# Observational Evidence for a Thick Disk of Dark Molecular Gas in the Outer
Galaxy

Michael P. Busch Department of Physics and Astronomy, Johns Hopkins
University, 3400 North Charles Street, Baltimore, MD 21218, USA  NSF GRFP
Fellow  LSSTc DSFP Fellow  Philip D. Engelke Department of Physics and
Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD
21218, USA  Frontier Technology Inc., 100 Cummings Center Suite 450G, Beverly,
MA, 01915, USA  Ronald J. Allen  Department of Physics and Astronomy, Johns
Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA  Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
David E. Hogg  National Radio Astronomy Observatory, 520 Edgemont Road,
Charlottesville, VA 22903, USA

###### Abstract

We present the serendipitous discovery of an extremely broad ($\Delta
V_{LSR}\sim 150$ ${\rm km\;s^{-1}}$), faint ($T_{mb}<10\textrm{mK}$), and
ubiquitous 1667 and 1665 MHz ground-state thermal OH emission towards the 2nd
quadrant of the outer Galaxy ($R_{gal}$ > 8 kpc) with the Green Bank
Telescope. Originally discovered in 2015, we describe the redundant
experimental, observational, and data quality tests of this result over the
last five years. The longitude-velocity distribution of the emission
unambiguously suggests large-scale Galactic structure. We observe a smooth
distribution of OH in radial velocity that is morphologically similar to the
HI radial velocity distribution in the outer Galaxy, showing that molecular
gas is significantly more extended in the outer Galaxy than previously
expected. Our results imply the existence of a thick ($-200<z<200$ pc) disk of
diffuse ($n_{H_{2}}$ $\sim$ 5 $\times$ 10-3 cm-3) molecular gas in the Outer
Galaxy previously undetected in all-sky 12CO surveys.

Galaxy: disk — ISM: molecules — ISM: structure — local interstellar matter —
radio lines: ISM — surveys

††software: Astropy (The Astropy Collaboration et al., 2013), dustmaps (M.
Green, 2018), GBTIDL (Garwood et al., 2006)

††thanks: To whom correspondence should be addressed<EMAIL_ADDRESS>Deceased as of the date of publication.

##  1 Introduction

Molecular Hydrogen (H2) is the most abundant molecule in the Universe (Draine,
2011). It is important for cooling of the interstellar medium and thus
regulates star-formation in galaxies (Glover & Abel, 2008). It is a catalyst
for interstellar chemistry. H2 can be observed using the higher J rotational
states in warm molecular gas (Goldsmith et al., 2010), but these require
energies usually not available in cold molecular gas, where the bulk of the H2
lies. This means that despite the importance of H2 to most areas of
astrophysics, it remains difficult to study.

The carbon monoxide (CO) molecule has been used historically to trace the
overall distribution of molecular gas in our Galaxy and other galaxies
(Bolatto et al., 2013), due to its substantial relative brightness and
abundance. Surveys for the lowest rotational transition of 12CO (J=1-0) at 2.6
mm have made invaluable contributions to our knowledge of the overall
distribution, morphology and mass content of molecular gas (Heyer & Dame,
2015).

There remain difficulties in using CO as a tracer for the total H2 content.
The CO line is commonly optically thick, and thus an uncertain and indirect
conversion factor, X(CO), is used to convert between a CO line intensity to a
H2 column density. Secondly, a growing body of observational research using
indirect total gas tracers (Dust, $\gamma$-ray) have shown that there exists a
significant portion of H2 in diffuse regions that are not observed in CO
surveys (Blitz et al., 1990; Grenier et al., 2005; Ackermann et al., 2010;
Planck Collaboration et al., 2011). Wolfire et al. (2010) coined the term
‘dark molecular gas’ to describe the gas not observed by CO, and theorized
that up to 30% of H2 can be missed by CO in the surface layers of molecular
clouds.

The physical conditions of the dark gas have been mysterious for much of the
past decade, but recent research has shown that the missing portion of gas is
not predominately explained by optically thick HI (Murray et al., 2018), which
was a promising possible avenue for explaining the dark gas (Fukui et al.,
2015). The indirect tracers of gas content such as dust emission and
$\gamma$-rays are suggestive that the content is in fact molecular gas not
traced by CO, but only a molecular tracer could provide such confirmation.

Additionally, these tracers cannot provide radial velocity measurements and
hence we lack structure information about the distribution of the dark gas in
the plane of the Galaxy. Using a molecule, and particularly for chemical
reasons, an abundant hydride (Neufeld & Wolfire, 2016) such as CH (Weselak et
al., 2010; Xu & Li, 2016; Jacob et al., 2019, 2020) or OH (Weinreb et al.,
1963, 1965; Heiles, 1968; Dickey et al., 1981; Wannier et al., 1993; Allen et
al., 2012; Dawson et al., 2014; Allen et al., 2015; Xu et al., 2016; Engelke &
Allen, 2018; Nguyen et al., 2018; Rugel et al., 2018; Engelke & Allen, 2019;
Busch et al., 2019) can provide both the unambiguous conclusion that the dark
gas is indeed diffuse molecular gas, and the radial velocity information from
a spectral line to describe the Galactic distribution of the gas by utilizing
the velocity field of the Galaxy (Brand & Blitz, 1993) Studies of the large-
scale distribution of gas have been completed mainly using HI (Kalberla et
al., 2005; Peek et al., 2011; Winkel et al., 2015) or CO (Heithausen et al.,
1993; Heyer et al., 1998; Dame et al., 2001; Heyer & Dame, 2015), and thus the
diffuse (or “dark“) molecular gas, and its mass content and Galactic
structure, are inherently excluded by these studies (Blitz et al., 1990;
Goldsmith et al., 2010; Li et al., 2015; Xu et al., 2016; Li et al., 2018).

The situation also has been uncertain in the outer Galaxy, where an analysis
of the diffuse Galactic $\gamma$-ray distribution from the Fermi LAT satellite
has suggested that “vast amounts of missing gas in the outer Galaxy are also
possible” (Ackermann et al., 2010). This is called the ‘CR (cosmic ray)
gradient problem‘, as the observed decline of CR emissivity is inconsistent
with CR propagation models. The presence of a warm ($T\sim$ 100-200 K),
extremely diffuse ($n\sim$ 5-20 cm-3) phase of molecular gas at large galactic
radius was suggested by Papadopoulos et al. (2002) as an explanation for
several observational facts.

These findings led us to the conclusion that an additional large-scale
molecular gas tracer is needed besides CO, one that could provide an
independent estimate of molecular gas content without being subject to the
same set of properties and deficiencies as signals from the CO molecule.
Desirable features of such a reliable, alternative molecular tracer would
include being optically thin, and containing low critical densities for
excitation, so as to better detect lower density, more diffuse molecular gas
not traced by CO. Such a tracer would provide information on column density,
structure, and kinematics, as well as possibly other physical properties of
the gas (Busch et al., 2019; Engelke et al., 2020; Petzler et al., 2020).

The OH molecule, first observed in radio by Weinreb et al. (1963), is one
promising alternative molecular tracer. It produces four optically thin
spectral lines centered near 18 cm, with critical densities three orders of
magnitude below those of the CO(1-0) transition. As such, OH 18 cm emission
lines provide a tracer for molecular gas including diffuse, low density
regions, and can yield direct calculation of column densities using equations
of radiative transfer.

Since 2012, we have demonstrated in a series of papers Allen et al. (2012,
2013, 2015); Busch et al. (2019) that OH successfully functions as an
alternate tracer for molecular gas in observations towards a quiescent region
of the Outer Galaxy; further observations in the vicinity of the W5 star-
forming region extend the use of OH as an alternate tracer to star-forming
regions and further develop the techniques of the field Engelke & Allen (2018,
2019); Engelke et al. (2020). These projects confirmed the viability of OH as
a tracer for molecular gas and studied the structure, column density, volume
density, and kinematics of molecular gas in these regions.

We proposed another project in 2014 for the Robert C. Byrd Green Bank
Telescope (GBT), aimed specifically at the Perseus Arm towards the Outer
Galaxy, with the intention of analyzing several outstanding questions in the
field of Galactic structure from the standpoint of molecular gas as traced by
OH. During the course of this observing run, an observation at
$l=108.0^{\circ}$, $b=3.0^{\circ}$ performed by RJA in September 2015 and
reduced and analyzed by PDE, showed evidence of a broad, low bump-like feature
between the narrower, taller features associated with the Perseus Arm and
other clumps of molecular gas, and most notably visible near $\sim-95$ km/s.
The feature did not go away when baseline fitting procedures were performed,
and at first we contemplated the possibility that the feature was an
observation of the Outer Arm. However, the feature was not amenable to
Gaussian fits, and comparisons to individual peaks in the HI spectrum at that
coordinate, which typically contain corresponding features, did not match. The
initial conclusions were that the spectral feature was not Gaussian in shape,
and did not correspond to any specific HI peaks, or that there was a source of
error in the analysis. More observations were planned and carried out on the
coordinate to improve the signal-to-noise on the detection during the fall of
2015 as well as observations at adjacent coordinates to see if the feature
appeared in them or not. The result was an unambiguous broad feature between
the spectral peaks, which continued to evade subtraction by baseline fitting
procedures. At this point we hypothesized that the feature could be a baseline
artifact resulting from reflections in the structure of the GBT.

Further work was carried out beginning in 2018, when MPB and RJA performed
observations with the 20m telescope at the Green Bank Observatory (GBO) 111The
Green Bank Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc. at the
same coordinates to determine whether the broad spectral feature was an
artifact of the GBT or a real astronomical detection, and we developed several
redundant methods to analyze the baselines attempting to demonstrate that the
feature could be an artifact. Nevertheless, these observations and analyses
have bolstered the evidence for a detection of a broad, diffuse OH signal in
between the arms of the Galaxy. From here, we performed new observations at
several other pointings in the vicinity to map out the extent of this broad
Galactic feature in Galactic longitude; the full results of that survey will
appear in a follow-up paper.

In this paper, we present the serendipitous discovery of extremely broad,
faint OH emission towards the Outer Galaxy, seemingly underlying the narrower
($\Delta V_{LSR}=$ 10 ${\rm km\;s^{-1}}$), brighter ($T_{mb}>$ 5mK) line
emission from discrete molecular clouds in spiral arms. The velocity extent
and morphology suggest that the H2 content traced by the OH is in the form of
extremely diffuse molecular gas in the geometry of a thick disk. We compute
the profile integrals in HI and OH and transform those to column densities
under a number of assumptions. We then compute the abundance ratio OH/HI, and,
using dust reddening as a calibration for the total column density of gas in
this direction, we calculate the abundance ratio of OH/H2. Under several
assumptions, we also calculate preliminary mass estimates.

##  2 Observations and Data

In this study, we present observations of the OH molecule at the main line
transitions of 1667 and 1665 MHz using two telescopes, the 100m Green Bank
Telescope and the 20m Telescope, at the GBO. We utilize the “SFD“ (Schlegel,
Finkbeiner, Davis) dust map (Schlegel et al., 1997; Schlafly & Finkbeiner,
2010) through a query from the Python package dustmaps (M. Green, 2018).

###  2.1 100m Green Bank Telescope

We have observed the ground state, $\Lambda$-doubling OH emission lines at 18
cm with the 100m Robert C. Byrd Green Bank Telescope, located in Green Bank,
West Virginia, with multiple observational projects over the past five years
in order to probe the characteristics of the OH emission from diffuse
molecular gas in the outer Galaxy. The project IDs: AGBT14B_031A,
AGBT14B_031B, AGBT15B_004, and AGBT19A_453 were all observed for various OH
emission projects in the outer Galaxy, but later were reanalyzed to probe the
robustness of the result presented. The first profile showing the broad
emission feature was under Project ID AGBT15B_004 at $l=108^{\circ}$,
$b=3^{\circ}$, and was thought to be a ‘baseline feature‘ that was difficult
to subtract.

A follow-up Project, AGBT19A_484, was undertaken during the summer of 2019
with the GBT in order to directly observe the source geometry of the extremely
broad OH emission as a function of Galactic longitude and latitude for the
first time. These observations were completed by using frequency-switching by
2 MHz centered on 1617, 1665, and 1720 MHz (O’Neil, 2002) using the VEGAS
spectrometer in the L-band range (Prestage et al., 2015). The observations
cover all four ground-state OH emission lines, although the 1612 band is
commonly unusable due to radio frequency interference (RFI). The beam size of
the GBT at 18 cm is 7.6’.

###  2.2 20m GBO Telescope

After an initial discovery in 2015 in a GBT spectrum towards
$l=108.000^{\circ},b=3.000^{\circ}$, we hypothesized that the broad signal was
a faint baseline artifact caused by multipath reflections in the structure of
the GBT itself and endeavored to prove that the OH signal was erroneous. This
type of instrumental artifact was discovered during the commissioning of the
GBT (Fisher et al., 2003), where ripples affect the Y linear polarization more
than the X linear polarization due to the geometry of the telescope structure.

To test this hypothesis we observed the same position with the 20m telescope
at the GBO, and an additional position separated by a degree on the sky. The
20m telescope has completely different structural reflections from the GBT due
to its geometry. The detection of the broad, faint OH emission with two
different telescopes eliminated the possibility that the broad OH signal was a
systematic inherent in the 100m dish of the GBT.

We observed the extended OH emission with the 20m telescope at the Green Bank
Observatory during the summer of 2018. Observing time on the 20m telescope for
this project was purchased using funds from the Director’s Discretionary funds
at Space Telescope Science Institute (STScI). The telescope operated in the
L-band (1.3-1.8 GHz) range and provided two spectral windows with one centered
on the 1667 and 1665 MHz lines, and the other on the 1720 MHz satellite line.
The beam size of the 20m at 18 cm was 45’. While the observations with the GBT
used frequency switching On/Off observations, the stability of the 20m
receiver allowed for total power observations.

###  2.3 Dust and HI

To trace the total gas column density $N_{H}$, we use the all-sky dust
reddening map from Schlegel et al. (1997); Schlafly & Finkbeiner (2010); the
well-known “SFD“ dust map. We integrate the entire sightline towards the outer
Galaxy, hence the more updated 3D dust maps from Green et al. (2017) are not
an appropriate estimation of the total gas column due to the lack of stellar
measurements a few kpc outside the solar circle. Therefore, we opt to use the
2D SFD dust map. This map has an angular resolution of 6’.1, which is
comparable to the GBT beam ($\sim$ 7’.8). We query the 2D SFD dustmap for the
coordinates $l=108.000^{\circ},b=3.000^{\circ}$, which returns a dust
reddening value, E(B-V) of 1.86 mag.

We also use HI data to estimate the atomic hydrogen column density N(H). We
use the HI profile towards $l=108.000^{\circ},b=3.000^{\circ}$ from the
Leiden-Argentine-Bonn (LAB) survey Kalberla et al. (2005) and HI4PI database
(Ben Bekhti et al., 2016). Additionally, for an estimate of the optical depth
of HI towards this sightline, needed for an accurate estimate of N(H), we use
two HI absorption-emission pairs from Strasser et al. (2007), to calculate the
median HI optical depth along the entire line of sight.

##  3 Data Reduction

The GBT and 20m OH data were reduced using the GBTIDL software (Garwood et
al., 2006). The quality of this data is generally very high, and each
polarization of each 10 minute scan was reviewed for the presence of radio
frequency interference and problems from instrumental effects. In this
section, we review the experimental and observational tests of the broad OH
emission that heightened our confidence in the result.

###  3.1 Separating the “ON” and “OFF” Spectrum

The observations in AGBT15B_004 were experimented on in several stages of the
frequency switching process in order to test different parts of the
intermediate frequency (IF) system to make sure the signal was not introduced
by some previously unknown systematic that was unaccounted for. Our
observations are completed with in-band frequency switching (see e.g. O’Neil,
2002); the reference measurement of ON-OFF is completed by shifting the
central frequency 2 MHz and differencing the spectrum. When we perform in-band
frequency switching, the science signal is still present in the “OFF”
measurement, and the final reduced signal is compiled by shifting and
averaging the “OFF-ON” and “ON-OFF” measurements to decrease the noise in the
final RMS by $\sqrt{2}$. This process is demonstrated in Fig. 1.

In each separate ON-OFF and OFF-ON signal, we should expect two signals from
the two OH lines, and the so-called “ghosts” of the subtracted signals from
the shifted frequency. Whichever phase is “ON” or “OFF” is arbitrary. However,
it is important that we check if the broad feature is introduced in one part
of the IF by appearing in one of the “phases” and not the other. If it were to
appear in “ON” but not “OFF”, this would raise the alarm of an unknown
systematic. We show in Fig. 1 that this is not the case, and the positive and
negative signals of the OH emission are clearly visible in both reference
spectra. The final product of the frequency switched measurement is shown in
the bottom panel of Fig. 1, where the “OFF-ON” measurement is shifted and
averaged together with the “ON-OFF” measurement.

In this final spectrum, we would expect the two signals as well as four
‘ghosts’, or negative, subtracted signals. Higher order polynomial fits are
required for such wide, frequency-switched measurements, as opposed to the
linear baseline fits typically used in position-switched measurements.

Figure 1: The top panel is the “ON-OFF” phase and “OFF-ON” phase plotted
separately. The second spectrum is a shift and average of the top two spectra,
typical of frequency-switched measurements (O’Neil, 2002).

###  3.2 Baseline Fitting

We were immediately suspicious that the broad OH signal was an erroneous
signal introduced by high-order polynomial baseline fitting. We performed
baseline fits for a wide range of polynomial orders to test the robustness of
this method. The broad emission feature is detected in spectra with polynomial
baseline fits from order 8 - 23. The precision of the measurement of the
profile integral from the GBT spectra is then constrained by the fitting
procedure, which introduces $\sim$ a few percent systematic error. To test the
robustness of the data pipelines; we created three different data pipelines
utilizing different common signal reduction techniques including: Fourier
filtering (or bandstop filtering), ridge regression (or L2 Regularization),
and the standard general orthogonal polynomial regression used in GBTIDL. None
of the data pipelines generated different results.

In the case of our GBT observations, since the emission feature is very wide
in velocity, and the data are frequency-switched, linear baselines cannot be
removed. A new data fitting procedure in GBTIDL had to be created to deal with
such wide profiles. Discontinuous regions of the bandpass that do not have
‘ghosts’ (negative signal) introduced by the frequency-switching, or signal
from the 1667 and 1665 MHz feature, were used ($N_{chans}\sim 6000$) to fit a
single high-order polynomial ranging from order 17-23 based on a few different
criteria: a) there should be no positive velocity emission due to galactic
structure, b) the chosen polynomial fit does not preferentially subtract or
under-subtract signal from the 1667 or 1665 emission near local-gas
velocities, resulting in a flat baseline, and c) the rms in the highlighted
baseline regions is minimized until no further improvement occurs. This
procedure produced the most consistent results for the GBT spectrum and flat
baselines in all regions where no signal or ‘ghost’ were expected.

The pipelines were also used on the 20m spectra to produce the broad OH signal
with identical results to the GBT spectra. The only noticeable difference in
the spectra was the ‘flattening’ of ‘spiral arm’ emission from discrete clouds
due to beam dilution. The size of the beam at 18 cm for the GBT is 7.6’,
whereas the size of the beam for the 20m is approximately 45’. While the
surface brightness is larger in the larger beam, the similar surface
brightness of the broad OH emission feature in both spectra indicate that the
filling factor of the emitting gas is high, suggesting that the source
geometry of the OH emission is extremely diffuse molecular gas that is volume
filling.

In addition to replicating the GBT result with the 20m telescope, we also
opted to forgo frequency-switching so that the baseline fitting of the 20m
spectra would be less complex. Baseline fits of order 2-3 were fit to the
total power spectrum and clearly showed the broad OH feature. This result
demonstrated that detection of the broad feature does not rely on the higher
order (i.e. 17-23) polynomial fits, which conceivably could have introduced
spurious artificial signals into the GBT spectrum. In the 100 hour 20m OH
spectrum, the RMS noise is $\sim$ 0.3 mK at a velocity resolution of 1 ${\rm
km\;s^{-1}}$.

###  3.3 Separating the Linear Polarization

Figure 2: The broad OH emission feature as seen in the GBT spectra. Top Panel:
A stacked 40 hr spectrum of XX linear polarization. Middle Panel: A stacked 40
hr spectrum of YY linear polarization. Bottom Panel: The resulting residual
spectrum from XX-YY. The residual noise is possibly from an additional known
spectral ripple in the YY polarization due to the geometry of the GBT itself
(Fisher et al., 2003).

The emission appears to be unpolarized. This is an important check because at
cm wavelengths at the GBO observers will sometimes experience weak, transient
and usually polarized radio frequency interference (RFI) that is time-
dependent. We expect any real thermal OH emission to be non-polarized because
it is due to particle motions and collisions in gas that result in dipole
oscillations among the four ground-state lines. We checked that we could
reproduce the characteristics of the broad and faint OH emission features in
1667 and 1665 MHz in the independent orthogonal linear polarizations (XX and
YY). While the YY polarization experiences more baseline noise due to the
aforementioned multi-path scattering, we were able to replicate the spectrum
and calculate a residual spectrum between the polarization which shows a
spectrum with zero emission from the velocity range of the broad OH feature.
If the emission were only detected in one polarization, that would indicate
that it would be some type of systematic error that we were previously not
accounting for.

In this experiment, we stacked five GBT OH spectra towards
$l=108.000^{\circ}$, $b=3.00^{\circ}$, keeping the linear polarizations
separate throughout the reduction. Then, the spectra are smoothed to a
velocity resolution of 1 ${\rm km\;s^{-1}}$. The XX and YY spectra are
displayed in the top two panels of Fig. 2. The residual spectrum of XX-YY is
also shown to demonstrate that the two polarizations behave differently at a
very small level. The resulting integration time on the spectra in Fig. 2 is
approximately 40 hours with a RMS noise of $T_{RMS}$ = 300 $\mu$K. The
systematic from the YY spectrum is at the level of 0.5-0.6 mK, the brightness
temperature of the 1667 MHz emission is higher at $T_{mb}=1-3mK$, and the 1665
MHz emission is about half of that, allowing a detection of this feature at
high S/N, especially if only using XX data. We rule out the possibility that
the emission from the this feature is a polarized systematic baseline ripple
in the GBT.

##  4 Results

In this section we will describe the two main spectra presented in this paper,
the nature of the OH main lines, and how column densities of HI, OH and H2 are
computed.

Figure 3: The observed broad emission in spectra towards the Outer Galaxy at
$l=108^{\circ}$, $b=3^{\circ}$ comparing the results from the 100m GBT and the
20m telescope at the GBO. The top panel shows the HI emission at 1420 MHz (21
cm) observed with the GBT. The middle panel shows the OH spectra observed at
1667 MHz (blue) and at 1665 MHz (green) using the 100m GBT, while the bottom
panel shows the OH spectra observed at 1667 MHz (red) and 1665 MHz (yellow-
brown) using the 20m telescope at the GBO. Note that while the telescope
resolution varies from 7.6’ at the GBT to 45’ at the 20m telescope, the broad
emission is detected with both of these telescopes. The dashed lines are the
$3\sigma$ statistical detection limit, corresponding to 3 mK for the 100m
telescope and 2 mK for the 20m telescope.

###  4.1 GBT Spectra

It became apparent that many sightlines could be stacked from multiple OH
projects because of the extended nature of the broad OH emission, which
appeared to be beam filling. This allowed for artificially extremely long
integration times (up to 300 hours). Depending on how many positions are
averaged together, the rms noise in the spectra was between $\sim$ 300-800
$\mu$K. The GBT spectra shown in the middle panel of Fig. 3 come from an 80
hour, stacked OH spectrum centered at 1666 MHz, which was then re-centered for
the 1667 and 1665 MHz lines respectively. The positions used in this spectra
are seven ten-hour exposures around the coordinates ($l=108\arcdeg$,
$b=3\arcdeg$). The main beam efficiency of the 100m at L-band is estimated to
be $\eta_{mb}$ = 0.95, which we used to convert antenna temperature to main
beam temperature.

###  4.2 20m Spectra

After averaging 100 hrs of observations at two separate positions
($l=108\arcdeg$, $b=3\arcdeg$ and $l=109\arcdeg$, $b=2.5\arcdeg$), separated
by a degree on the sky, and smoothing to 1 ${\rm km\;s^{-1}}$, the rms noise
in the spectrum was $\sim$ 300 $\mu$K. We have aligned the local gas emission
in the two spectra using a LSR (local standard of rest) velocity correction
calculator kindly provided by F. Ghigo222https://www.gb.nrao.edu/
fghigo/gbt/setups/radvelcalc.html. The resulting OH spectrum at 1667 and 1665
MHz are shown in the bottom panel of Fig. 3. The main beam efficiency of the
20m is estimated to be $\eta_{mb}$ = 0.77 (F. Ghigo, private communication),
which we used to convert antenna temperature to main beam temperature.

###  4.3 1667 and 1665 MHz OH Lines

On a GBT spectrum with 10 hours of integration time, the resulting rms at a
velocity resolution of 1 ${\rm km\;s^{-1}}$ is close to 1 mK. Typical
brightness temperatures per channel are $T_{mb}\sim 1-3$mK, and thus the
profile integral over the large velocity width ($\Delta v_{LSR}\sim
150\mbox{${\rm km\;s^{-1}}$}$) of the emission results in a signal-to-noise
(S/N) of approximately 25. The emission also appears clearly in the two main
lines of the OH ground state ($\Delta F=0$) at 1667 and 1665 MHz. The profile
integrals of the observed emission regularly appear to be in the 9:5 LTE ratio
set by the quantum mechanical transition strengths (Townes & Schawlow, 1955;
Turner & Heiles, 1971). This ratio indicates that collisional excitation of OH
is the dominant radiation mechanism in the gas, as the energy levels approach
the typical Boltzmann distribution as described by an excitation temperature
$T_{ex}$, yet the brightness temperature is not near the actual gas
temperature.

###  4.4 Column Density of OH

The expression for the total molecular gas column density is (Mangum &
Shirley, 2015):

$N_{tot}=\frac{3h}{8\pi^{3}|\mu_{lu}|^{2}}\frac{Q_{rot}}{g_{u}}\exp\left(\frac{E_{u}}{kT_{ex}}\right)[\exp\left(\frac{h\nu}{kT_{ex}}\right)-1]^{-1}\int\tau_{\nu}d\nu$
(1)

where $Q_{rot}$ is the rotational partition function, the statistical sum over
all rotational energy levels in the molecule (for the OH hyperfine,
$\Lambda$-doubling ground states, this sum is 16), and $g_{u}$ is the
degeneracy of the energy level $u$.

In the analysis of OH emission line profiles, this equation can be converted
to mostly observables. Consider a molecular cloud located in the ISM at a
distance S along a line of sight from the Sun towards the outer Galaxy. The
beam-averaged column density of OH molecules $\left<N(OH)\right>$ along the
line of sight through the cloud is (Liszt et al., 2010):

$\left<N(\mathrm{OH})\right>=C_{67}\left[\frac{T_{ex}^{67}}{T_{ex}^{67}-T_{c}}\right]\int\Delta
T_{b}^{67}(v)dv;$ (2)

where $\Delta T_{b}^{67}(v)$ is the (main beam) brightness temperature of the
OH emission profile from the cloud as observed in one of the 18-cm OH
transitions (in this example the transition that gives rise to the 1667 MHz
line), minus an estimate of the underlying radio continuum brightness at the
same radial velocity (the ‘baseline’); $T_{ex}^{67}$ is the excitation
temperature for the 1667 line; $T_{c}$ is the brightness of the Galactic
continuum emission at 1667 MHz incident on the back surface of the cloud, and
the integration over velocity includes all the molecular emission thought to
arise in that particular cloud. This equation assumes that the Rayleigh-Jeans
approximation applies, that the optical depth of the cloud in the OH line is
small, that all the OH molecules are in the ground rotational-vibrational
state, and that the 4 ground-state levels are populated according to LTE.
Under these conditions, the constant $C_{\nu}$ consists of:

$C_{\nu}=\frac{8\pi k\nu_{ul}\Sigma
g_{i}}{hc^{2}A_{ul}g_{u}}\times(\nu_{ul}/c)$ (3)

where $\nu_{ul}$ is the transition line frequency, $A_{ul}$ is the Einstein A
coefficient for that transition, $\Sigma g_{i}$ is the sum of the statistical
weights of the four levels giving rise to the 18cm lines (=16), $g_{u}$ is the
statistical weight of the upper energy level that gives rise to the specific
line (e.g. = 5 for the 1667 line), and k, h, and c designate the usual
physical constants. This equation has been grouped into two terms; the first
is from Goss (1968), and the second provides the conversion from line widths
in Hertz (used by Goss) to the currently more conventional units of doppler
velocity. Using the entries for the line frequencies and Einstein A
coefficients found in the Splatalogue data
base333http://www.cv.nrao.edu/php/splat/ appropriate for the 1667 MHz line,
the value of $C_{67}$ is $2.257\times 10^{14}$ ${\rm cm^{-2}}$ for $\Delta
T_{b}^{67}$ in Kelvins and velocity in ${\rm km\;s^{-1}}$.

In this analysis, we assume an excitation temperature of 5.1K $\pm$ 1K, as the
excitation temperature is usually found to be within 1-2K above the continuum
temperature (Li et al., 2018; Engelke et al., 2020), although we stress
caution because the real excitation temperature of the gas may be lower,
closer to the Galactic background as found in Li et al. (2018). The possible
range of excitation temperatures and continuum temperatures alone introduces
systematic uncertainty in the resulting N(OH) values, which we explore below.
To obtain an estimate of the continuum background brightness temperature,
$T_{c}$, we estimate the synchrotron contribution from the 408 MHz continuum
map of the Canadian Galactic Plane Survey (CGPS, Taylor et al. (2003)) by
adopting a temperature spectral index of 2.8 (Reich & Reich, 1988):

$T_{c}=2.7+T_{c,408}(1667/408)^{-2.8},$ (4)

which results in a Tc of approximately 4K. An approved GBT program for this
coming year will attempt to observe this dark disk in absorption and emission
pairs, which should unambiguously provide an accurate measurement of $T_{ex}$
in the future. The resulting column density is: $N($OH$)$ = (7.41 $\pm$ 0.17)
$\times$ 1014 cm-2, which includes statistical error. For a sensitivity
calculation on systematic errors, we calculate the possible ranges of OH
column densities using 4.1K < $T_{c}$ < 4.5 K, and 4.6K < $T_{ex}$ < 6.1 K.
This range results in: $N($OH$)$ = (1.0 $\pm$ 0.017 statistical $\pm$ 0.4
systematic) $\times$ 1015 cm-2. Note that the systematic uncertainty due to
the unknown values of $T_{c}$ and $T_{ex}$ dominate. We remind the reader here
that the continuum temperature and excitation conditions are likely not
constant through the entire outer Galaxy, and thus a more precise treatment in
the future will be warranted when more data are available.

Figure 4: Left: The atomic (top) and molecular (bottom) content towards the
sightline $l=108^{\circ}$, $b=3^{\circ}$ as traced by HI and OH with a 7.6’
beam. Middle: The same at $l=138^{\circ}$, $b=3^{\circ}$. Right: The same at
$l=158^{\circ}$, $b=3^{\circ}$. Notice that because $l$ is approaching
$180^{\circ}$, the velocity of the emission begins to crowd towards 0 ${\rm
km\;s^{-1}}$due to the geometry of the Galactic velocity field.

###  4.5 Column Density of HI

An accurate measurement of the column density of HI is needed to calculate the
presumed amount of H2 in the sightline from the gas content as traced by dust
reddening. We estimate the optical depth of HI, $\tau$, as a constant value
across the sightline by examining two absorption-emission pairs from Strasser
et al. (2007) that we read from their online data: l = 108.446°, b = 4.0498°;
and l = 108.753°, b = 2.5772°. Although $\tau$ changes along the line of
sight, we take the median value from these sight lines as a constant for the
calculations of the column density of HI at $\textit{l}=108\arcdeg$,
$\textit{b}=3\arcdeg$ because they are close in proximity and representative
of the diffuse ISM in this direction. The median across the line of sight in
these two directions results in a $\tau$ = 0.103 $\pm$ 0.03, which we use as
the $\tau$ in the following HI column density calculations.

We use Equation 5 in Dickey & Benson (1982) to calculate the column density of
HI using the LAB survey HI spectrum at l = 108°, b=3°:

$\left<N(\mathrm{HI})\right>=C_{0}\int T_{mb}(v)\frac{\tau}{1-e^{-\tau}}dv$
(5)

where $C_{0}$ = 1.82 $\times$ 1018
$\frac{\mathrm{cm^{-2}}}{\mathrm{K}\mbox{${\rm km\;s^{-1}}$}}$. This
calculation assumes an isothermal medium along the line of sight, with
constant HI spin temperatures; which is likely somewhat incorrect because
there are different environmental regions along the entire line of sight
towards the outer Galaxy. However, because our estimate of the total gas
content, $N_{H}$ is from dust IR emission, and hence does not contain radial
velocity information, we cannot reliably decompose the HI and OH spectrum into
their different components if the H2 content is of pertinent interest. The
resulting HI column density is $N($H$)$ = (1 $\pm$ 0.3) $\times$ 1022 cm-2.
The resulting N(OH)/N(H) ratio is $\sim$ a few $\times$ 10-7.

The HI4PI server444https://www.astro.uni-bonn.de/hisurvey/AllSky_gauss/
returns an HI column of $N($H$)$ = 9.58 $\times$ 1021 cm-2. Our column density
differs (albeit, within error) because we restrain our column density
calculation only to the Milky Way disk $(-150$ ${\rm km\;s^{-1}}$< VLSR <
$100$ ${\rm km\;s^{-1}}$$)$ and correct for $\tau$, whereas the HI4PI N(H)
maps calculate their optically-thin column densities along their entire radial
velocity range, $-600$ ${\rm km\;s^{-1}}$< $V_{LSR}$ < $600$ ${\rm
km\;s^{-1}}$. This may include not only the Milky Way disk material, but also
HI gas in the MW halo, intermediate and high velocity gas complexes, and
extra-galactic HI objects.

###  4.6 The OH Abundance Ratio

While several estimates of the N(OH)/N(H2) abundance ratio exist for several
Galactic environments (Liszt & Lucas, 1996; Nguyen et al., 2018; Rugel et al.,
2018; Jacob et al., 2019), it is unclear which value is appropriate to use for
this sightline, as most of these estimates were calculated towards discrete
clouds with absorption sources, and typically near the solar neighborhood. We
estimate our own abundance ratio in this sightline by using dust reddening as
a tracer for the total gas content, to see if it is consistent with the latest
abundance ratio calculations from the THOR survey (Rugel et al., 2018) and
using CH as a calibration tracer (Jacob et al., 2019). We closely followed the
prescription of this method as laid out in Nguyen et al. (2018). Reddening is
caused by dust grains through the absorption and scattering of light, defined
as:

$E(B-V)=\frac{A_{V}}{R_{V}}=1.086\frac{\kappa}{R_{V}}r\mu m_{H}N_{H}$ (6)

Figure 5: The observed 1667 and 1665 MHz OH broad emission toward the outer
Galaxy ($l=108$, $b=3$), as observed by the 100m GBT and 20m telescope at the
GBO. The dashed lines are the 3$\sigma$ (corresponding to 2mK for the 20m, 3mk
for the 100m) statistical detection limit. The HI spectrum is from the LAB
survey (Kalberla et al., 2005). The CO spectrum is from the FCRAO spectrum
(Heyer et al., 1998), kindly provided by M. Heyer.

where $A_{V}$ is the dust extinction, $R_{V}$ is an empirical coefficient,
$\kappa$ is the emissivity per cross section, $r$ is the dust-to-gas mass
ratio, $\mu$ is the molecular weight (=2), $m_{H}$ is the weight of a hydrogen
atom and $N_{H}$ is the total gas content, defined as: $N_{H}=N(H)+2N(H_{2})$.
$R_{V}$ is typically assumed to be 3.1 (Cardelli et al., 1989). While it has
been shown that $R_{V}$ varies on Galactic scales, it does not vary
significantly (Schlafly et al., 2016). The SFD dustmap assumes a value of 3.1,
which we use in this paper.

In this analysis we assume that the total gas content is a linear combination
of atomic and molecular hydrogen, $N_{H}=N(\mathrm{H})+2N(H_{2})$. We subtract
the column density of atomic hydrogen and assume that the residual is
molecular hydrogen. The uncertainties in this method include the uncertainty
in the dust reddening, the uncertainty in the $N_{H}$/E(B-V) ratio, and the
uncertainty in the optical depth, $\tau$, of the HI.

Using the opacity corrected HI column density from Sect. 4.5, we calculate a
column density of protons, NH, of 1 $\times$ 1022, and H2 of N(H2) = 3
$\times$ 1020 cm-2. Using this to calculate an abundance ratio, N(OH)/N(H2) =
1.5-4 $\times$ 10-6, which is consistent the range of abundance ratios from
the literature (Nguyen et al., 2018; Rugel et al., 2018; Jacob et al., 2019).
Our calculations and the predictions from theory are also consistent with
expectations for diffuse regions, $A_{v}$ < 0.1 mag (Neufeld & Wolfire, 2016).

##  5 Discussion

###  5.1 The large-scale Galactic structure of the OH Emission

In 2019 we undertook a GBT survey in longitude and latitude to probe the
structure of the broad OH emission under Project ID AGBT19A_484. It was
quickly discovered that the emission experiences the same expected velocity
structure of the large-scale HI in the outer Galaxy. That is, the velocity
distribution of the emission approaches zero with increasing $l$, as shown in
Fig. 4. This unambiguously implies that the broad OH profile is indeed a
large-scale structure in the Galaxy that obeys the geometry of the velocity
field (Burton & te Lintel Hekkert, 1985). The HI and OH emission morphology
appear to behave the same in velocity space, implying a cospatial nature of
this phase of molecular gas with atomic gas in the outer galaxy. The HI
associated with the Galactic disk is known to extend outward from the center
farther than optically observed portions of the Galaxy such as stars, as well
as known molecular components detected with CO. As such, this result extends
the distance outward in the Galaxy in which molecular gas is found. At the
same longitude and latitude, we demonstrate that this gas is by definition
dark, because it lacks a similar detection in a sensitive CO spectrum (Heyer
et al., 1998), see Fig. 5.

While the longitude-velocity dependence is suggestive of a large-scale
structure, we also checked for structure between $-5^{\circ}<b<5^{\circ}$
towards $l=105^{\circ}$ by stacking in latitude the data from AGBT15B_004 in
longitude. We were able to detect the broad OH emission to some level in every
spectrum. It was apparent, however, that the OH emission is very small
($T_{mb}$ = 1 mK) at high latitudes, $b=-5^{\circ}$ and $b=5^{\circ}$. This
suggests that the feature has some scale height of $\sim$ 200 pc. In a future
paper we will investigate this further. Additionally, it appears that the gas
has a bi-modal distribution around $b=1^{\circ}$. This may be because of the
Galactic warp in this direction (Yuan & Wallace, 1973), which warps the
stellar and gas disk above the Galactic plane.

###  5.2 The Implied Volume Density of the Diffuse Molecular Disk

Owing to Galactic rotation, the distance $s$ along the line of sight and the
radial velocity $v$ at that point are approximately linearly related in the
outer Galaxy. This suggests that we can estimate the $volume\ density\ n_{OH}$
at any point $s$ along the line of sight by differentiating equation 2 with
respect to distance $s$ along the line of sight to obtain:

$\displaystyle n_{s}(OH)$ $\displaystyle\approx$ $\displaystyle
C_{67}\left[\frac{T_{ex}^{67}}{T_{ex}^{67}-T_{c}(l\geq
L)}\right]d/ds\int\Delta T_{b}^{67}(v)dv$ (7) $\displaystyle\approx$
$\displaystyle C_{67}\left[\frac{T_{ex}^{67}}{T_{ex}^{67}-T_{c}(l\geq
L)}\right]\Delta T_{b}^{67}(v)dv/ds|_{s=S}$ (8)

where $dv/ds$ is in units of ${\rm km\;s^{-1}}$/cm for the value of $C_{67}$
adopted here. To determine an approximate value for $dv/ds$ appropriate for
any given Galactic longitude, we use the values of $S$ for a given value of
$v$ returned by the VLBI “Revised Kinematic Distance Calculator“ developed by
the “BeSSeL“ collaboration555http://bessel.vlbi-astrometry.org/home; for
additional background information see the review paper by Reid & Honma (2014).
This approach allows us to estimate values for $s(v)$ and $ds/dv$ which are
less prone to local perturbations in the velocity field of the outer Galaxy.
For instance, at $l=108.0\arcdeg$, $b=+3.0\arcdeg$, and e.g. at $v=-80$ ${\rm
km\;s^{-1}}$(the approximate radial velocity of the outer arm of the Galaxy at
this value of $l$) we obtain the linear approximation $s=a-bv$ with $a\approx
0$ and $b\approx 0.0875$ kpc/(${\rm km\;s^{-1}}$) at a distance of $\approx
7.0$ kpc from the Sun. As an illustration, the volume density at this location
is 1.8 $\times$ 10-9 OH molecules ${\rm cm^{-3}}$per mK of 1667 MHz main-beam
brightness temperature, assuming we are far enough in the outskirts of the
Galaxy to set $T_{c}(s\geq S)=3$K (i.e. slightly above CMB alone), and
choosing $T_{ex}^{67}=4.1$K (a reasonable value between the ones found in
Engelke & Allen (2018) and Li et al. (2018)). Using the abundance ratio
N(OH)/N(H2) = 2 $\times$ 10-6, the corresponding H2 volume density is $\sim$ 7
$\times$ 10-3 H2 molecules ${\rm cm^{-3}}$at this location. By changing the
excitation and continuum temperatures within reasonable assumptions ($\pm$ 2
K, for either), the volume density could be higher or lower by a factor of 10.
This is consistent with the volume density of the diffuse molecular gas phase
suggested by Papadopoulos et al. (2002), and consistent with the average
molecular volume density of the ISM found by Bohlin et al. (1978).

Figure 6: The observed HI, OH and CO spectrum from Fig. 5, transformed into
estimated number of protons traced by each tracer. Estimated number of protons
for the atomic gas component is calculated directly from the HI signal;
estimated number of protons for the molecular gas component is calculated from
the OH signal using our assumed value of N(OH)/N($H_{2}$) of 2 $\times$ 10-6,
and from the CO signal using the X(CO) value 2 $\times$ 1020 cm-2 (K ${\rm
km\;s^{-1}}$)-1 from (Bolatto et al., 2013). We see that diffuse molecular gas
(as traced by OH) may be a component in mass equal to the atomic gas traced by
HI, while much of the molecular gas appears to be CO-dark.

###  5.3 Implications for Diffuse Molecular Gas in the Outer Galaxy

An estimate of the H2 mass implied by the broad OH emission from the diffuse
OH emission can be calculated using several reasonable assumptions. It is,
however, important that we treat this number with caution because the
uncertainties involved in this method amount to roughly an order of magnitude.
Since we calibrate our OH/H2 abundance ratio for this sightline in an
integrated fashion with dust reddening, we have no knowledge of the evolution
of the OH/H2 abundance ratio in radial velocity (and hence, distance towards
the Outer Galaxy). With the assumption of excitation and continuum temperature
above, we move forward by transforming our column density of OH to H2 using
the abundance ratio calculated in Sect. 4.6. The uncertainty in the abundance
ratio is the dominant error in the following calculation. While we calculate
an N(OH)/N(H2), it is subject to all of the assumptions implicit in the dust
reddening method, and we should also not be using one number for the abundance
ratio for the sightline, as there are multiple environments in the spectrum.
Nevertheless, if we use the derived abundance ratio, and the X(CO) number we
can transform the Tmb measurements of the HI, OH and CO spectrum in proton
units. This exercise, displayed in Fig. 6, shows that diffuse molecular gas
(as traced by OH) may be a component in mass equal to the atomic gas traced by
HI in the outer Galaxy.

We use a simple geometric model of the outer Galaxy: a cylinder with a hole in
the middle corresponding to the solar circle (R = 8 kpc). We calculate the
volume of this cylinder as $V=\pi h(r_{1}^{2}-r_{2}^{2})$, where we take the
height of the dark molecular disk as 200 pc, which is motivated by
observations but will be further investigated in a future paper. In broad
strokes, the z-distribution of OH was investigated at the “Perseus Arm” gas
velocity ($V_{LSR}=-60$ ${\rm km\;s^{-1}}$), by fitting a Gaussian we were
able to measure the scale height of the gas features. We take this as the
scale height because it appears that the disk of emission peaks in surface
brightness at this feature. We assume a line of sight disk length of R = 25
kpc, which is approximately the extent of the HI disk. The resulting volume of
this hollow cylinder is 1 $\times$ 1067 cm3. Using a Tmb = 4 mK, the number
density is 7 H2 molecules per liter, the mass in the outer Galaxy can be
calculated as: $M=2m_{p}Vn_{\mathrm{H_{2}}}$ $\sim$ 1 $\times$ 108 M⊙ The main
uncertainties here are: the excitation temperature of OH and the abundance
ratio of OH to H2 the abundance ratio being the dominant contribution to the
uncertainty.

The current literature values of OH/H2 cluster around $\sim$ 1 $\times$ 10-7
(Liszt & Lucas, 1996; Nguyen et al., 2018; Rugel et al., 2018; Jacob et al.,
2019); a possible value of the mass using this ratio (which results in 150 H2
molecules per liter) is $\sim$ 2 $\times$ 109 M⊙, which is about equal to
current estimates of CO-traced H2 mass in the Galaxy at $\sim 1\times
10^{9}M_{\odot}$ (Heyer & Dame, 2015), and within an order of magnitude of the
HI mass, at $\sim 8\times 10^{9}M_{\odot}$ (Kalberla & Kerp, 2009).

If we use our calculated value for the abundance ratio for the lower limit of
mass, and the literature value of the abundance ratio for the upper limit of
the mass, the resulting range of molecular disk mass values is $\sim$ 108-9
M⊙. The existence of a mass of dark H2 in our Galaxy equal to the CO-bright-
traced H2 has been inferred from the indirect observations of $\gamma$-ray
(Grenier et al., 2005) and dust (Planck Collaboration et al., 2011). We
provide with this measurement concrete evidence of the existence of these
diffuse molecules and, with the radial velocity distribution, the geometry of
the diffuse molecular gas as a thick disk beyond the Solar circle. Note that
the newly discovered diffuse molecular disk contains CO-dark molecular gas in
addition to the previously reported clouds of CO-dark molecular gas found
during our OH observing program in the arms of the Galaxy (Allen et al., 2015;
Busch et al., 2019). Moreover, it is interesting to examine to what extent the
diffuse molecular disk might affect the total baryonic mass of the Galaxy.
Using an estimate of stellar mass of $\sim 6\times 10^{10}M_{\odot}$ (Licquia
& Newman, 2015; McMillan, 2017) and an estimate of the known mass of the
interstellar medium of $\sim 1\times 10^{10}M_{\odot}$ (Kalberla & Kerp,
2009), we roughly estimate that this newly discovered diffuse molecular disk
could increase the baryonic mass estimate of the Galaxy by $\sim$ 0.1 $\%$ to
$\sim$ 2 $\%$. We anticipate that further sensitive surveys of this broad OH
emission throughout the Galaxy will refine the measurement of the mass of the
diffuse molecular disk in the future.

The scale height of the diffuse OH emission corresponds extremely well with
the thick diffuse molecular gas disk found in M51 in extended CO emission
(Pety et al., 2013). This comparison raises the question as to why the CO
sightlines in the Milky Way are CO-dark, when the diffuse molecular gas in
certain other galaxies others are not. One possible explanation for the
detectable CO emission in M51 is the geometric effect of M51 being face-on.
This orientation could result in enough radiative trapping in the z-direction
for the emission from the diffuse component to be detectable at a sensitivity
of $\sim$ 1 K ${\rm km\;s^{-1}}$. We show in Busch et al. (2019) that CO is
subthermally excited below its critical density of 1000 cm-3, and below this
threshold the surface brightness of CO is heavily dependent on gas volume
density. While the CO emission in Pety et al. (2013) appears to come from
diffuse molecular gas of n $\sim$ 300-500 cm-3, the OH emission may yet still
be tracing an even more diffuse component in addition to the CO-traced diffuse
molecular gas because its critical density is n $\sim$ 1 cm-3. A future study
intended to search for OH emission from diffuse disks in external galaxies
known to contain CO-bright disks, as well as several that do not contain
detected CO disks, could provide more information about the differences
between conditions in the disks of these galaxies and the properties that lead
to a diffuse disk being CO-bright or CO-dark. Such a study could also
demonstrate whether OH traces a larger total mass of diffuse molecular gas in
these disks even for galaxies that already contain a detectable disk in CO.

Papadopoulos et al. (2002) lays out a theoretical basis for both H2 formation
and shielding at large Galactic distances. They argue that both the
ingredients (HI and dust grains) and environmental conditions exist for H2
formation well inside an HI disk. The existence of abundant dark H2may also
shed light on the puzzling star formation behaviours in HI dominated regions
(Krumholz, 2013). Star formation correlates with H2 down to the lowest column
densities that can be detected by CO, but then also begins to correlate with
HI in the outer parts of the Galaxy. If there were a thick disk of diffuse
molecular gas in the outer Galaxy, this may help explain the star formation in
the HI dominated portions of the outer Galaxy, as opposed to alternative
explanations such as high-velocity runaway stars (Andersson et al., 2021).

Modeling and observations in future papers will have to study the distinction
between the OH and CO-traced diffuse molecular gas disks in order to fully
understand the transition zone between CO-bright and CO-dark molecular gas.

##  6 Conclusions

We have discovered an extremely broad OH emission profile towards the outer
Galaxy that is cospatial with the HI velocity distribution. We have convinced
ourselves that it is a real astronomical signal by observing the feature in:
both 1667 and 1665 MHz OH lines, two different telescopes with vastly
different beams and associated instrumental effects, and using three different
data pipelines with which to reduce the data.

This result indicates that the molecular disk of the Galaxy is significantly
more extended than previously suspected. We believe that the geometry of the
molecular gas traced by the emission is in the form of an extremely diffuse
disk, with a thickness observed in the Galactic latitude range
$-5^{\circ}<b<5^{\circ}$. The signal is indicative of large-scale structure
because it obeys the velocity field of the Galaxy.

A conceptual model of the outer Galaxy using our derived value of the volume
density of $\sim$ tens of H2 molecules per liter (7-70 $\times$ 10-3 cm-3)
results in a molecular mass value for the dark molecular disk of $\sim$ 108-9
M⊙, with the order of magnitude uncertainty resulting from the adopted range
of values of the abundance ratio, N(OH)/N(H2).

We plan multiple follow up surveys to probe the latitude and longitude
dependence of the OH emission, and to characterize the structure of the
diffuse molecular gas using OH emission. We hope this discovery will motivate
similar searches for extremely diffuse molecular gas using dark molecular gas
tracers on a large, Galactic scale. We anticipate there is much to learn about
Galactic structure by using an optically thin, widely observed molecular gas
tracer in the Galaxy.

We would like to dedicate this discovery to Ronald J. Allen, who passed away
during the drafting of this manuscript. He was an amazing mentor, a brilliant
astronomer, and a good friend to many in our profession. He led much of the
early research in tracing the large scale structure of diffuse molecular gas
through blind, sensitive emission surveys of Galactic OH, which this work was
built upon. We will miss him dearly. We are grateful to the scientific staff
and telescope operators at the Green Bank Observatory for their advice and
assistance with the operation of the GBT, in particular Karen O’Neil, Jay
Lockman, Toney Minter, Ron Maddalena, and Amanda Kepley, and for the
development and support of the GBTIDL data analysis software, especially Jim
Braatz and Bob Garwood. We are especially grateful to Frank Ghigo of Green
Bank Observatory, for a tremendous investment of his time and skills in the
required maintenance of the 20m telescope and the development of a data
pipeline for the observations required for this paper. We are grateful to the
anonymous referee, whose comments improved the contents of this paper. Michael
P. Busch is grateful to Josh Peek, Claire Murray, Susan Kassin, Christine
Chen, Tobias Marriage, David Neufeld, Colin Norman, and Nadia Zakamska for
their help and guidance through a difficult time in his graduate school career
after RJA’s passing. MPB is supported by a National Science Foundation
Graduate Research Fellowship under grant No. 1746891. MPB thanks the LSSTc
Data Science Fellowship Program, which is funded by LSSTc, NSF Cybertraining
Grant No. 1829740, the Brinson Foundation, and the Moore Foundation; their
participation in the program has benefited this work. This research has been
supported by the Director’s Research Fund at STScI.

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# A Invertible Dimension Reduction of Curves on a Manifold

Sheng Yi
North Carolina State University
Hamid Krim
North Carolina State University
Larry K. Norris
North Carolina State University

###### Abstract

In this paper, we propose a novel lower dimensional representation of a shape
sequence. The proposed dimension reduction is invertible and computationally
more efficient in comparison to other related works. Theoretically, the
differential geometry tools such as moving frame and parallel transportation
are successfully adapted into the dimension reduction problem of high
dimensional curves. Intuitively, instead of searching for a global flat
subspace for curve embedding, we deployed a sequence of local flat subspaces
adaptive to the geometry of both of the curve and the manifold it lies on. In
practice, the experimental results of the dimension reduction and
reconstruction algorithms well illustrate the advantages of the proposed
theoretical innovation.

##  1 Introduction

For many pattern recognition problems, the feature space is nonlinear and of
high dimensionality and nonlinearity, for example, face recognition, image
classification and shape recognition. Many dimension reduction techniques [1]
are therefore naturally introduced to ease the modeling, computation and
visualization of the feature space. Dimension reduction has been widely used
with different vision techniques such as contour tracking [3], face
recognition [4] and object recognition [2].

One of the key points of dimension reduction is how to represent the geometry
of the original space in a lower dimensional space [9, 10]. Most of the
previous dimension reduction research [1] focuses on embedding the points in
the original space into a lower dimensional space. Beyond the point-wise
dimension reduction, the dimension reduction for a sequence of points or an
evolving curve in the original feature space is more important in applications
where features of interest are time varying such as, face expression [4],
events indexing in video sequence as a shape process [11] and human activity
as a shape sequence [3]. To the best of our knowledge, only a few works [3, 2]
address the problem of how to represent a curve in the original feature space
in a lower dimensional space.

The dimension reduction of high dimensional curves may be categorized into two
classes. One is to reduce the whole curve space to a lower dimension as in [3,
12, 13]. The other one is to reduce a particular the curve to a corresponding
lower dimensional curve as in [2]. The drawback of the first one is the
Euclidean assumption by using an approximation with a spline function.
Moreover, as in [3] by directly applying PCA to the spline coefficients, the
global dimension reduction does not consider the intrinsic dimensionality of a
particular high dimensional curve. A curve sequence like walking, for example,
is much simpler than a dancing sequence. Thus the dynamics of walking might be
represented in a much lower dimensional space in comparison to other complex
activities. The second category of dimension reduction as in [2], considered
the geometry of a particular curve in the original space, which, due to the
chosen implementation of the Whitney embedding theorem, is computationally
heavy on account of the iterative subspace search. At the same time, there
exists no invertible mapping from the embedding space to the original one,
thus limiting its extension to a generative process.

Motivated by the importance of a lower dimensional representation of a
manifold-valued curve and the limitations of the current techniques, in this
paper a novel dimension reduction of a shape manifold valued curve is
proposed. In contrast with previous curve dimension reduction techniques [3,
2], the proposed method has the following advantages:

  1. 1.

the dimension reduction is adaptive to the Riemannian geometry of a shape
space.

  2. 2.

the computation is linear in the length of the curve times the dimension of
the shape space.

  3. 3.

with a proper dimensionality, there exists a reconstruction as a subspace
parallel transport on a manifold.

To introduce the idea of the proposed dimension reduction framework for the
manifold valued curves, this paper follows the moving frame formulation in [6,
7]. According to [7, 6], every curve on a manifold may be represented by a
differential equation,

$X_{t}=X_{0}+\int_{0}^{t}\sum_{i=1}^{dim(M)}V_{i}(X_{t})dZ_{i}(t)$ (1)

where, $V(X_{t})$ is a vector field on a manifold, and $Z_{t}$ is a curve in a
Euclidean space. Under such representation, to introduce a lower dimensional
representation of $X_{t}$, a method is proposed to find the optimal vector
field $V(X_{t})$ such that:

  1. 1.

The sequence of $V(X_{t})$ is path dependent and uniquely determined by
$X_{t}$ and initial condition $V(X_{0})$ by a efficient computation.

  2. 2.

The resulting $dZ_{t}$ from $X_{t}$ and $V(X_{t})$ have a distribution that
lies in a lower dimensional subspace.

The first requirement on $V(X_{t})$ means that $X_{t}$ can be represented by
only the initial condition $V(X_{0}),X_{0}$ and the embedded curve $Z_{t}$.
According to [6] such property yields the 1-1 correspondence between $X_{t}$
and $Z_{t}$. The second requirement is more intuitive that among all the
$V(X_{t})$ satisfied the first requirement, the optimal one is selected such
that the variance of tangents is best represented in a subspace.

The first requirement is satisfied by select $V(X_{t})$ that is a horizontal
vector field [7] defined by the Levi-Civita connection as in Section 2.2 with
a $L^{2}$ metric induced to the shape manifold $M$ from the ambient space.
According to [6, 7], under a defined connection, for $X_{t}\in M$ and initial
condition $V(X_{0})$, there exists a unique selection of moving frame
$V(X_{t})$. Thus given $V(X_{0})$ the resulting curve in Euclidean space have
one-one correspondence to $X_{t}$. The reason of using the Levi-Civita
connection have two folds. The first one is that it is a non-flat connection.
In contrast to the other category: flat connection, it is path dependent. Such
property is important for dimension reduction because it provides more
adaptivity to the geometry of the path on the manifold. The second reason is
that in comparison with the general nonflat connections defined by a PDE on
manifold, such particular Levi-Civita connection in Section 2.2 is consistent
with the metric of the shape manifold in [5] that this paper is based on, and
is computationally more efficient for a known ambient space and normal space
of the manifold, as is the case of our shape manifold.

The second requirement is achieved by the subspace representation resulting
from maximizing the variance of the corresponding curve development $dZ(t)$
with different choices of $V(X_{0})$. In theory, without knowing the initial
condition $V(X_{0})$ the Levi-Civita connection only determines the vector
field $V(X_{t})$ up to a group action within the fiber of the Frame bundle
(the total space of frames of the tangent space of a manifold) [7]. Among
different choices for $V(X_{t})$ along the fiber in the frame bundle, a
optimal frame is selected such that the previously discussed second
requirement is satisfied to make the resulted $Z{t}$ better representable in a
lower dimensional space. A
$\tilde{V}(X_{0})=\\{x_{0},e_{1},e_{2},\cdots,e_{dim(M)}\\}$ is selected by
carried out PCA of all the parallel transported tangents along $X_{t}$.let
$\\{e_{1},e_{2},\cdots,e_{L}\\}$ are the eigenvectors corresponding to the $L$
largest Eigenvalues in the PCA calculation. Then A subspace spanned by the
subset $\\{e_{1},e_{2},\cdots,e_{L}\\}$ is constructed to better represent the
variance of all the tangents along $X_{t}$ in a lower dimensional space.

To summarize, a curve in the shape manifold $M$ is represented as

$X_{t}=X_{0}+\int_{0}^{t}\sum_{i=1}^{L}\tilde{V}_{i}(X_{s})d\tilde{W}_{i}(s)$
(2)

where $\tilde{V}_{i}(X_{t})$ can be represented by $L$ curves in $R^{3}$ and
$\tilde{Z}_{i}(t)$ is represented by a curve in $R^{L}$. Our proposed
dimensional reduction achieves the following points,

  1. 1.

a lower dimensional representation of curves on a manifold.

  2. 2.

The moving frame representation can be generated to a stochastic
representation of random process on a manifold. Thus the propose method can be
well adapted to a generative modeling in a lower dimensional space.

  3. 3.

the computation is linear in the dimension of $X_{t}$.

  4. 4.

there exists a reconstruction from the lower dimensional representation to the
original curve on the shape manifold.

The rest of the paper is organized as following. Section 2 provide necessary
background knowledge of shape manifolds and the geometric calculation on it.
In Section 3 and 4, the dimension reduction framework is introduced, which
includes the curve embedding and the curve reconstruction.

##  2 Preliminary

In this section, we first introduce in Subsection 2.1 the shape manifold that
the dimension reduction is based on. Then in Subsection 2.2 a fundamental
moving frame representation of curves on manifold is briefly introduced.

###  2.1 Shape Manifold

According to [5], a planar shape is a simple and closed curve in
$\mathbb{R}^{2}$,

$\alpha(s):I\rightarrow\mathbb{R}^{2},$ (3)

where an arc-length parameterization is adopted. A shape is represented by a
_direction index function_ $\theta(t)$. With such a parameterization,
$\theta(s)$ may be associated to the shape by

$\frac{\partial\alpha}{\partial s}=e^{j\theta(s)}.$ (4)

The ambient space of the manifold of $\theta$ is an affine space based on
$\mathbb{L}^{2}$. Thus

$\theta\in A(\mathbb{L}^{2}).$ (5)

The restriction of a shape is that it must be a closed and simple curve, and
invariant over rigid Euclidean transformations. The shape manifold $M$ is
defined by a level function $\phi$ as

$\phi(\theta)=\left(\int_{0}^{2\pi}\theta
ds,\int_{0}^{2\pi}\cos(\theta)ds,\int_{0}^{2\pi}\sin(\theta)ds\right).$ (6)
$M=\phi^{-1}(\pi,0,0).$ (7)

One of the most important properties of $M$ is that the tangent space $TM$ is
well defined. Such a property not only simplifies the analysis, but also makes
the incremental computation possible,

$T_{\theta}M=\\{f\in\mathbb{L}^{2}|f~{}\bot~{}span\\{1,\cos(\theta),\sin(\theta)\\}\\}.$
(8)

In addition, an iterative projection is proposed in [5] to project the point
in ambient space back to the shape manifold $M$. The idea is that each time
$\theta$ is updated as $\theta+d\theta$, where $d\theta$ is orthogonal to the
level set $\phi^{-1}(\phi(\theta))$. The $d\theta$ is calculated as
$d\phi^{-1}((\pi,0,0)-\phi(\theta))$. For the detailed form of the Jacobian of
$d\phi$, one could refer to [5].

The problem of this manifold for our stochastic modeling is the infinite
dimension. The mapping of random process in [6] from a manifold to a flat
space is only defined on a finite dimensional manifold. Therefore, a Fourier
approximation of the shape manifold we discussed above is developed, such that
the dimension is reduced to a finite number.

###  2.2 Moving Frame Geometry

Let $X_{t}$ be a curve on a manifold $M$, then following the moving frame
representation [8], the tangent $dX_{t}$ of manifold valued curve $X_{t}$ may
be written as:

$dX_{t}=\sum_{i}V_{i}(X_{t})dZ_{t},$ (9)

where $\\{V_{i}(X_{t})\\}_{i=1,2,\cdots,dim(M)}$ is a frame of the tangent
space at $X_{t}$, which is denoted as $T_{X_{t}}M$. Consequently $dZ_{t}$ may
be understood as a linear coefficient of $dX_{t}$ under the representation of
$V(X_{t})$. Such moving frame representation is widely used in geometry
studies of curves. For example in the Frenet theorem [8] the geometry of curve
is uniquely identified by the curvature of $V(X_{t})$ up to a rigid Euclidean
transformation. Recently in computer graphics, a rotation minimizing moving
frame [17] are developed to avoid the singularity of original moving frame
used in Frenet theorem.

In this paper, the moving frame representation is utilized for the similar
purpose of representing a curve on a manifold. In contrast to the previous
designs of moving frame, in this paper we elaborate to propose a adaptive
moving frame to represent a high dimensional curve in a lower dimensional
space. Vector field $V(X(t))$ are developed as a sequence of parallel frames
along $X_{t}$. The parallelism is defined under a Levi-Civita connection. The
advantages of such innovation are briefly introduced in the following:

  1. 1.

Once $V(X_{t})$ are parallel according to the curve development theory in [6],
$X_{t}$ can be represented uniquely as $(X_{0},V_{0},Z_{t})$. Such property
yields the invertibility of the proposed dimension reduction of $X_{t}$. In
other words, by the representative curve $Z_{t}$ in a lower dimensional space,
$X_{t}$ can be reconstructed when the initial conditions $(X_{0},V_{0})$ are
available.

  2. 2.

Under Levi-Civita connection, the parallel frames are calculated according to
metric of the manifold $M$. Such property implies that the shape of $Z_{t}$
will reflect well the shape of $X_{t}$ on $M$. For example, if $X_{t}$ is a
geodesic curve, then in the proposed method $Z_{t}$ will be a straight line in
Euclidean space

  3. 3.

The particular shape manifold [5] we adopted in this paper have a known
tangent and normal space in the ambient space which greatly simplified the
calculation of the moving frames under Levi-Civita connection.

The Levi-Civita connection may be viewed as the Christoffel Symbol $\Gamma$
that defines the directional derivative in $M$.

$D_{\partial x_{i}}\partial x_{j}=\sum_{k}\Gamma_{i,j}^{k}\partial x_{k}$ (10)

where $x_{i}$ is the coordinate function of an arbitrary point $m$ in $M$. The
tangents of $M$ can be written as linear combination of $\\{\partial
x_{i}\\}_{i=1,2,\cdots,dim(M)}$. For example at $X_{0}$,
$V_{i}(X_{0})=\sum_{j}a_{j}^{i}\partial x_{j}$ and
$dX_{0}=\sum_{j}b_{j}\partial x_{j}$. Thus according to Equation
$(\ref{eq.direc_direv})$, we can analytically calculate the derivative of
$V_{i}(X_{t})$ along $X_{t}$ in direction $dX_{t}$, which is denoted as
$D_{\frac{\partial X_{t}}{\partial t}}(V_{i}(X_{t}))$. $V(X(t))$ is parallel
if $\forall i$,

$D_{\frac{\partial X_{t}}{\partial t}}(V_{i}(X_{t}))=0.$ (11)

However in practical problems, it is usually impossible to chart the manifold.
In other words, the coordinate functions $x$ of $M$ are usually unknown. In
this paper, the shape manifold [5] we introduced in Section 2.1 have some nice
properties that allows us to calculate $D_{\frac{\partial X_{t}}{\partial
t}}(V_{i}(X_{t}))$ in terms of a projection of Euclidean calculus in the
ambient space onto the tangent space of the manifold $M$.

$D_{\frac{\partial X_{t}}{\partial
t}}(V_{i}(X_{t}))=\it{Proj}_{T_{X_{t}}M}\left(\frac{\partial\bar{V}_{i}(X_{t})}{\partial
t}\right),$ (12)

where $\bar{V}_{i}$ is the vector representing $V_{i}$ in the ambient space of
$M$. Since the ambient space is an affine space based on $\mathbb{L}^{2}$,
thus $\bar{V}_{i}$ is calculable as a real vector in $\mathbb{L}^{2}$. The
$\it{Proj}_{T_{X_{t}}M}$ is a mapping of vectors in $\mathbb{L}^{2}$ onto the
tangent space of $M$.

##  3 Dimension Reduction

The dynamics of some common human activities such as walking and running, are
relatively simpler in comparison to a complex activity like dancing. Thus it
is nature to speculate that the simple activity could be represented in a
lower dimensional subspace on the shape manifold. To learn such sub-manifold,
a dimension reduction is proposed for a curve on a shape manifold. As
described in the previous section, first the horizontal vector field
$V(X_{t})$ is defined by a Levi-Civita connection. Then Principle Component
Analysis (PCA) is carried out to compute the optimal vector field
$\tilde{V}(X_{t})=\\{x_{t},e_{1},e_{2},\cdots,e_{L}\\}$ such that all the
parallel transported tangents along $X_{t}$ could be represented in a lower
dimensional space.

The shape sequence on a manifold $M$ can be represented by a differential
equation as,

$X_{t}=X_{0}+\int_{0}^{t}\sum_{i=1}^{dim(M)}V_{i}(X_{t})dZ_{i}(t),$ (13)

where, $V(X_{t})$ is selected to be the horizontal vector field on a manifold
and $Z_{t}$ is the curve development in a Euclidean space. Such a formulation
have the property according to [7, 6] that the resulting $Z_{t}$ is 1-1
corresponded to $X_{t}$ with the initial condition $V(X_{0}),X_{0}$.

The idea of the proposed dimension reduction is to reduce the dimension for a
manifold valued curve $X_{t}$ by learning a sequence of subspaces
$\tilde{V}(X_{t})$ on the shape manifold $M$, such that the resulting $dZ_{t}$
have a distribution that concentrate in a lower dimensional subspace.

A Levi-Civita connection is adapted on the shape manifold according to the
$L^{2}$ metric induce from the ambient space. According to the definition of
covariant derivative under Levi-Civita connection as in Section 2.2, the
horizontal vector field $V(X_{t})$ is the solution of the following
differential equation.

$\forall i,D_{\frac{\partial X_{t}}{\partial t}}(V_{i}(X_{t}))=0,$ (14)

with initial condition $(X_{0},V(X_{0}))$.

The subspace moving frames $\tilde{V}(X_{t})$ for dimension reduction is
constructed by select the optimal initial condition
$V(X_{0})=\tilde{V}(X_{0})$ . First parallel transport all the tangents
$\frac{\partial X_{t}}{t}$ to the tangent space at $X_{0}$. With the moving
frame formulation in Equation $(\ref{eq.mov_frame})$, the parallel
transportation results is a set $\tau$

$\tau=\\{\sum_{i=1}^{dim(M)}V_{i}(X_{t})dZ_{i}(t)\\}_{t=1,2,\cdots,N}$ (15)

According to [7], the set $\tau$ is invariant to the choice of $V(X_{0})$.
Thus in the first step an arbitrary initial condition is assigned.

Then Let the $\\{e_{1},e_{2},e_{3},\cdots,e_{L}\\}$ be the eigenvectors that
corresponding to the $L$ largest eigenvalues of vectors in $\tau$. The optimal
initial condition is constructed as

$\tilde{V}_{i}(X_{0})=e_{i}.$ (16)

The solution $\tilde{V}(X_{t})$ of Equation $(\ref{eq.pde})$ with initial
condition $(X_{0},\tilde{V}(X_{0}))$ is the moving frames we used to represent
the $X_{t}$ by the corresponding $\tilde{Z}_{t}$ in a lower dimensional space
$R^{L}$

In the following, we provide the solution to the differential equation
$(\ref{eq.pde})$. According to Equation $(\ref{eq.proj_deri})$ and normal
space expression of the shape manifold $M$ in Section 2.1, the above equation
can be solved as the following. Let $\\{B_{1}(t),B_{2}(t),B{3}(t)\\}$ be the
orthogonalization of $\\{1,cos(X_{t}),sin(X_{t})\\}$, which is the basis of
normal space of the shape manifold $M$. $\forall i$,

$\frac{\partial V_{i}(X_{t})}{\partial
t}=a_{1}B_{1}(t_{0})+a_{2}B_{2}(t_{0})+a_{3}B_{3}(t_{0}).$ (17)

In the numerical calculation of the Euclidean derivatives
$\left(\frac{V_{i}(X_{t})}{\partial t}\right)$ are implemented as follows. For
a small enough $h>0$,

$\left(\frac{\partial V_{i}(X_{t})}{\partial
t}\right)=\frac{V_{i}(X_{t+h})-V_{i}(X_{t})}{h}.$ (18)

Substitute the Equation $(\ref{eq.direv_app})$ into the covariant derivative
Equation $(\ref{eq.cond1})$, $V_{i}(X_{t+h})$ could be written as,

$V_{i}(X_{t+h})=V_{i}(X_{t})+a_{1}B_{1}(t_{0})+a_{2}B_{2}(t_{0})+a_{3}B_{3}(t_{0}).$
(19)

Since $V_{i}(X_{t+h})\in T_{X_{t+h}}M$, the parameter $a_{i}$ should satisfy
that,

$<V_{i}(X_{t})+a_{1}B_{1}(t_{0})+a_{2}B_{2}(t_{0})+a_{3}B_{3}(t_{0}),B_{1}(t_{0}+h))>=0,$
$<V_{i}(X_{t})+a_{1}B_{1}(t_{0})+a_{2}B_{2}(t_{0})+a_{3}B_{3}(t_{0}),B_{2}(t_{0}+h)>=0,$

and

$<V_{i}(X_{t})+a_{1}B_{1}(t_{0})+a_{2}B_{2}(t_{0})+a_{3}B_{3}(t_{0}),B_{3}(t_{0}+h)>=0.$

From the above three equations, we have,

$\left(\begin{array}[]{c}a_{1}\\\ a_{2}\\\ a_{3}\end{array}\right)=-A^{-1}V,$
(20)

where $A=$

$\left(\begin{array}[]{ccc}B_{1}(X_{t})B_{1}(X_{t+h})&B_{2}(X_{t})B_{1}(X_{t+h})&B_{3}(X_{t})B_{1}(X_{t+h})\\\
B_{1}(X_{t})B_{2}(X_{t+h})&B_{2}(X_{t})B_{2}(X_{t+h})&B_{3}(X_{t})B_{2}(X_{t+h})\\\
B_{1}(X_{t})B_{3}(X_{t+h})&B_{2}(X_{t})B_{3}(X_{t+h})&B_{3}(X_{t})B_{3}(X_{t+h})\\\
\end{array}\right).$

So the parameter $a_{i}$ could be solved in a linear fashion. Thus the shape
sequence can be represented as follows,

$X_{t}=X_{0}+\int_{0}^{t}\sum_{i=1}^{L}\tilde{V}_{i}(X_{t})d\tilde{Z}_{i}(t),$
(21)

where $d\tilde{Z}_{i}(s)=<\frac{\partial X_{t}}{\partial
t},\tilde{V}_{i}(X_{t})>$ is a curve development in $R^{L}$.

Figure 2 illustrates the $R^{3}$ embedding of the shape sequence in Figure 1.
The curvature of the embedded curve in $R^{3}$ in Figure 2 represent well the
change of original shape sequence along time.

Figure 1: The original shape sequence for activity: Running

Figure 2: The embedding of a shape sequence in Figure 1 into $R^{3}$

In the following, the additional different activities from database [16] is
considered, such as Jumping, Siding, Skipping and Waving. The dimension
reduction result in Figures 3,5,4,6, also well visualized the evolution on the
shape manifold $M$.

Figure 3: (a) The original shape sequence; (b) The embedding into $R^{3}$; (c)
The reconstruction of the shape sequence from the embedding curve in $R^{3}$.

Figure 4: (a) A original shape sequence; (b) The embedding into $R^{3}$; (c)
the reconstruction of the shape sequence from its corresponding embedding
curve in $R^{3}$.

Figure 5: (a) A original shape sequence; (b) The embedding into $R^{3}$; (c)
The reconstruction of the shape sequence from its corresponding embedding
curve in $R^{3}$.

Figure 6: (a) A original shape sequence; (b) The embedding into $R^{3}$; (c)
The reconstruction of the shape sequence from its corresponding embedding
curve in $R^{3}$.

##  4 Reconstruction from the Lower Dimensional Representation

As introduced in the previous sections, one of the claimed advantages of the
proposed dimension reduction is that it is possible to generate the original
shape sequence from the dimension reduction results. Such an invertible
property makes our lower dimensional shape process embedding a compelling way
to learn the generative model of the shape sequence.

Let $\tilde{Z}(t)$ be the embedded shape process in $R^{L}$. A approximation
of the original sequence on the shape manifold $M$ may be generated by the
following differential equation:

$X_{t}=X_{0}+\int_{0}^{t}\sum_{i=1}^{L}\tilde{V}_{i}(X_{0})d\tilde{Z}_{i}(t)$
(22)

where the horizontal vector field $\tilde{V}_{i}(X_{t}))$ is generated
uniquely by the differential equation 14 with the initial frame
$X_{0},\tilde{V}_{i}(X_{0})$.

Numerically Equation $(\ref{eq.gen1})$ is implemented as a difference
function,

$X_{k}=f(X_{k-1})+\sum_{i=1}^{L}\tilde{V}_{i}(X_{0})d\tilde{Z}_{i}(k-1))$ (23)

where $f$ is the iterative projection from the ambient space to the shape
manifold as in [5], which is also briefly introduced in Section 2.1.

Figure 7 illustrates the reconstruction result of the shape sequence of
activity: running. The reconstruction is from the embedding curve
$\tilde{Z}(t)\in R^{3}$ and the initial condition $X_{0}$,
$\tilde{V}_{i}(X_{0})$ to the shape manifold $M$.

Figure 7: The reconstruction result for original sequence in Figure 1 with
embedding result in Figure 2

The Figure 7 demonstrates that the reconstruction approximates well the
original shape sequence. More reconstruction results of the shape sequence of
other human activities are shown in Figures 3,5,4,6.

##  5 Conclusion

Dimension reduction of points on a high dimensional manifold is well studied
in the past decades. Only a few works, however, address the problem of
dimension reduction of curves on a manifold. In this paper a novel dimension
reduction technology is proposed for shape dynamics with the following
advantages that, as far as we know, no previous work has ever achieved:

  1. 1.

The subspace is learned nonlinearly according to the geometry of the
underlying manifold.

  2. 2.

The computation is linear in the size of data.

  3. 3.

There exists an efficient reconstruction from the lower dimensional
representation to the original curve on a shape manifold.

  4. 4.

The proposed dimension reduction technique provides both analytical and
practical foundations for generative modeling of shape dynamics.

This work also first apply the differential geometry tools such as moving
frame representation and Levi-Civita connection to dimension reduction of
curves on a manifold. It provides a new perspective on the dimension reduction
with a moving frame formulation to naturally characterize the nonlinear
features of a manifold valued dynamics in a linear subspace.

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