# Atmospheric characterization of the super-Jupiter HIP 99770 b with KPIC

Yapeng Zhang 51 Pegasi b fellow  Department of Astronomy, California Institute
of Technology, Pasadena, CA 91125, USA  Jerry W. Xuan Department of Astronomy,
California Institute of Technology, Pasadena, CA 91125, USA  Dimitri Mawet
Department of Astronomy, California Institute of Technology, Pasadena, CA
91125, USA  Jason J. Wang Center for Interdisciplinary Exploration and
Research in Astrophysics (CIERA) and Department of Physics and Astronomy,
Northwestern University, Evanston, IL 60208, USA  Chih-Chun Hsu Center for
Interdisciplinary Exploration and Research in Astrophysics (CIERA) and
Department of Physics and Astronomy, Northwestern University, Evanston, IL
60208, USA  Jean-Bapiste Ruffio Department of Astronomy & Astrophysics,
University of California, San Diego, La Jolla, CA 92093, USA  Heather A.
Knutson Division of Geological & Planetary Sciences, California Institute of
Technology, Pasadena, CA 91125, USA  Julie Inglis Division of Geological &
Planetary Sciences, California Institute of Technology, Pasadena, CA 91125,
USA  Geoffrey A. Blake  Division of Geological & Planetary Sciences,
California Institute of Technology, Pasadena, CA 91125, USA  Yayaati Chachan
CITA National Fellow  Department of Physics and Trottier Space Institute,
McGill University, 3600 rue University, H3A 2T8 Montreal QC, Canada  Trottier
Institute for Research on Exoplanets (iREx), Université de Montréal, Canada
Katelyn Horstman Department of Astronomy, California Institute of Technology,
Pasadena, CA 91125, USA  Ashley Baker  Department of Astronomy, California
Institute of Technology, Pasadena, CA 91125, USA  Randall Bartos  Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Dr.,Pasadena, CA 91109, USA  Benjamin Calvin Department of Physics &
Astronomy, 430 Portola Plaza, University of California, Los Angeles, CA 90095,
USA  Sylvain Cetre  W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela,
HI, USA  Jacques-Robert Delorme W. M. Keck Observatory, 65-1120 Mamalahoa Hwy,
Kamuela, HI, USA  Greg Doppmann  W. M. Keck Observatory, 65-1120 Mamalahoa
Hwy, Kamuela, HI, USA  Daniel Echeverri Department of Astronomy, California
Institute of Technology, Pasadena, CA 91125, USA  Luke Finnerty Department of
Physics & Astronomy, 430 Portola Plaza, University of California, Los Angeles,
CA 90095, USA  Michael P. Fitzgerald Department of Physics & Astronomy, 430
Portola Plaza, University of California, Los Angeles, CA 90095, USA  Nemanja
Jovanovic Department of Astronomy, California Institute of Technology,
Pasadena, CA 91125, USA  Joshua Liberman Department of Astronomy, California
Institute of Technology, Pasadena, CA 91125, USA  James C. Wyant College of
Optical Sciences, University of Arizona, Meinel Building 1630 E. University
Blvd., Tucson, AZ. 85721  Ronald A. López Department of Physics & Astronomy,
430 Portola Plaza, University of California, Los Angeles, CA 90095, USA  Evan
Morris  Department of Astronomy & Astrophysics, University of California,
Santa Cruz, CA95064, USA  Jacklyn Pezzato  Department of Astronomy, California
Institute of Technology, Pasadena, CA 91125, USA  Ben Sappey Department of
Astronomy & Astrophysics, University of California, San Diego, La Jolla, CA
92093, USA  Tobias Schofield  Department of Astronomy, California Institute of
Technology, Pasadena, CA 91125, USA  Andrew Skemer  Department of Astronomy &
Astrophysics, University of California, Santa Cruz, CA95064, USA  J. Kent
Wallace Jet Propulsion Laboratory, California Institute of Technology, 4800
Oak Grove Dr.,Pasadena, CA 91109, USA  Ji Wang Department of Astronomy, The
Ohio State University, 100 W 18th Ave, Columbus, OH 43210 USA  Clarissa R. Do
Ó Department of Physics, University of California, San Diego, La Jolla, CA
92093, USA

###### Abstract

Young, self-luminous super-Jovian companions discovered by direct imaging
provide a challenging test of planet formation and evolution theories. By
spectroscopically characterizing the atmospheric compositions of these super-
Jupiters, we can constrain their formation histories. Here we present studies
of the recently discovered HIP 99770 b, a 16 $M_{\mathrm{Jup}}$ high-contrast
companion on a 17 au orbit, using the fiber-fed high-resolution spectrograph
KPIC ($\mathcal{R}\sim 35,000$) on the Keck II telescope. Our K-band
observations led to detections of H${}_{2}\textrm{O}$ and CO in the atmosphere
of HIP 99770 b. We carried out free retrieval analyses using petitRADTRANS to
measure its chemical abundances, including the metallicity and C/O ratio,
projected rotation velocity ($v\sin i$), and radial velocity (RV). We found
that the companion’s atmosphere has C/O $=0.55_{-0.04}^{+0.06}$ and [M/H]
$=0.26_{-0.23}^{+0.24}$ (1$\sigma$ confidence intervals), values consistent
with those of the Sun and with a companion formation via gravitational
instability or core accretion. The projected rotation velocity $v\sin(i)<7.8$
km s-1 is small relative to other directly imaged companions with similar
masses and ages. This may imply a near pole-on orientation or effective
magnetic braking by a circumplanetary disk. In addition, we added the
companion-to-primary relative RV measurement to the orbital fitting and
obtained updated constraints on orbital parameters. Detailed characterization
of super-Jovian companions within 20 au like HIP 99770 b is critical for
understanding the formation histories of this population.

††facilities: Keck (KPIC)††software:  numpy (Harris et al., 2020), scipy
(Virtanen et al., 2020), matplotlib (Hunter, 2007), astropy (Astropy
Collaboration et al., 2018), petitRADTRANS (Mollière et al., 2019),
PyMultiNest (Buchner et al., 2014), orvara (Brandt et al., 2021), corner
(Foreman-Mackey, 2016),

##  1 Introduction

Young, self-luminous super-Jovian companions discovered by direct imaging
surveys have posed significant challenges to planet formation theories. These
objects typically straddle the mass boundary between planets and brown dwarfs
($\sim$13 $M_{\mathrm{Jup}}$) and orbit their primary stars at large
separations ($>$10 au). Their formation mechanisms remain under debate as they
are not easily compatible with either in-situ core accretion or gravitational
instability (Pollack et al., 1996; Boss, 1997; Chabrier, 2003; Kratter &
Lodato, 2016). The long timescale of core accretion in the outer
protoplanetary disk makes it challenging to assemble massive cores to trigger
runaway accretion before the gas disk dissipates (Lambrechts & Johansen,
2012), while disk instability and cloud fragmentation are expected to form
more massive objects like brown dwarfs or stellar companions (Zhu et al.,
2012; Forgan & Rice, 2013; Offner et al., 2023). For some wide-orbit
companions at hundreds of au, formation at closer orbits followed by
scattering events may be effective (Veras et al., 2009). Previous studies on
orbital architectures of these super-Jupiters ($<20$ $M_{\mathrm{Jup}}$)
versus those of brown dwarfs ($>20$ $M_{\mathrm{Jup}}$) found evidence for
distinct distributions in their semimajor axes and orbital eccentricities
(Nielsen et al., 2019; Bowler et al., 2020; Nagpal et al., 2023; Do Ó et al.,
2023). This suggests that the two populations likely have distinct formation
mechanisms, but it is unclear whether there is a well-defined boundary
separating them.

Atmospheric characterization provides an important avenue to distinguish
between competing formation mechanisms for substellar companions. Elemental
abundances including the metallicty [M/H] and carbon-to-oxygen (C/O) ratios
have been suggested as probes for planet formation (e.g. Öberg et al., 2011;
Madhusudhan, 2012; Turrini et al., 2021). In general, formation via
gravitational instability or cloud fragmentation is thought to result in
compositions akin to stars, while core accretion can result in a variety of
compositions depending on the accretion mechanisms, formation location
relative to disk icelines, migration histories (e.g. Madhusudhan et al., 2014;
Mordasini et al., 2016; Bitsch et al., 2019), among other factors. Although
companions formed via gravitational collapse can also be enriched through late
accretion, the impact on their compositions is not expected to be significant
because of the high envelope masses (Inglis et al., 2024).

Super-Jupiters on wide orbits are excellent targets for directly probing
planetary emission at high signal-to-noise, therefore allowing for robust
constraints on their chemical compositions. As measurements of atmospheric C/O
and metallicity in directly imaged super-Jupiters have accumulated over the
past few years, we begin to build a sample large enough to test the
hypotheses. Spectral surveys of substellar companions with masses of $10-30$
$M_{\mathrm{Jup}}$ such as Hoch et al. (2023); Xuan et al. (2024a) suggest
that they display broadly solar compositions, which are chemically compatible
with formation via gravitational instability or core accretion beyond CO
icelines, where solids have approximately stellar C/O ratios. Although the
planetesimal accretion scenario is less likely for wide-orbit massive gas
giants, pebble accretion, whose efficiency increases in the outer disk, is not
precluded (Lambrechts & Johansen, 2012; Bitsch et al., 2015, 2019).

At an individual system level, linking the atmospheric composition of a planet
to its formation history is complicated by a chain of processes, including
dust and ice/vapor chemistry evolution in disks, planet migration, late solid
enrichment, and mixing of planet’s core and envelope (Mollière et al., 2022).
These convoluted processes make inferring a planet’s formation history from
its current atmosphere difficult. The challenging nature of this task calls
for combining different lines of evidence from atmospheric tracers such as
refractory-to-volatile ratios (Lothringer et al., 2021; Chachan et al., 2023)
and isotopologue ratios (Zhang et al., 2021a, b) as well as dynamical
indicators such as spin and orbital architecture (Bryan et al., 2018, 2020b;
Xuan et al., 2020). High-resolution spectroscopy of super-Jovian companions
provides information on these various aspects simultaneously, enabling a more
comprehensive understanding of the current state of planetary system and its
potential connection to formation pathways and evolutionary history.

Despite extensive studies of wide-orbit companions (e.g., Zhang et al., 2021b;
Palma-Bifani et al., 2023; Petrus et al., 2024; Inglis et al., 2024; Xuan et
al., 2024a), there are relatively few published spectral characterization of
companions with orbital separation within 20 au, which represents a critical
region of the parameter space for understanding population-level trends. Most
super-Jovian companions with compositional measurements are located beyond 50
au, with only a handful of exceptions such as $\beta$ Pic b (GRAVITY
Collaboration et al., 2020; Landman et al., 2024), HR 8799 cde (Konopacky et
al., 2013; Mollière et al., 2020; Ruffio et al., 2021; Wang et al., 2022;
Nasedkin et al., 2024), 51 Eri b (Brown-Sevilla et al., 2023; Whiteford et
al., 2023), and AF Lep b (Zhang et al., 2023; Palma-Bifani et al., 2024).
Planets with cooler temperatures and/or smaller separations are more
challenging to characterize because of the predominant stellar speckle
contamination. Techniques that combine both the spatial and spectral resolving
capability have proven to be powerful for enhancing the contrast limit of
detection and characterization (Snellen et al., 2015; Hoeijmakers et al.,
2018; Ruffio et al., 2023; Agrawal et al., 2023). New instruments coupling
adaptive optics systems with high-resolution spectrographs, such as the Keck
Planet Imager and Characterizer (KPIC, Mawet et al., 2017), HiRISE on VLT
(Vigan et al., 2023), and REACH on Subaru (Kotani et al., 2020), have recently
enabled high resolution spectroscopy of faint companions on close-in orbits
(e.g., HR 8799 planets; Wang et al., 2021).

The recently discovered 16 $M_{\mathrm{Jup}}$ HIP 99770 b is among only a
handful of super-Jovian companions with an orbital separation within 20 au
(Currie et al., 2023). Its current location is expected to be within the CO
iceline of the protoplanetary disk around A-type host stars (Qi et al., 2015).
In core-accretion scenarios, the C/O ratio likely varies as a function of
companion’s metallicity because of the partitioning of solid versus gas
compositions. Therefore, companions at smaller separations are likely to
display a larger variety of atmospheric compositions, providing intriguing
case studies for testing formation models by measuring both the C/O and
metallicity. In contrast, beyond the CO iceline, the companion’s C/O ratio is
expected to be solar for a wide range of metallicity because most volatiles
are condensed out. We note, however, if the companion has a solar metallicity,
it is difficult to distinguish between core accretion and gravitational
instability as both mechanisms can plausibly lead to this outcome, regardless
of the planet’s birth location.

In this paper, we present the atmospheric characterization of HIP 99770 b
using KPIC observations. When combined with its spin velocity and orbital
properties, this allows us to place constraints on its formation and
evolution. The paper is organized as follows. In Section 2, we introduce the
target system HIP 99770. The KPIC observations and data reduction are
explained in Section 3. Then, we carry out free retrieval analyses with the
observations. The forward model and retrieval framework are described in
Section 4. We present the molecular detections and retrieval results in
Section 5. We discuss the measured chemical abundances, spin velocity, and
their implications for planet formation in Section 6. Finally, we summarize
the findings in Section 7.

##  2 HIP 99770 system

HIP 99770 b is a recently discovered super-Jovian companion with a joint
direct imaging and precision astrometry detection (Currie et al., 2023). We
summarize the properties of the HIP 99770 system in Table 2. The 1.8
$M_{\odot}$ A-type primary star HIP 99770 has an effective temperature of
$\sim$8000 K, an age of 40-400 Myr, and a distance of 40.74 pc (Gaia
Collaboration et al., 2021; Currie et al., 2023). The primary star shows
evidence of Hipparcos-Gaia astrometric acceleration induced by an orbiting
companion, which was confirmed by direct imaging with the Subaru Coronagraphic
Extreme Adaptive Optics Project (SCExAO/CHARIS) and Keck/NIRC2 at an angular
separation of 0.44″ (Brandt, 2021; Currie et al., 2023). The combination of
the relative astrometry and acceleration leads to a dynamical mass of
$16.1^{+5.4}_{-5.0}$ $M_{\mathrm{Jup}}$ for the companion, or a companion-to-
star mass ratio of $\sim 8\times 10^{-3}$, at a semimajor axis of
$16.9^{+3.4}_{-1.9}$ au and an orbital eccentricity of $0.25^{+0.14}_{-0.16}$
(Currie et al., 2023). The companion’s luminosity and dynamical mass are
consistent with an age of 115-200 Myr.

Currie et al. (2023) carried out spectral characterization of HIP 99770 b
using a CHARIS JHK broadband spectrum at 1.16-2.37 $\mu$m and Keck/NIRC2 L
band photometry, which suggested a spectral type of L7-L9.5 near the L/T
transition, an effective temperature of $T_{\mathrm{eff}}$$\sim$ 1300-1600 K,
and a surface gravity of $\log g=$4-4.5. Evolutionary models (Baraffe et al.,
2003; Spiegel & Burrows, 2012) predict a radius of 1.1-1.2 $R_{\mathrm{Jup}}$
while the spectral analysis tends to imply a smaller radius of up to 1.05
$R_{\mathrm{Jup}}$ (Currie et al., 2023). A comparison to HR 8799 d’s spectral
shape shows that HIP 99770 b’s spectrum is less flat, indicating moderate
cloudiness.

Table 1: Properties of the HIP 99770 system

Property | Value | References
---|---|---
HIP 99770
$\alpha_{2000.0}$ | 20:14:32.032 | 1
$\delta_{2000.0}$ | +36:48:22.7 | 1
Distance (pc) | $40.74\pm 0.15$ | 1
$v_{\mathrm{sys}}$ (km s-1) | $-20.52\pm 0.40$ | 1
$v\sin i$ (km s-1) | $81.1\pm 1.5$ | 1
Age (Myr) | 40 (Argus membership) | 2
| 115-414 (astroseismology) | 2
Mass ($M_{\odot}$) | $1.85\pm 0.19$ | 2
SpT | A5-A6 | 3
$T_{\mathrm{eff}}$(K) | 8000 | 3
$K_{s}$ (mag) | $4.42\pm 0.02$ | 2
$L_{p}$ (mag) | $4.40\pm 0.05$ | 2
$\rm[C/H]$ | $0.18\pm 0.09$ | 4
$\rm[O/H]$ | $0.01\pm 0.09$ | 4
HIP 99770 b
SpT | L7-L9.5 | 2
$T_{\mathrm{eff}}$(K) | $1400^{+200}_{-150}$ | 2
$K_{s}$ (mag) | $15.66\pm 0.09$ | 2
$L_{p}$ (mag) | $14.52\pm 0.12$ | 2
$a$ (au) | $16.9^{+3.4}_{-1.9}$ | 2
eccentricity | $0.25^{+0.14}_{-0.16}$ | 2
Mass ($M_{\mathrm{Jup}}$) | $16.1^{+5.4}_{-5.0}$ | 2
Radius ($R_{\mathrm{Jup}}$) | $1.0-1.1$ | 2
$\log g$ (cgs) | $4.0-4.5$ | 2
$v\sin i$ (km s-1) | $3.5_{-1.9}^{+1.7}$ | 5
$\rm[M/H]$ | $0.26_{-0.23}^{+0.24}$ | 5
C/O | $0.55_{-0.04}^{+0.06}$ | 5

References. — (1) Gaia Collaboration et al. (2021), (2) Currie et al. (2023),
(3) Murphy & Paunzen (2017), (4) Erspamer & North (2003); Hinkel et al.
(2014), (5) this work.

##  3 Observations and Data Reduction

We observed the system with KPIC (Mawet et al., 2017; Delorme et al., 2021;
Echeverri et al., 2022) on UT 2023 June 16 and 21. KPIC uses a fiber injection
unit (FIU) located downstream of the Keck II adaptive optics system to inject
light into single-mode fibers, which are connected to the NIRSPEC high-
resolution ($\mathcal{R}\sim 35,000$) spectrograph (McLean et al., 1998;
Martin et al., 2018) to disperse light onto the detector. Among the four
illuminated fibers, one of them is aligned to the position of the companion to
obtain its spectra. The use of single-mode fibers aids in the suppression of
the stellar speckle and sky background, and ensure a stable Gaussian-like line
spread function (LSF) during observations. A more detailed description of the
instrument can be found in Delorme et al. (2021); Wang et al. (2021).

Table 2: KPIC Observations of HIP 99770 b

Date |  $t_{\mathrm{int}}$ (min) | Airmass | Throughput | S/N
---|---|---|---|---
2023-06-16 | 90 | 1.1 | $3.4\%$ | 1.0
2023-06-21 | 60 | 1.1 | $3.3\%$ | 0.8

Note. — The S/N per wavelength channel for the companion is estimated from the
residuals (see Fig. 1) that include both the background thermal noise and the
photon noise due to stellar speckles.

Our KPIC observations of the HIP 99770 b are summarized in Table 2. We used an
AB nodding scheme that alternately aligns the companion on one of two fibers
to facilitate sky/background subtraction. Eighteen frames of 300s exposure
time were recorded on June 16 and 11 exposures of 600s on June 21. The total
integration time on HIP 99770 b was 90 minutes and 60 minutes, respectively.
We also switched to the primary star once every hour for modeling telluric
absorption and instrument response. Each night we took short observations on
an M giant star, HIP 95771 and HIP 81497, respectively, which display deep and
narrow spectral lines, therefore convenient for the calibration of wavelength
solution.

We reduced the data using the public KPIC Data Reduction
Pipeline111https://github.com/kpicteam/kpic_pipeline, including steps of nod-
subtraction, bad pixel removal, spectral order tracing, optimal spectrum
extraction, and wavelength calibration. We refer readers to Wang et al. (2021)
for an in-depth description. The K-band observations span the wavelength range
of 1.9-2.5 $\mu$m with a spectral resolution of $\mathcal{R}\sim$35,000. We
focused our analysis on the three reddest spectral orders from 2.29 to 2.49
$\mu$m, covering the main near-infrared absorbing molecules such as CO and
H${}_{2}\textrm{O}$ in the companion’s atmosphere, as shown in Fig. 1.

As a result of the companion’s small separation (0.44″) and high contrast
($\Delta K_{\mathrm{mag}}=11.3$), the extracted spectra are dominated by the
diffracted starlight, which varies across wavelengths. Therefore, the
continuum of the companion’s spectrum cannot be recovered. Taking into account
the photon noise from the stellar speckles, we reach a signal-to-noise (S/N)
of $\sim$1 per pixel for the companion’s signal in 1.5-hour integration on the
first night and S/N $\sim$0.8 on the second night.

Figure 1: High-pass filtered K-band (2.29-2.49 µm) spectra of the HIP 99770
system taken with one of KPIC fibers. The observations are shown in black
lines. Overplotted are the best-fit full model as obtained with the retrieval
analysis. The spectrum is dominated by the stellar contribution with a small
fraction of emission from the companion, as shown in the offset pink lines. We
annotated the positions of absorption features from H${}_{2}\textrm{O}$ and CO
with short bars on the top of each panel. The residuals (data minus the full
model) are shown with scattered dots in gray.

##  4 Retrieval Analysis

###  4.1 Spectral Model of HIP 99770 b

The spectral model of the companion consists of three components, including
the temperature profile, the chemistry model, and the cloud model. The models
are set up as follows.

####  4.1.1 Temperature Model

We parametrize the temperature-pressure (T-P) profile using the gradients
$\Delta\ln T/\Delta\ln P$ across 7 pressure knots set at $10^{-5}$,
$10^{-3.5}$, $10^{-2}$, $10^{-1}$, $10^{0}$, $10^{1}$, and $10^{2}$ bar, and
one absolute temperature value, $T_{0}$, at 100 bar. This follows the approach
in Zhang et al. (2023); Xuan et al. (2024b). We set broad prior ranges with a
uniform distribution to ensure the flexibility of the temperature model (see
Table 3). The full T-P profile is determined by a spline interpolation onto 60
layers evenly spaced in log pressure between $10^{-5}$ and 100 bar.

####  4.1.2 Chemistry Model

We explore two different chemistry models: one a free retrieval and the other
a disequilibrium model. The disequilibrium chemistry model first assumes
chemical equilibrium to determine the chemical abundances at a given pressure,
temperature, carbon-to-oxygen ratio (C/O) and metallicity ([M/H]), by
interpolation of a precomputed table (Mollière et al., 2017, 2020). The model
then parameterizes the disequilibrium chemistry of major species (CO, CH4, and
H${}_{2}\textrm{O}$) by enforcing constant volume mixing ratios (VMRs) at
altitudes above a certain quenching pressure level ($P_{\mathrm{quench}}$),
which approximates the effect of rapid vertical mixing in disequilibrium
chemistry (Zahnle & Marley, 2014). The free chemistry model allows the
abundance of each chemical species to vary while assuming a vertically
constant profile. In the case of chemical quenching at a deep atmospheric
region below the photosphere, and modest vertical sampling of the atmosphere,
this constant VMR profile can be a reasonable approximation, which is
therefore expected to lead to similar retrieval results as the disequilibrium
chemistry model.

####  4.1.3 Cloud Model

We adopt the condensate cloud model from Ackerman & Marley (2001) with MgSiO3
as the cloud opacity source – the expected dominant cloud species in L dwarfs
(Cushing et al., 2006). The cloud is characterized by four parameters: the
mass fraction of the cloud species at the cloud base
$X_{0}^{\mathrm{MgSiO_{3}}}$, the settling parameter $f_{\mathrm{sed}}$
(controlling the thickness of the cloud above the cloud base), the vertical
eddy diffusion coefficient $K_{\mathrm{zz}}$ (effectively determining the
particle size), and the width of the log-normal particle size distribution
$\sigma_{g}$. Following Mollière et al. (2020), the location of the cloud base
$P_{\mathrm{base}}$ is determined by intersecting the condensation curve of
the cloud species with the T-P profile of the atmosphere.

###  4.2 Fringing removal

Inspecting the observational residuals, i.e., the observed spectra minus the
best-fit model as obtained through the retrieval analysis, we identified
sinusoidal fringing features in the data. These fringing signals are believed
to result from the entrance window of NIRSPEC and KPIC dichroics. Although the
fringing due to the NIRSPEC entrance window is static, the KPIC fringing
signal shows temporal variation depending on the angle of incidence into the
optics and the change of optical properties of the material with ambient
temperature (Finnerty et al., 2022). We found significant fringing in the
periodogram of the residual data with a period of $\sim 4.5$ Å originating
from the KPIC dichroics, especially in the second epoch. To model the
fringing, we used the function as described in Xuan et al. (2024b):

$t=\bigg{[}1+F\sin^{2}\bigg{(}\frac{2\pi nd}{\lambda}\bigg{)}\bigg{]}^{-1},$
(1)

where $n$ is the index of refraction of the material as a function of
wavelength $\lambda$. $F$ and $d$ are two free parameters determining the
amplitude and period of the fringing signal. We fit this function to the
observational residuals using least-squares optimization to obtain the optimal
fringing parameters in each spectral order and fiber. Then, we applied this
optimized fringing formula to the full spectral model during retrieval
analysis. We note that the fringing signals do not bias the measurement of
chemical composition as we obtained consistent results either with or without
the treatment of fringing.

###  4.3 Retrieval Framework

Given the model setup detailed in Section 4.1, we compute synthetic spectra of
the companion using the radiative transfer code petitRADTRANS (pRT, Mollière
et al., 2019). The model accounts for the Rayleigh scattering of H2 and He,
the collision-induced absorption of H2-H2 and H2-He, and the scattering and
absorption cross sections of crystalline, irregularly shaped MgSiO3 cloud
particles. We use the line-by-line mode of pRT to calculate the emission
spectra at high spectral resolution. To speed up the calculation, we
downsample the original opacity tables (with $\lambda/\Delta\lambda\sim
10^{6}$) by a factor of 3. This downsampling factor has been tested to ensure
that it does not bias the retrieval results (Zhang et al., 2021b; Xuan et al.,
2022). We include opacity from H${}_{2}\textrm{O}$ (Polyansky et al., 2018),
${}^{12}\textrm{CO}$, ${}^{13}\textrm{CO}$ (Li et al., 2015), CH4 (Hargreaves
et al., 2020), CO2 (Rothman et al., 2010), and NH3 (Coles et al., 2019) in our
model.

Subsequently, the synthetic high-resolution spectrum is radial-velocity (RV)
shifted by the systemic and barycentric velocity, and rotationally broadened
by $v\sin i$ using the method from Carvalho & Johns-Krull (2023). The model
spectrum at native resolution is then convolved with a Gaussian kernel and
binned to the wavelength grid of the observed spectrum to match the resolving
power of the instrument ($\lambda/\Delta\lambda\sim 35,000$). The width of the
LSF is measured from the point spread function of the stellar observations in
the cross-dispersion direction. We apply a scaling factor
$\mathcal{R}_{\mathrm{scaling}}$ to the width to account for potential
inaccuracy in the measured values. Then, the model spectrum is multiplied by
the telluric transmission and instrument response, which is obtained by
dividing the spectrum of the featureless A-type primary star by a PHOENIX
stellar model (Husser et al., 2013). As the spectrum is contaminated by low-
order stellar speckle noise, we discard the continuum information by high-pass
filtering of both the observation and model using a 150-pixel ($\sim$3 nm)
wide Gaussian filter. We note that the retrieval results are robust to the
choice of the kernel width.

To construct the full forward model and compare it directly to observations,
we took the linear combination of the companion’s component $\mathbf{m}_{p}$
with the observed stellar spectrum $\mathbf{m}_{\ast}$ following Ruffio et al.
(2019); Wang et al. (2021); Landman et al. (2024). The full model is
calculated as follows:

$\mathbf{M}=\mathbf{m}\,\mathbf{c}=\begin{bmatrix}\mathbf{m}_{\ast}&\mathbf{m}_{p}\end{bmatrix}\begin{bmatrix}\mathbf{c}_{\ast}\\\
\mathbf{c}_{p}\end{bmatrix},$ (2)

where the coefficients $\mathbf{c}$ of the two linear components can be
analytically marginalized by solving the linear equation

$\mathbf{d}=\mathbf{m}\,\mathbf{c}+\eta,$ (3)

with $\mathbf{d}$ being the observed spectrum and $\eta$ the random noise. The
least square solution for the linear equation can be found by solving

$\mathbf{c}^{T}\mathbf{m}^{T}\Sigma_{0}^{-1}\mathbf{m}=\mathbf{d}^{T}\Sigma_{0}^{-1}\mathbf{m},$
(4)

where $\Sigma_{0}$ is the covariance matrix with the diagonal items populated
by observation uncertainties. Here, we do not account for the non-diagonal
covariance matrix because of the low S/N of the observations. We test for the
effects of correlated errors by modeling them with Gaussian Processes (GP)
following de Regt et al. (2024), and find no difference in the constraints on
the free parameters (see Fig. 9 in Appendix). This demonstrates that the
uncorrelated noise dominates the uncertainty.

The log-likelihood of the model $\mathbf{M}$ is formulated as:

$\displaystyle\ln\mathcal{L}=$
$\displaystyle-\frac{1}{2}\bigg{[}N\ln(2\pi)+\ln(|\Sigma_{0}|)+N\ln(s^{2})$
(5)
$\displaystyle+\frac{1}{s^{2}}(\mathbf{d}-\mathbf{M})^{T}\Sigma_{0}^{-1}(\mathbf{d}-\mathbf{M})\bigg{]},$

where $N$ is the number of data points, and $s$ is an error bar inflation
parameter, which accounts for the underestimated uncertainties of the
observations. Following Ruffio et al. (2019), the optimal $s$ for each model
can be calculated by effectively scaling the reduced $\chi^{2}$ to 1:

$s^{2}=\frac{1}{N}(\mathbf{d}-\mathbf{M})^{T}\Sigma_{0}^{-1}(\mathbf{d}-\mathbf{M}).$
(6)

The error bar inflation parameter can essentially account for the model’s
systematic errors compared to the observations, which ensures that these
uncertainties are propagated to the posteriors of the free parameters. The
log-likelihood is evaluated in each spectral order and fiber before being
combined, allowing for different linear coefficients and error bar inflation
factors optimized for each order.

In total, the retrieval with disequilibrium chemistry model has 22 free
parameters and the free chemistry model has 23 free parameters, as summarized
in Table 3. For the Bayesian inference process, we use the nested sampling
tool PyMultiNest (Buchner et al., 2014), which is a Python wrapper of the
MultiNest method (Feroz et al., 2009). The retrievals are performed in
importance nested sampling mode with a constant efficiency of 5%. It uses 2000
live points to sample the parameter space and derives the posterior
distribution of free parameters.

Table 3: Priors and posteriors of the retrievals.

Parameter | Prior | Disequilibrium | Free
---|---|---|---
$M_{p}$ ($M_{\mathrm{Jup}}$) |  $\mathcal{N}$(16.1, 5.0) | $17.2\pm 4.2$ | $17.3\pm 4.2$
$R_{p}$ ($R_{\mathrm{Jup}}$) |  $\mathcal{U}$(0.7, 1.3) | $0.94_{-0.16}^{+0.21}$ | $0.93_{-0.15}^{+0.22}$
$v\sin i$ km s-1 |  $\mathcal{U}$(0.1, 20) | $3.5_{-1.9}^{+1.7}$ | $3.4_{-1.9}^{+1.7}$
RV km s-1 |  $\mathcal{U}$(-30, 30) | $17.9\pm 0.3$ | $17.9\pm 0.3$
$\epsilon_{\mathrm{limb}}$ |  $\mathcal{U}$(0, 1) | $0.5\pm 0.3$ | $0.5\pm 0.3$
$\mathcal{R}_{\mathrm{scaling}}$ |  $\mathcal{U}$(0.8, 1.2) | $0.96_{-0.10}^{+0.12}$ | $0.95_{-0.10}^{+0.12}$
$\rm[M/H]$ |  $\mathcal{U}$(-1.5, 1.5) | $0.26_{-0.23}^{+0.24}$ | $0.22\pm 0.23$
C/O |  $\mathcal{U}$(0.1, 1.5) | $0.55_{-0.04}^{+0.06}$ | $0.64\pm 0.05$
log$X^{\mathrm{H_{2}O}}$ |  $\mathcal{U}$(-12, -1) | - | $-2.47\pm 0.21$
log$X^{\mathrm{CO}}$ |  $\mathcal{U}$(-12, -1) | - | $-2.03\pm 0.23$
log$X^{\mathrm{CH_{4}}}$ |  $\mathcal{U}$(-12, -1) | - | $-5.2_{-4.4}^{+0.6}$
log$X^{\mathrm{NH_{3}}}$ |  $\mathcal{U}$(-12, -1) | - | $-7.5_{-2.9}^{+2.5}$
log$X^{\mathrm{{}^{13}CO}}$ |  $\mathcal{U}$(-12, -1) | - | $-4.5_{-4.2}^{+0.6}$
log(${}^{13}\textrm{CO}$/${}^{12}\textrm{CO}$) |  $\mathcal{U}$(-12, -1) | $-2.7_{-5.6}^{+0.8}$ | -
log$P_{\mathrm{quench}}$ (bar) |  $\mathcal{U}$(-5, 2) | $0.21_{-0.83}^{+1.06}$ | -
$T_{0}$ (K) |  $\mathcal{U}$(1500, 5000) | $3904\pm 785$ | $4094\pm 724$
$(d\ln T/d\ln P)_{0}$ |  $\mathcal{U}$(0.02, 0.05) | $0.035\pm 0.01$ | $0.035\pm 0.01$
$(d\ln T/d\ln P)_{1}$ |  $\mathcal{U}$(0.03, 0.07) | $0.05\pm 0.01$ | $0.05\pm 0.01$
$(d\ln T/d\ln P)_{2}$ |  $\mathcal{U}$(0, 0.15) | $0.07\pm 0.05$ | $0.08\pm 0.05$
$(d\ln T/d\ln P)_{3}$ |  $\mathcal{U}$(0, 0.5) | $0.22\pm 0.09$ | $0.29\pm 0.12$
$(d\ln T/d\ln P)_{4}$ |  $\mathcal{U}$(0., 0.5) | $0.12\pm 0.05$ | $0.13\pm 0.06$
$(d\ln T/d\ln P)_{5}$ |  $\mathcal{U}$(0., 0.5) | $0.22\pm 0.10$ | $0.24\pm 0.09$
$(d\ln T/d\ln P)_{6}$ |  $\mathcal{U}$(0, 0.5) | $0.20\pm 0.15$ | $0.17\pm 0.14$
log($X_{0}^{\mathrm{MgSiO_{3}}}$) |  $\mathcal{U}$(-12, -1) | $-7.4\pm 3.0$ | $-7.4\pm 3.0$
$f_{\mathrm{sed}}$ |  $\mathcal{U}$(0, 10) | $6.0\pm 2.5$ | $6.0\pm 2.5$
log($K_{\mathrm{zz}}$) |  $\mathcal{U}$(5, 13) | $6.0\pm 2.6$ | $6.0\pm 2.6$
$\sigma_{g}$ |  $\mathcal{U}$(1.05, 3) | $2.0\pm 0.6$ | $2.0\pm 0.6$

Note. — $\mathcal{U}$(a, b) represents a uniform distribution and
$\mathcal{N}$(a, b) represents a normal distribution. The last two columns
show posteriors with 1$\sigma$ uncertainties from the disequilibrium chemistry
model and free chemistry model.

##  5 Retrieval results

We jointly fit two epochs of KPIC observations by generating the model
spectrum and calculating the likelihood for each epoch and summing up the two
likelihood values. We show retrieval results for two model setups:
disequilibrium and free chemistry. The posterior distributions of the free
parameters in these models are summarized in Table 3. In general, these models
result in consistent constraints on the atmospheric properties. We also carry
out analyses for the two epochs independently. The retrieved parameters are
consistent within 1-2$\sigma$, as shown in Fig 8. The first epoch provides
slightly tighter constraints because of its better S/N (see Table 2).

###  5.1 Cross-correlation detection

We carried out a cross-correlation analysis to show the detection of molecules
in HIP 99770 b. The cross-correlation functions (CCF) were computed as

$\mathrm{CCF}(v)=\frac{1}{s^{2}}\mathbf{F}(v)^{T}\Sigma_{0}^{-1}\mathbf{R},$
(7)

where $\mathbf{R}$ is the observed spectra minus the best-fit disequilibrium
chemistry model with the abundance of a specific molecule being set to zero;
$\mathbf{F}$ is the molecular template which was computed by differencing the
best-fit model and the best-fit model without the contribution of that
molecule; $v$ is the radial velocity shift between the model and data. Both
$\mathbf{R}$ and $\mathbf{F}$ were high-pass filtered using a median filter
with a width of 100 pixels. Then, the CCFs of individual orders were combined
into a master CCF, as shown in Fig. 2. The noise of the CCF was estimated by
subtracting the model’s auto-correlation function (ACF) and taking the
standard deviation at $|v|>150$ km s-1. We detected H${}_{2}\textrm{O}$ and CO
with an S/N of 14 and 23, respectively. We checked other molecules, including
CH4, NH3, ${}^{13}\textrm{CO}$, H2S, and CO2, and found no significant
detections (see Fig.7 in Appendix).

Figure 2: Cross-correlation detection of H${}_{2}\textrm{O}$ and CO in KPIC
observations of HIP 99770 b. The cross-correlation functions (CCF) are scaled
by the noise to show the detection S/N on the y-axis. The dashed lines show
the auto-correlation function (ACF) of each molecular template.  Figure 3:
Retrieved temperature structure and chemical abundances of HIP 99770 b. Left
panel: best-fit T-P profile with 1$\sigma$, 2$\sigma$, and 3$\sigma$ envelope
are shown in blue. The red dashed line is self-consistent T-P profile of
$T\mathrm{eff}=1400$ K and $\log g=4.5$ from the Sonora model (Marley et al.,
2021). The blue dotted line represents the flux-weighted emission contribution
of the model. The gray dashed line shows the condensation curve of the MgSiO3
cloud. The middle panel shows retrieved volume mixing ratios (VMRs) of the
major molecules with 1$\sigma$ envelopes. The dashed lines show the results of
the disequilibrium chemistry model; the dotted lines represent the equilibrium
case when the chemical quenching is manually turned off. The error bars denote
the retrieved VMRs in the free chemistry model. The right panels compare the
retrieved C/O and metallicity of the disequilibrium and free chemistry models.
The free chemistry model retrieves a slightly higher C/O ratio is because of
the hidden oxygen in cloud condensates, such as enstatite.

###  5.2 C/O and Metallicity

The retrieved chemical abundances and metallicity [M/H] and C/O ratios using
two models are shown in Fig. 3. Using the disequilibrium chemistry model, we
obtained a C/O=$0.55_{-0.04}^{+0.06}$ and [M/H]=$0.26_{-0.23}^{+0.24}$
(1$\sigma$ confidence intervals). The empirical level of systematic
uncertainty of KPIC measurements is $\sim$ 20% for C/O ratio and 0.2 dex for
metallicity, as estimated from benchmark brown dwarf companions, HR 7672 B and
HD 4747 B (Wang et al., 2022; Xuan et al., 2022). These are roughly consistent
with our results on HIP 99770 b. The free chemistry model resulted in a
derived C/O of $0.64\pm 0.05$ and metallicity of $0.22\pm 0.23$, in line with
the disequilibrium chemistry model. In comparison, the free chemistry model
retrieved consistent VMRs for the major molecules as shown in Fig. 3. The
slightly different C/O ratio derived from the free chemistry model can be
attributed to the cold trapping of oxygen in clouds (Woitke et al., 2018).
Condensates such as MgSiO3 take a fraction of oxygen out of the gas phase. The
chemical model predicts a mass fraction of $\sim 2.4\times 10^{-3}$ for
MgSiO3, which was not constrained with our analysis (see Section 5.4). Taking
this amount of clouds into account, we estimated that the actual C/O ratio in
the free chemistry model would be $\sim 0.55$, consistent with the result from
the disequilibrium chemistry model.

Regarding Bayesian evidence, the disequilibrium chemistry model slightly
outperforms the free chemistry model at 3$\sigma$ significance (Benneke &
Seager, 2013). The values are summarized in Table 4. We found that the log
likelihood of the best-fit disequilibrium model is the same as the free
chemistry model. This suggests that both models fit the observations equally
well, and the slightly lower Bayesian evidence of the free chemistry model is
due to the increased number of free parameters. Additionally, we tested
imposing a Gaussian prior of $1.1\pm 0.1$ $R_{\mathrm{Jup}}$ on the radius of
the companion following the evolutionary model (Baraffe et al., 2003; Currie
et al., 2023) and found that the retrieval results remained entirely
consistent.

###  5.3 Disequilibrium Chemistry

The disequilibrium chemistry model retrieves a quench pressure of
$P_{\mathrm{quench}}\sim 0.2-22.4$ bar (1$\sigma$). This quench pressure is
needed in order to explain the observed under-abundance of CH4 relative to CO
in the atmosphere. In contrast, turning off chemical quenching results in a
transition from CO-dominated to a CH4-dominated composition at pressures below
0.1 bar given the T-P profile (see Fig.3). The KPIC observations do not
support this equilibrium case, as the VMR of CO in our free chemistry
retrieval is constrained to a higher value than the equilibrium prediction and
we obtain a 3$\sigma$ upper limit of $<4\times 10^{-5}$ on the VMR of CH4.

We re-ran the retrieval assuming chemical equilibrium without quenching and
compared the resulting fit to the nominal model with quenching. The
equilibrium retrieval converged to a more isothermal T-P profile (i.e., hotter
upper layers) in order to avoid the transition to a CH4-dominated atmosphere.
We calculated the Bayes factor for these two models and found that the nominal
disequilibrium chemistry model is favored over that for equilibrium at a
2.4$\sigma$ significance. Similar findings have been observed in several late
L-type super-Jupiters such as HR 8799 cde (e.g. Konopacky et al., 2013; Barman
et al., 2015; Mollière et al., 2020) and brown dwarfs (e.g. Xuan et al., 2022;
de Regt et al., 2024).

###  5.4 Clouds

The cloud-related parameters were not constrained in our retrieval analysis
(see Table 3 and Fig. 6). We calculated the Bayesian evidence for the clear
versus cloudy models and found the clear model is slightly preferred at
2.7$\sigma$ significance because of fewer free parameters. We retrieved
consistent chemical abundances from the cloudy and cloud-free models. While
clouds are expected in atmospheres of late L-type brown dwarfs (Burrows et
al., 2006), the reason for our data’s insensitivity to clouds may be twofold.
First, the limited S/N of the observations may not be sufficient to
distinguish the small differences in line contrasts caused by the cloud
opacity. Second, the silicate cloud base in our model is located at $\sim$10
bar, below the photosphere of the atmosphere in K band, as shown in Fig. 3.
Hence, the impact of cloud opacity is not significant at the pressure levels
probed by the high-resolution K-band observations (see also Zhang et al.,
2021b; Xuan et al., 2022; Landman et al., 2024; Inglis et al., 2024).

Figure 4: Comparison of the orbital fitting of the HIP 99770 system with and
without the relative RV measurement in orange and blue, respectively. The
corner plot shows posterior distributions of parameters including companion’s
mass $M_{\mathrm{sec}}$, semimajor axis $a$, eccentricity $e$, inclination
$i$, argument of periastron $\omega$, and longitude of ascending node
$\Omega$. The upper right panel shows the relative RV over time from random
draws of the orbital solutions in both fits. Adding our RV measurement to the
fitting helps improve constraints on $i$, $\omega$, and $\Omega$. The updated
1$\sigma$ constraints are labeled on top of each column.

###  5.5 Rotation and RV

The projected rotation velocity of HIP 99770 b is measured to be
$v\sin(i)=3.5_{-1.9}^{+1.7}$ km s-1 in our retrievals. However, considering
the spectral resolution of KPIC ($\mathcal{R}\sim 35,000$), the instrument
broadening dominates over the rotational broadening of the target, meaning
that the rotation is not resolved in our data. The likelihood of the best-fit
retrieval model shows no difference whether including spin as a free parameter
or not. Therefore, we report a 3$\sigma$ upper limit of $v\sin(i)<7.8$ km s-1
as derived from the posterior distribution. This is also consistent with the
spectral resolution of KPIC. Adopting an isotropic distribution of
inclination, we estimated its spin velocity to be $<10$ km s-1, which
corresponds to a rotation velocity versus break-up velocity
$v/v_{\mathrm{break}}\sim 0.06$. We further discuss this constraint in Section
6.1.

We measured a radial velocity of $-17.9\pm 0.3$ km s-1for HIP 99770 b. This
allows us to compute the relative RV between the star and companion and add to
the orbital constraints of the system. Since the A-type host star is almost
featureless in K band, we cannot measure the stellar RV in our KPIC data. The
stellar RV is not fully consistent in the literature, likely because the fast
rotation of the star ($\sim 80$ km s-1, Gaia Collaboration et al., 2021) makes
it challenging to measure the RV accurately. We adopted the RV measurement of
HIP 99770 from Gaia DR3 of $-20.5\pm 0.4$ km s-1 (Gaia Collaboration et al.,
2021). This led to a relative RV of $2.6\pm 0.5$ km s-1 between the companion
and primary. Adding this relative RV constraint, we carried out orbital
fitting using the orvara code (Brandt et al., 2021) following (Currie et al.,
2023) to refine the orbital properties of HIP 99770 b. We note that here we
used the default 1/$M_{\mathrm{sec}}$ prior for the secondary mass, which is
responsible for the lower $M_{\mathrm{sec}}$ ($\sim 13$ $M_{\mathrm{Jup}}$) as
was also mentioned in Currie et al. (2023). We compared the posteriors of the
orbital fitting with and without the relative RV measurement in Fig. 4.
Although the addition of relative RV did not significantly change the overall
orbital constraints, it assisted in ruling out some solutions and improving
the constraint on the argument of periastron $\omega$, longitude of ascending
node $\Omega$, and orbital inclination $i=152_{-11}^{+9}$ deg. Additional
epochs of RV measurements in five-year baseline will further strengthen the
constraining power on other parameters, such as the secondary mass and orbital
eccentricity.

##  6 Discussion

###  6.1 Rotation velocity

Putting our upper limit on rotation velocity into context, we note that the
spin of HIP 99770 b is at the low end compared to literature measurements of
other super-Jovian companions as shown in Fig. 5. Previous studies suggest
that younger companions generally have lower rotation rates, which are
expected to increase with age as their radii contract following angular
momentum conservation (Bryan et al., 2020a; Vos et al., 2020). In contrast,
field brown dwarfs display a larger scatter in spin (Hsu et al., 2021).
Although the age of HIP 99770 b is older than 40 Myr, its rotation velocity
(assuming isotropic inclination distribution) is comparable to that of very
young companions ($<10$ Myr) such as GQ Lup b (Schwarz et al., 2016), DH Tau
b, and HIP 79098 b (Xuan et al., 2024a), and is lower than those at older
ages. If the trend of increasing spin with increasing age holds, it may
indicate a nearly pole-on orientation for HIP 99770 b. If we assume that its
spin and orbit are aligned ($i\sim 152$), the spin velocity is $<15$ km s-1 or
$v/v_{\mathrm{break}}\sim 0.09$, making it more comparable to other super-
Jovian companions.

On the other hand, the slow spin may be indicative of effective magnetic
braking by the circumplanetary disk (CPD) (Batygin, 2018; Ginzburg & Chiang,
2020; Wang et al., 2021). Massive companions are expected to effectively
ionize the CPD and interactions with magnetic fields then act to spin down the
companions. Therefore, the slow spin of HIP 99770 b may indicate that it
hosted an unusually long-lived CPD and/or that it formed early via
gravitational instability, both of which allow for a longer time for the
companion to spin down. The large scatter in the measured spins of super-
Jupiter companions may represent the consequences of mixed formation pathways
and histories (Bryan et al., 2020a). Future rotation period measurements with
light curves will help break this degeneracy with inclination. More accurate
measurements of rotation velocities using high-resolution, large spectral
grasp instruments such as VLT/CRIRES+ and Keck/HISPEC will be essential for
understanding the population-level trend.

Figure 5: Rotation velocities (assuming isotropic inclination) as a fraction
of break-up velocities for super-Jovian companions. Our measurement of HIP
99770 b is denoted with a red diamond. The measurements of other companions
are adopted from the literature, including Bryan et al. (2020a); Wang et al.
(2021); Xuan et al. (2024a). The data are color-coded by the companion masses.
The blue data points show the spin of Jupiter and Saturn in the solar system.

###  6.2 Implication for Formation

We retrieve an atmospheric [M/H] $=0.26\pm 0.24$ and C/O $=0.55\pm 0.06$
(1$\sigma$ confidence intervals) for HIP 99770 b, consistent with the solar
values. Previous studies reported stellar abundances of [C/H]$\sim$0.18 $\pm$
0.09 and [O/H]$\sim$0.01 $\pm$ 0.09 (Erspamer & North, 2003; Hinkel et al.,
2014). This corresponds to a C/O ratio of $0.8_{-0.2}^{+0.3}$, roughly
consistent with solar to super-solar value of $0.59\pm 0.08$ (Asplund et al.,
2021). The uncertainties in the stellar composition result from the large
rotational broadening of the primary and the deficit of oxygen lines in its
spectrum. A more accurate stellar C/O constraint is needed in order to draw
unambiguous conclusion on the formation of the companion. For consistency, we
simply assume a solar composition for the primary star HIP 99770 in the
following discussion.

The companion’s current semimajor axis of 17 au is well within the typical
location of the CO iceline in protoplanetary disks around early-type stars (Qi
et al., 2015). This makes it an interesting case for testing formation models.
Given our constraints on its atmospheric composition, the companion is
compatible with a formation via either core accretion or gravitational
instability. Based on the static picture of chemistry in smoothed
protoplanetary disks (Öberg et al., 2011), the core accretion scenario is
expected to result in either: 1) a substellar C/O and metal-rich atmosphere
because of the accretion of oxygen-enhanced solids, 2) an elevated C/O
combined with substellar metallicity due to gas-dominated accretion, or 3) a
stellar C/O when the accreted material corresponds to an overall stellar
metallicity for the atmosphere. In practice, protoplanetary disks are not
static and have complicated substructures (Öberg et al., 2021). Dust drift and
chemical evolution of gas and dust can alter the distribution of disk
elemental abundances over time (Mollière et al., 2022). Planet formation
models coupled with disk chemistry evolution and dust migration are needed to
investigate these questions further. Qualitatively, disk observations support
the simplified picture that the gas-phase oxygen is strongly depleted as more
O-bearing species are locked in the ice phase beyond the snowline (Le Gal et
al., 2021; Bosman et al., 2021). Therefore, planets formed under such
conditions are expected to follow the scenarios outlined above.

In contrast, gravitational instability should lead to similar stellar and
companion atmospheres since it is thought to occur relatively early when the
disk is massive and the reservoir of solids is pristine (Schib et al., 2021).
Although the companion may undergo late enrichment by solid accretion after
the initial collapse, this would not significantly alter the composition due
to the high envelope mass; for example, to enhance the metallicity by one dex
would require accreting $\sim$2 $M_{\mathrm{Jup}}$ of solids (Inglis et al.,
2024). Similarly, we do not expect a large enrichment of metal in super-
Jupiter atmospheres in core accretion models. Therefore, the solar composition
of HIP 99770 b does not preclude either formation scenarios. If the companion
was born ex-situ, it may have formed in the outer disk followed by type II
migration (Dürmann & Kley, 2015; Robert et al., 2018). Such planet migration
could potentially boost the accretion of planetesimals, leading to a slight
metal enrichment, as we constrain for HIP 99770 b.

The chemical composition of HIP 99770 b also fits in the overall trend for
super-Jupiter companions. Xuan et al. (2024a) carried out retrieval analyses
in a sample of 10-30 $M_{\mathrm{Jup}}$ companions using KPIC observations.
They found the companions generally had solar compositions, hinting at
formation via gravitational collapse or plausible core accretion beyond CO
iceline. In contrast, lower mass ($<10$ $M_{\mathrm{Jup}}$) companions appear
to display a larger range in C/O and metallicity (e.g, Zhang et al., 2023;
Whiteford et al., 2023; Landman et al., 2024; Nasedkin et al., 2024). By
comparing the C/O ratio in a sample of directly imaged companions with
transiting exoplanets, Hoch et al. (2023) suggested that there are two
distinct populations with a boundary at $\sim$5 $M_{\mathrm{Jup}}$, implying a
transition of formation pathways near a mass range of
$5-10$$M_{\mathrm{Jup}}$. However, more work is needed to determine whether
there exists a clear mass boundary for distinct formation channels, as such
compositional measurements get challenging towards lower-mass and closer-in
companions. The comparison between directly imaged super-Jupiters and
transiting exoplanets with similar masses will be informative in
distinguishing whether these populations are separated by different formation
pathways or migration histories (Kempton & Knutson, 2024). It also calls for
the joint understanding of trends in the orbital architectures of these
systems (Nielsen et al., 2019; Bowler et al., 2020; Nagpal et al., 2023; Do Ó
et al., 2023) to unravel their formation.

##  7 Conclusion

We carried out detailed spectral characterization of the recently discovered
super-Jupiter companion HIP 99770 b using the fiber-fed high resolution
spectrograph KPIC ($\mathcal{R}\sim 35,000$). The K-band observations led to
clear detections of H${}_{2}\textrm{O}$ and CO in the atmosphere of HIP 99770
b. We used atmospheric retrievals to constrain its atmospheric composition,
including its C/O and metallicity, projected rotation velocity ($v\sin i$),
and radial velocity (RV).

  * •

We found the companion’s atmosphere has C/O $=0.55\pm 0.06$ and [M/H]
$=0.26\pm 0.24$ (1$\sigma$ confidence interval), which are consistent with
stellar values. It joins the ensemble of super-Jovian companions showing
broadly solar compositions. This is compatible with a formation via either
gravitational instability or core accretion.

  * •

We found that the ratio of CH4 to CO in the companion’s atmosphere is lower
than predicted by equilibrium chemistry models, and place a lower bound of 0.2
bars on the quench pressure.

  * •

Although the companion is expected to be cloudy, we found that our models do
not require clouds. This may be because of the limited S/N, or or because the
clouds are located below the photosphere for our $K$ band observations. The
presence or absence of clouds could be better constrained by expanding the
wavelength range of these observations, which would increase our sensitivity
to the wavelength-dependent scattering signal from clouds.

  * •

We added the companion-to-primary relative RV measurement to the orbital
fitting and obtained updated constraints on the companion’s orbital $i$,
$\omega$, and $\Omega$.

  * •

The projected rotation velocity $v\sin(i)<7.8$ km s-1 is small compared to
other directly imaged companions with similar ages and masses. This may
indicate a nearly pole-on orientation or effective magnetic braking by a
circumplanetary disk.

Located within 20 au and straddling the deuterium-burning mass boundary, HIP
99770 b represents an intriguing target to investigate the link between the
present atmosphere and formation history. The convoluted formation and
evolution processes make it challenging to retrace the origin of planets
unambiguously. Modeling studies on the formation of these well-characterized
super-Jovian companions are crucial for advancing our knowledge. In the
future, higher S/N spectroscopic observations will allow for tighter
constraints on the metallicity by probing elements beyond C and O. The
estimates of solid budgets will massively benefit the inference of formation
history (Lothringer et al., 2021; Chachan et al., 2023). Other formation
tracers, such as carbon isotope ratios, will help pin down the formation
location relative to the CO iceline, therefore allowing us to lift the
degeneracy between formation mechanisms (Zhang et al., 2021a). Detailed
characterization of similar companions will play an essential role in charting
the parameter space to understand the formation of the super-Jovian
population.

Y.Z. is thankful for support from the Heising-Simons Foundation 51 Pegasi b
Fellowship (grant #2023-4298). Y.C. acknowledges support from the Natural
Sciences and Engineering Research Council of Canada (NSERC) through the CITA
National Fellowship and the Trottier Space Institute through the TSI
Fellowship. Funding for KPIC has been provided by the California Institute of
Technology, the Jet Propulsion Laboratory, the Heising-Simons Foundation
(grants #2015-129, #2017-318, #2019-1312, #2023-4598), the Simons Foundation,
and the NSF under grant AST-1611623. The data presented herein were obtained
at the W. M. Keck Observatory, which is operated as a scientific partnership
among the California Institute of Technology, the University of California and
the National Aeronautics and Space Administration. The Observatory was made
possible by the generous financial support of the W. M. Keck Foundation. The
authors wish to recognize and acknowledge the very significant cultural role
and reverence that the summit of Mauna Kea has always had within the
indigenous Hawaiian community. We are most fortunate to have the opportunity
to conduct observations from this mountain. The computation was carried out on
the Caltech High Performance Cluster.

##  Appendix A Retrieval results

We summarize the Bayesian evidence of various retrieval models mentioned in
Section 5. Fig. 6 shows the posterior distributions of free parameters in the
baseline retrieval with disequilibrium chemistry model. In Table 4, we list
the Bayesian evidence of various retrieval models and their comparisons to the
baseline model. The cross-correlation functions of molecular non-detections
are shown in Fig.7. We carry out independent retrievals on two epochs of KIPC
data and find broadly consistent posterior constraints as shown in Fig. 8. We
also explore the effect of correlated noise on the retrieval results by
implementing Gaussian Processes. The corner plot of the posteriors is shown in
Fig.9.

Figure 6: Retrieved posterior distributions of free parameters and
temperature-pressure profile in disequilibrium chemistry model using KPIC
observations of HIP 99770 b.  Figure 7: CCFs of non-detected molecules
including ${}^{13}\textrm{CO}$, CH4, NH3, H2S, and CO2.  Table 4: Summary of
Bayesian evidence of various retrieval models

Model | $\ln Z$ | $\Delta\ln Z$ | Significance ($\sigma$)
---|---|---|---
Disequilibrium chemistry | -93748.5 | - | -
Free chemistry | -93751.8 | -3.3 | 3.1
Equilibrium chemistry | -93750.0 | -1.5 | 2.4
Disequilibrium chemistry, no cloud | -93746.2 | 2.3 | 2.7

Note. — $\Delta\ln Z$ represents the difference of Bayesian evidence $\ln Z$
of each alternative model compared to the nominal disequilibrium chemistry
model.

Figure 8: Comparison of independent retrieval analyses on two epochs of
observations. The results are consistent within 1-2$\sigma$.  Figure 9:
Retrieved posterior distributions of free parameters with Gaussian processes
(GP). GP_l is the length scale of GP in unit of km s-1, GP_a0 to GP_a11
represents the amplitude of GP (in unit of typical data uncertainties) for
each order in each fiber and epoch. The constraints are fully consistent with
the model without GP.

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# Region-Conditioned Orthogonal 3D U-Net
for Weather4Cast Competition

Taehyeon Kim Shinhwan Kang Hyeonjeong Shin Deukryeol Yoon
Seongha Eom Kijung Shin Se-Young Yun
KAIST AI
Seoul, Korea
{potter32, shinhwan.kang, hyeonjeong1<EMAIL_ADDRESS>
{doubleb, kijungs<EMAIL_ADDRESS>

###### Abstract

The Weather4Cast competition (hosted by NeurIPS 2022) required competitors to
predict super-resolution rain movies in various regions of Europe when low-
resolution satellite contexts covering wider regions are given. In this paper,
we show that a general baseline 3D U-Net can be significantly improved with
region-conditioned layers as well as orthogonality regularizations on
1$\times$1$\times$1 convolutional layers. Additionally, we facilitate the
generalization with a bag of training strategies: mixup data augmentation,
self-distillation, and feature-wise linear modulation (FiLM). Presented
modifications outperform the baseline algorithms (3D U-Net) by up to 19.54%
with less than 1% additional parameters, which won the 4th place in the core
test leaderboard.

##  1 Introduction

Precipitation forecasting is one of the most arduous problem in forecasting
the meteorological conditions such as air quality, solar, temperature, and
wind velocity. Accurate forecasting can prevent enormous economic and social
damages from a variety of applications: large-scale crop management,
autonomous driving systems, and air traffic control. While Numerical Weather
Prediction (NWP) is a general method for predicting the climate changes based
on the calculation of physics-based simulations, its performance for short-
term rain prediction (i.e., less than 6 hours) is still inaccurate despite
lots of computational efforts. Recently, deep learning techniques have
attracted huge attention from the weather research community for such short-
term precipitation forecasting [9, 10, 13, 14, 5]. Specifically, among these
techniques, Espeholt et al. [5] develop an end-to-end deep learning method
that outperforms High Resolution Rapid Refresh (HRRR) [3], which is the start-
of-the-art method used in United States.

Weather4Cast 2022 [6, 7] is a competition for designing the best deep
learning-based precipitation forecasting model where competitors attempted to
forecast super-resolution rainfall events for the next 8 hours at 15-minute
intervals from low-resolution satellite radiances over various regions in
Europe. For the stage2 task in which our team participated, the desired model
for competitors is to predict the 7 Europe regions across two years (2019 and
2020) when the training dataset is composed of spectral satellite imagery
which covers larger areas with low-resolutions having 11 input variables for
each pixel. While the satellite imagery data demands for the spatio-temporal
modelling, the studies for conventional methods are still under-explored for
being robust towards such spatio-temporal shifts.

To tackle the challenge of spatio-temporal shifts, we propose a Region
Conditioned Network (RCN) to inject the regional information into the output
of 3D residual U-Net’s encoder architecture, which is the variation of 3D
U-Net [4]. With given spectral satellite contexts of different regions, RCN
can extract the region-conditioned context and such contexts linearly modulate
the output of the 3D U-Net. In addition, by penalizing the orthogonality
regularization for the 1$\times$1$\times$1 convolutional layer, the network
can capture more fined-grained representations for the super-resolution
prediction so that it yields the better score. We also stabilize the training
from the spatio-temporal shifts of the dataset. Lastly, we adapt a bag of
training strategies such as mixup [15], self-distillation, and feature-wise
linear modulation (FiLM) [2, 11]. More precisely, we add FiLM layers to a
backbone model for fine-tuning the layers for each region of each year while
freezing other backbones except the FiLM layers. We provide more details about
RCN and training strategies in Section 3.

Our contributions can be summarized as follows:

  * •

Effective: We utilize two concepts: (1) region-conditioned network and (2)
orthogonality regularization on 1$\times$1$\times$1 convolutional layers. With
additional training strategies, our solution outperforms a baseline up to
19.54% with less than 1% additional parameters.

  * •

Applicable: Our approaches can be adapted to any other deep neural networks.

  * •

Reproducible: We provide our source code at [1].

##  2 Overview of Weather4Cast Challenge and Provided Data

###  2.1 Weather4Cast Challenge

The main objective of Weather4Cast competition is to predict future super-
resolution rainfall events (i.e., rain or no-rain) from lower-resolution
satellite radiance. In this competition, competitors are required to provide a
model predicting rainfall events until eight hours in 32 time slots for given
4 time slots of a proceeding hour. As the given data is composed of multiple
regions in Europe across two years, a key is to learn the robust model under
spatio-temporal shifts.

The challenge comprised two different tasks: (1) stage1: predicting 3
different regions for 1 year (2019) and (2) stage2: 7 different regions for 2
years (2019 and 2020). Additionally, the rain rate threshold for the latter
task is 0.2 while the former one is 0.0001. The solution for the stage2 task
can bring beneficial meteorological meanings while it is a harder challenge
which increases the sparsity of the rain events to be predicted. In this
paper, our solution targets for the stage2 task.

###  2.2 Dataset

The dataset is provided with satellite imagery including 11 observed physics-
information, positional information, and observed rainfall amounts. The
detailed explanations are as follows:

  * •

Regions: The dataset consists of 3 different regions for 1 year (2019) in
stage1, and it is extended to 7 regions for 2 years (2019 and 2020) in stage2.

  * •

Input variables: Each spectral satellite imagery include 11 variables which
are slightly noisy satellite radiances covering visible, water vapor, and
infrared bands: IR_016, IR_039, IR_087, IR_097, IR_108, IR_120, IR_134,
VIS006, VIS008, WV_062, and WV_073. Detailed information for each context is
not provided.

  * •

Sequential information: Each input image covers 15 minutes where each pixel
corresponds to 12km $\times$ 12km area while each pixel for the output
indicates 2km $\times$ 2km area.

  * •

Rainfall amount: Pixel-wise rainfall information is provided as a float value.
The precipitation ratio is in Table 1.

  * •

Static information: Metadata contains the information of latitude, longitude,
and height for each pixel.

Table 1: Statistics over different regions: boxi0015, boxi0034, and boxi0076. Region | boxi0015 | boxi0034 | boxi0076
---|---|---|---
No-rain | 0.810 | 0.810 | 0.892
Rain | 0.190 | 0.190 | 0.108

###  2.3 Evaluation Metrics

We evaluate the predictive performance in terms of Critical Success Index
(CSI) score [12], F1-score, accuracy, and Intersection over Union (IoU). In
particular, CSI-score is the common evaluation metric in precipitation
forecasting. It is the total number of correct event forecasts divided by the
sum of the total number of storm forecasts and the number of misses, i.e.,

$\textsc{CSI}=\frac{\text{TP}}{\text{TP}+\text{FN}+\text{FP}}$ (1)

where TP, FN, and FP are true positive, false negative, and false positive,
respectively.

Figure 1: An overview of our solution in the task of predicting the 7 regions
at 2019 and 2020.

##  3 Method

This section provides the solution of KAIST AI. Overall training consists of 3
steps: (1) train the backbone (Residual U-Net) with orthogonal
1$\times$1$\times$1 convolutional layers as well as region-conditioned, (2)
fine-tune the backbone with FiLM Transfer approach [2] for each region of a
certain year, and (3) predict the output via thresholding (Figure 1).

###  3.1 Baseline: 3D U-Net

We choose the baseline model as a 3D U-Net [4], which utilizes the same layers
with the convolutional encoder-decoder architecture for the volumetric
segmentation task, on the region ‘boxi0015’, ‘boxi0034’, ‘boxi0076’ in 2019
with DiceBCEloss [16]. As Table 2 shows, the baseline performance can be
improved more as the batch size increases. Interestingly, such baseline models
make fairly accurate predictions for different regions even without the use of
region information as an input. However, to make super-resolution predictions
more accurate, conditioning for regions is needed during the propagation.

Table 2: A preliminary survey of the 3D U-Net architecture on the validation dataset. We set the regions to ‘boxi0015’, ‘boxi0034’, and ‘boxi0076’ in 2019. Batch size | CSI-score | F1-score | Accuracy | IoU | Loss | Precision | Recall
---|---|---|---|---|---|---|---
16 | 0.3130 | 0.4668 | 0.7302 | 0.3130 | 0.7670 | 0.3310 | 0.8321
32 | 0.3221 | 0.4767 | 0.7499 | 0.3221 | 0.7638 | 0.3457 | 0.8043
48 | 0.3303 | 0.4934 | 0.7348 | 0.3303 | 0.7629 | 0.3454 | 0.8790
Figure 2: An overview of our modified U-Net architecture. Each blue box
corresponds to the residual convolutional unit and each green block denotes
the residual transposed convolutional unit. During the propagation, the
region-conditioned context is added to the last output of the encoder while
the shortcut from the encoder unit to the corresponding decoder unit is
transformed with orthogonal 3D 1$\times$1$\times$1 convolutional opertors as
well as FiLM layer. The arrow denotes the propagation of a multi-channel
feature map.

###  3.2 Region Conditioning

To inject region information into feature maps during the propagation, we
propose a new Region Conditioned Network (RCN) to generate region-conditioned
context (Figure 2). RCN is a method of adding an auxiliary region conditioner
by using two layered fully-connected networks with a ReLU activation function.
Here, we transform the region categorical variables into one-hot vector.
Because the given dataset is comprised of satellite contexts for 7 different
regions in Europe, the length of one-hot vector is 7. We extract the region-
conditioned contexts including scale $\mathbf{\gamma}$ vector and bias
$\mathbf{\beta}$ vector as an output of RCN with a categorical input and
formulate the feature map as follows:

$\begin{gathered}\mathbf{\gamma},\mathbf{\beta}\leftarrow\textsc{Region
Conditioned Network}(\mathbf{x_{r}})\\\
\mathbf{\tilde{x_{r}}}\leftarrow\mathbf{\gamma}\odot\mathbf{x_{r}}+\mathbf{\beta}\end{gathered}$
(2)

where $\mathbf{x_{r}}$ is the last representation output of the encoder
architecture. For the detailed computation of the $\mathbf{\tilde{x_{r}}}$,
$\gamma$ is element-wisely multiplied with $\mathbf{x_{r}}$ in a pointwise
manner ($\odot$), and $\beta$ is added similarly.

###  3.3 Orthogonal 1$\times$1$\times$1 Convolution and Residual Unit

Orthogonal convolutional kernel is a class of the advanced normalization
techniques to preserve the magnitude of the propagation signal as well as to
reduce the redundant features in the filter response. Because there is the
difference of resolution size between input and output, it is needed to
capture more fine-grained features from the latent representations. Inspired
by the orthogonal concept, we alleviate such issue by adding
1$\times$1$\times$1 convolution into the path from the encoder block to the
corresponding decoder block (Figure 2) and making those 1$\times$1$\times$1
convolutions soft orthogonal with the orthogoanlity regularization, termed as
Spectral Restricted Isometry Property+ (SRIP+) referring to as Kim and Yun
[8], as follows:

$\frac{\lambda}{|\mathcal{W}|}\sum_{\mathbf{W}\in\mathcal{W}}\sigma(\mathbf{W^{\top}W}-\mathbf{I}_{n})$
(3)

where $\mathbf{W}$ is a weight matrix of 1$\times$1$\times$1 convolutional
kernel, $\mathcal{W}$ is a set of 1$\times$1$\times$1 convolutional kernel’s
weight matrices, and $\sigma(\mathbf{W^{\top}W}-\mathbf{I}_{n})$ is the output
of the power method.

$\begin{gathered}u\leftarrow(\mathbf{W^{\top}W}-\mathbf{I}_{n})v,v\leftarrow(\mathbf{W^{\top}W}-\mathbf{I}_{n})u,\\\
\sigma(\mathbf{W^{\top}W}-\mathbf{I}_{n})\leftarrow\frac{\|v\|}{\|u\|}.\end{gathered}$
(4)

where the vector $v\in\mathbf{R}^{n}$ is randomly initialized with normal
distribution. The key difference from Kim and Yun [8] is whether the dimension
of the penalized weight is 5D convolution or 4D convolution. Although we can
not quantitatively/qualitatively confirm the visible latent feature-map
changes through orthogonality, we observe the improvement in CSI-score
performance in the validation set.

#### Residual Unit.

We design a 3D-Residual U-Net, which is a variant of the baseline 3D U-Net [4]
(Figure 2). The main difference between the baseline and ours is the block
type. We make a shortcut for each encoder and decoder block while an
1$\times$1$\times$1 convolutional layer is added if there is the difference of
number of channels between input and output.

###  3.4 Data Augmentation: Mixup

Since satellite imagery datasets rarely contain rainfall data, compared to the
abundant non-rainfall data, the model is easily biased towards the majority
class, i.e., non-rainfall. To mitigate the bias on the majority class and
encourage elaborated classification on the minority class, we applied Mixup
[15], a popular data augmentation technique. Mixup regularizes neural networks
by utilizing the convex combination of training data without large
computational overhead, and the effectiveness is proved over various image
classification tasks and semantic segmentation tasks [15]. Generally, mixup is
utilized on 4D datasets, i.e., ($x_{i}$, $y_{i}$) $\in\mathbb{R}^{B\times
C\times H\times W}$ where $B$ is batch size, $C$ is the number of channels,
$H$ is the height, and $W$ is the width, while it is under-explored on 5D
dataset which time-dimension is added. We formulate the augmented training
data in the same manner for the general Mixup, and the details are as follows:

$\displaystyle\tilde{x}=\lambda x_{i}+(1-\lambda)x_{j},$
$\displaystyle\tilde{y}=\lambda y_{i}+(1-\lambda)y_{j},$

where ($x_{i}$, $y_{i}$) $\in\mathbb{R}^{B\times C\times T\times H\times W}$
are training data and ground truth target, ($x_{j}$, $y_{j}$) are randomly
shuffled data of ($x_{i}$, $y_{i}$), and $\lambda\sim$ Beta($\alpha$,
$\alpha$). We fixed $\alpha=1$ after exploring some values. Interestingly, as
seen in Table 3, the model utilizing mixup achieved the visible performance
improvement in terms of F1-score, CSI-score, and IoU.

Table 3: Comparison of baseline and model utilizing mixup on the validation dataset. Methods | CSI-score | F1-score | Accuracy | IoU | Loss | Precision | Recall
---|---|---|---|---|---|---|---
Baseline (best) | 0.3303 | 0.4934 | 0.7348 | 0.3303 | 0.7629 | 0.3454 | 0.8790
Mixup | 0.3699 | 0.5397 | 0.7340 | 0.3699 | 0.8024 | 0.3877 | 0.8926

###  3.5 Self-Distillation

After training the region-conditioned backbone, it is re-trained with self-
distillation, without the supervision of ground-truth labels. Since the
ground-truth observation is sparse and noisy, its training is unstable as well
as over-confident. To release this concern, we apply the self-distillation
loss which is a well-known technique for smoothing the loss landscape to lead
to a flatten optima.

###  3.6 Feature-wise Linear Modulation (FiLM) Layer for further Fine-Tuning

As the last step, the self-distilled model is fine-tuned with each regional
data for each year (i.e., we have the 7$\times$2 architectures). Because the
pre-trained model can be under-performed for a specific region (or year) while
it captures the general features for all regions, the pre-trained model needs
further updates for personalization. For fine-tuning, we use the FiLM Transfer
(FiT) inspired by [2], which fixes the pre-trained backbone and fine-tunes
only FiLM adapter layers (Figure 2). We initialize the new learnable
parameters for linear modulation: scale $\mathbf{\gamma_{f}}$ and bias
$\mathbf{\beta_{f}}$ and modify the latent representations of the shortcut
path from encoder to decoder during fine-tuning:

$\begin{gathered}\mathbf{\tilde{x_{r}}}\leftarrow\mathbf{\gamma_{f}}\odot\mathbf{x_{r}}+\mathbf{\beta_{f}}\end{gathered}$
(5)

where the detailed computation of the $\mathbf{\tilde{x_{r}}}$ is the same
with Equation (2). Here, we do not use an auxiliary network like RCN, but
directly use scale and bias parameters as learnable parameters.

###  3.7 Thresholding

We generate the optimal precipitation output by controlling the threshold for
each region across a year. Generally, the model decides positive rain if the
corresponding probability is over 0.5. However, because all regions have
different precipitation distributions, i.e., different precipitation scales,
this characteristic leads to a sub-optimal due to different scales of
certainty on each region, even after the FiLM Transfer. Thresholding approach
is a ubiquitous technique to tune the prediction in a post-processing manner
[5]. More precisely, given $p\in[0,1]$, the points with probability higher
than $p$ is decided as positive rain. We explore the best threshold with
fixing bin into 0.1, i.e., $p=0.1,0.2,\cdots,0.9$. As a result, the relaxed
threshold enhances the rain generalization across several regions, and the
best performance can be achieved with the combination of thresholds
$p=0.1,p=0.2,$ and $p=0.4$ for different regions. Especially, ‘boxi0076’
region, which has little precipitation observations, is significantly
improved.

###  3.8 Training Details and Leaderboard Results

Table 4 includes the value of hyperparameters used in this work. Through the
combination of our proposed methods, we can achieve more generalized score
(Table 5).

Table 4: Hyperparameter settings.

Hyperparameter | Optimizer | Learning rate | Maximum epoch | Dropout rate | Patience | Batch Size
---|---|---|---|---|---|---
Value | AdamW | 1e-4 | 90 | 0.4 | 40 | 56

Table 5: The leaderboard score (i.e., IoU) of our solution for stage2 compared
to the baseline score submitted by the organizer.

Task | Region | boxi0015 | boxi0034 | boxi0076 | roxi0004
---|---|---|---|---|---
Year | 2019 | 2020 | 2019 | 2020 | 2019 | 2020 | 2019 | 2020
Core Test | Baseline | 0.223 | 0.163 | 0.149 | 0.298 | 0.268 | 0.070 | 0.204 | 0.296
Ours | 0.270 | 0.237 | 0.200 | 0.362 | 0.294 | 0.104 | 0.234 | 0.321
Core Heldout | Baseline | 0.299 | 0.193 | 0.243 | 0.238 | 0.155 | 0.369 | 0.272 | 0.251
Ours | 0.328 | 0.210 | 0.279 | 0.237 | 0.166 | 0.328 | 0.294 | 0.311
Task | Region | roxi0005 | roxi0006 | roxi0007 | Overall
Year | 2019 | 2020 | 2019 | 2020 | 2019 | 2020
Core Test | Baseline | 0.240 | 0.228 | 0.341 | 0.274 | 0.327 | 0.089 | 0.226
Ours | 0.281 | 0.272 | 0.384 | 0.336 | 0.361 | 0.133 | 0.271
Core Heldout | Baseline | 0.274 | 0.299 | 0.325 | 0.403 | 0.245 | 0.005 | 0.255
Ours | 0.287 | 0.318 | 0.389 | 0.417 | 0.248 | 0.028 | 0.274

##  4 Conclusion

In this work, we propose the region-conditioned orthogonal residual U-Net
model for precipitation forecasting. The performance of the proposed model
outperforms the 3D U-Net model by up to $19.54\%$. Our contributions can be
folded into three perspectives. Firstly, we renovate the original 3D U-Net
with effective techniques, that are region-conditioned layers and
orthogonality regularization on 1$\times$1$\times$1 convolutional layers.
Next, the generalization capability of the model can be enhanced by utilizing
common training techniques such as mixup data augmentation, self-distillation,
and feature-wise linear modulation. Lastly, our solution can be easily
reproduced, and thus our repository can offer a great entry point to
facilitate future developments in rainfall forecasting for data scientists.
These interesting approaches are not limited to 3D U-Net, but could be applied
to other architectures designed for precipitation forecasting, such as
ConvLSTM and MetNet.

## Acknowledgement

This work was supported by the Korea Meteorological Administration Research
and Development Program "Development of AI techniques for Weather Forecasting"
under Grant (KMA2021-00121) and Institute of Information & communications
Technology Planning & Evaluation (IITP) grant funded by the Korea
government(MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School
Program(KAIST)).

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