process: (1) Given a context (or
prompt) $x_{u}$ from user prompt data $D_{u}$, we retrieve a prompt $x_{t}$
from training distribution $P_{t}$, or joint distribution with language models
$P_{t}\cup P_{LM}$, which we will elaborate in Section 3. (2) Conditioned on
prompt $x_{t}$, we sample the latent representation $z\backsim f(x_{t})$ from
trained text encoder $f(\cdot)$.

Note that $f(\cdot)$ is a generalized text encoder, which can be any
pretrained text encoder such as RoBERTa [9], T5 [10], or CLAP [11]. As such,
the metric is suitable for any text encoder models, which makes the approach
rather generalized.

#### Distributional shift in prompts.

We propose to measure the KL reduction as it reflects the relative divergence
when user prompts are fed to the text encoder as opposed to the training set
prompts [12]. Instead of measuring divergence at the text level, we utilize
the encoded text’s feature distribution $Z$. Here we define the
Kullback–Leibler divergence ($\text{KL}(P||Q)$) between two encoded text
features $P(X)$ and $Q(X)$:

$\small\text{KL}(P||Q)=\frac{1}{|X|}\sum_{x\in X}P(x)\log\frac{P(x)}{Q(x)}.$
(1)

where each input $x$ consists of normalized scores bounded in $[0,1]$. The KL
value is then averaged across feature channels; and the prompt divergence
score $r_{div}$ is given as:

$\displaystyle r_{div}=\frac{1}{|Z|}\sum_{\hat{p}_{\text{train}}\in
Z,}\text{KL}(\hat{p}_{\text{train}}\|\hat{q}_{\text{user}})-\text{KL}(\hat{p}_{\text{train}}\|{p^{\prime}}_{\text{new}})$
(2)

where $\hat{p}_{\text{train}}$ is the original prompt feature distribution,
$\hat{q}_{\text{user}}$ is the converted prompt feature distribution, and
${p^{\prime}}_{\text{new}}$ is the empirical feature distribution of the
sampled prompt $x_{t}$. $Z$ is the set of extracted text features (or latent
code $z$) with text encoder $f(\cdot)$. We then examine the relationship
between the KL reduction induced by a specific prompt editing approach to the
resulting audio quality in terms of FAD scores in Figure 1. A smaller
$r_{div}$ value indicates that the prompt-induced audio distribution captures
the in-domain audio quality, under retrieved prompt distribution $P_{t}$.

Fig. 1:  Plot of average KL reduction on the n-gram feature space, defined as
how much the selected dataset reduces KL divergence to the target distribution
over just random sampling. The _retrieval_ uses the data samples from the
training prompt distribution, and the _user_ specifies the input from the user
prompt distribution. There is a strong correlation between KL reduction and
FAD reduction.  Fig. 2:  Diagram depicting the process of in-context prompt
editing for improved audio quality. Training set is first clustered via
K-means, then top-$M$ prompts are retrieved based on user queries, of which
the most similar prompt is then used as the exemplar for in-context prompt
editing with LLM. Prior to finding representative centroids, we apply de-
duplication to eliminate the nearly identical demonstrative examples in the
training set. This enable us to retain sufficient data to represent the data
distribution while improve retrieval efficiency.

##  3 In-Context Prompt Editing

Inspired by the recent successes in in-context learning [8], we formulate the
prompt editing process as follows: Let $x_{u}$ be a query input prompt,
written by the user, and consider $Y=\\{y_{1},\ldots,y_{m}\\}$ as the set of
refined audio samples. We edit $x_{u}$ by incorporating a task instruction $I$
and an in-context demonstration set $C=\\{c_{1},\ldots,c_{k}\\}$, which
consists of $k$ demonstration examples. Each $c_{i}$ is a caption retrieved
from training prompt set $D_{t}$, and the resultant in-context prompt is
formulated as $\tilde{x}=[I,C,x_{u}]$. We approximate the likelihood of the
audio $y_{j}$ being representative using a scoring function $f$ parameterized
by $\theta$ and applied to the entire input sequence:

$P(y_{j}|x_{u})\triangleq f_{\theta}(y_{j},\tilde{x};\theta)$ (3)

Rather than conditioning on an unseen user prompt, we draw the audio signal
from a surrogate distribution: $\hat{y}=\arg\max P_{y_{j}\in
Y}(y_{j}|\tilde{x};\theta)$. Given the challenges posed by distributional
shift arising from disparities between training and real distributions, we
present a framework for in-context prompt editing. The framework edit user
prompts into with demonstrative exemplars from the training prompt
distribution. Primarily, the process of editing in-context prompts based on a
collection of training prompts $D_{t}$ consists of two major steps:

  1. 1.

_De-duplication_ to improve retrieval efficiency, since the data $D_{t}$ can
be prohibitively large.

  2. 2.

_Retrieval of demonstrative exemplars_ for language model inference.

In what follows, we provide the details of these steps.

###  3.1 De-duplication

Retrieving prompts from large-scale datasets can result in resource-intensive
computations, especially when multiple pairs of similar documents are present
in the data. Thus, the goal of de-duplication is to eliminate duplicate or
nearly identical items from a large sample pool. To do so, we adopt MinHash
[13] for identifying demonstrative exemplars within the training dataset.
MinHash represents each document, denoted as $x_{i}$ and $x_{j}$, using sets
of $n$-grams, expressed as $d_{i}$ and $d_{j}$ respectively. The similarity
between these sets is measured using the Jaccard Index [14] to indicate the
overlap between the sets. We discard high Jaccard indexes which are highly
matched documents for similarity greater than $0.8$.

###  3.2 Retrieval of demonstrative exemplars

The retrieval process begins with K-means clustering using the Faiss library
[15] which is built for efficient similarity search. Each training text prompt
is projected into embeddings with the sentence encoder (SBERT [16]). For the
sake of ease of analysis, we use AudioCAPS [17] and BBC sounds [2] training
prompts as exemplars for in-context prompts in clustering.

For each user prompt $x_{u}$, we first perform similarity search with the
indexed clusters and obtain the top-$M$ closest neighbors –
$x_{1},x_{2},...,x_{k}$ – from the training set using the distances within the
sentence encoder’s embedding space. Utilizing these neighboring exemplars, the
in-context demonstration set $C$ is constructed, wherein each $x_{i}$
corresponds to $c_{i}$. We order the neighbors to satisfy $d(c_{1})\leq
d(c_{k})$ when $i<j$. This ranking provides a natural hierarchy of sentences
within the cluster, based on their contextual relevance to the user’s query.
Consequently, the top-$M$ candidate prompts are selected as illustrative
examples. We then structured the top candidate as in Figure 2.

Prompting Approach |  $\Delta r_{div}$ $\uparrow$ |  $\Delta$ CLAP $\uparrow$ |  $\Delta$ KL $\uparrow$ |  $\Delta$ FAD $\uparrow$ |  SUB $\uparrow$ |  OBJ $\uparrow$
---|---|---|---|---|---|---
User | - | - | - | - | 3.58 | 1.54
Random | 0.472 | 0.003 | 0.1013 | 0.590 | 3.65 | 2.56
LLM-only | 0.044 | 0.036 | 0.0649 | 0.943 | 3.61 | 2.58
exemplar (out-domain, K=100, random) | 0.444 | 0.019 | 0.0660 | 1.433 | 3.66 | 2.61
exemplar (in-domain, K=100, random) | 0.439 | 0.025 | 0.0469 | 1.803 | 3.72 | 2.62
exemplar (in-domain, K=100, farthest) | 0.464 | 0.046 | 0.0577 | 2.203 | 3.78 | 2.64
exemplar (in-domain, K=50, closest) | 0.520 | 0.042 | 0.0813 | 2.860 | 3.84 | 2.69
exemplar (in-domain, K=100, closest) | 0.594 | 0.047 | 0.1453 | 3.068 | 3.832 | 2.68
Table 1: The comparison between retrieval-based approaches and baseline TTA
generation models on the collected 1,525 user prompts (_User_). Other
approaches include random prompt retrieval from training set using user prompt
as queries (_Random_), and using LLM to edit the user prompt directly without
exemplars are demonstrations (_LLM_). The proposed approaches uses in-context
editing of prompt with exemplars from (1) _out-domain_ text drawn from
wiki-103 with fixed length window size of $10$, and from (2) _in-domain_
AudioCAPS and BBC sound prompts which AudioLDM learned from. We fix the
retrieved candidate to up to $K=100$ and experimented with various settings
within $K$.

##  4 Experimental Settings

We employ AudioLDM [2] to generate realistic speech and piano music audio
samples. We use LLaMa-70B [18] as the prompt editing models, which is a
decoder-only language model. We collected and evaluated our approaches $1525$
free-form user prompts (Open-prompts) as real-world test prompts; and
evaluated on AudioCAPS [17] to see if there is performance degradation if more
elaborate, expert-annotated prompts are used instead. No training was
performed except for the instruction-tuning of the large language models.
Audio samples are evaluated with CLAP [4] and FAD [3] for automatic text-audio
alignment and distance to clean audios respectively. Human evaluation was
performed in terms of subjective (SUB) and objective (OBJ) human evaluation
[2, 19] for audio quality assessment by five participants. Both SUB and OBJ
are rated on a scale of 5; and SUB is focused on audio quality and OBJ is
measured likewise on a scale of 5 for relevance to the edited prompts, where
scores are averaged over the participants.

Model | #T | TTR(%) |  $\Delta r_{div}$ $\uparrow$
---|---|---|---
Random | 148 | 9.56 | 0.474
Farthest | 147 | 9.61 | 0.422
Closest | 117 | 10.38 | 0.602

Table 2: #T denote as the number of tokens and TTR(%) refers to the type token
ratios and the prompt divergence ($\Delta r_{div}$).

##  5 Results and Analysis

We first demonstrate that retrieval approach (_exemplar_) synthesize better
quality audio samples than the original user prompt baseline (_User_) across
automatic metrics and human evaluation in Table 1, where consistent
improvement is observed across metrics, while $r_{div}$ is reduced with the
guidance of demonstration. To show that the improvement does not simply come
from LLM prompt editing, we also compare _exemplar (K=100, closest)_ with
_LLM_ , where text-audio alignment (CLAP) is increased by $+0.011$ and
distance to clean audios (FAD) is further improved by $+2.125$. Moreover, we
revisit the past hypothesis that the most similar exemplars are the best for
in-context editing by comparing _exemplar (K=100, farthest)_ and exemplar
(K=100, random), where exemplars are selected differently from the top-$M$
candidates. The closest _exemplars_ are more distinct examples with highest
token-type ratio in Table 2. Overall, we also found higher agreement of
exemplar-based editing as compared with other editing techniques. The greatest
audio improvement comes from the use of LLM editing with in-context learning,
even out-performing pure LLM technique by up to $+0.23$ in subjective
icashuman evaluation.

###  5.1 Correlation of prompt divergence with audio quality

Here we intend to show the impact of exemplars from the perspective of prompt
divergence metric in Table 1 and Figure 3. Primarily, the average KL reduction
is linearly proportional to the quality in terms of normalized FAD reduction
(N($\Delta$ FAD)), which measures the reduced distance to clean audio samples.
This allows us to deduce the usefulness of retrieved prompts as exemplars in
terms of LLM prompt editing. Further, we compared in- and out-domain prompts
in order to show that in-domain prompts as demonstrative exemplars are more
effective in driving up the audio qualities. We show in Table 1 this
comparison, and see that the domain relevance does indeed help in the editing
process, exhibiting a $+0.37$ improvement in FAD and the increases in both
human evaluations.

Fig. 3:  Plot of average KL reduction on the n-gram feature space, defined as
how much the retrieved prompt sets reduces KL divergence to the training
distribution. There is a strong correlation between KL reduction and the audio
quality in terms of FAD reduction.

###  5.2 On the inference efficiency

While in-context retrieval improves audio quality, one major concern remains
in terms of its efficiency since the method is employed at inference time. In
fact, we observe that on a Intel Xeon CPU with FAISS implementation, the
average search time for $K=100$ candidate is $2.13$ seconds; and only scale
approximately linearly with the total number of training samples. In general,
several factors including (1) number of clusters: a higher number of cluster
corresponds to a better performance, though the difference makes up a minor
degree in some instances, as the LLM editing is also crucial in this process.
(2) The number of retrieved candidates: since we re-compute the similarity of
each candidate with user query, the size of the pool will directly (linearly)
influence the speed of inference. (3) Size of dimension: The size of sentence
embedding is fixed at $384$ due to S-BERT.

##  6 Conclusions

In this work, we address the challenge of _distributional shift_ when the
text-to-audio generation models are conditioned on under-specified user
prompts. We propose to edit user prompts with demonstrative exemplars, where
training captions are used as demonstrations for the LLMs to better make the
edits. We observed consistent improvement in audio quality as the captions are
now closer in distribution to the training captions. Our approach is simple
and requires no retraining of models, and can be easily adopted to any text-
based audio pipelines.

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