Fingerprinting the effects of hyperfine structure on CH and OH far infrared spectra using Wiener filter deconvolution

Context. Despite being a commonly observed feature, the modification of the velocity structure in spectral line profiles by hyperfine structure complicates the interpretation of spectroscopic data. This is particularly true for observations of simple molecules such as CH and OH toward the inner Galaxy, which show a great deal of velocity crowding. Aims. In this paper, we investigate the influence of hyperfine splitting on complex spectral lines, with the aim of evaluating canonical abundances by decomposing their dependence on hyperfine structures. This is achieved from first principles through deconvolution. Methods. We present high spectral resolution observations of the rotational ground state transitions of CH near 2 THz seen in absorption toward the strong FIR-continuum sources AGAL010.62−00.384, AGAL034.258+00.154, AGAL327.293−00.579, AGAL330.954−00.182, AGAL332.826−00.549, AGAL351.581−00.352 and SgrB2(M). These were observed with the GREAT instrument on board SOFIA. The observed line profiles of CH were deconvolved from the imprint left by the lines’ hyperfine structures using the Wiener filter deconvolution, an optimised kernel acting on direct deconvolution. Results. The quantitative analysis of the deconvolved spectra first entails the computation of CH column densities. Reliable N(CH) values are of importance owing to the status of CH as a powerful tracer for H2 in the diffuse regions of the interstellar medium. The N(OH)/N(CH) column density ratio is found to vary within an order of magnitude with values ranging from one to 10, for the individual sources that are located outside the Galactic centre. Using CH as a surrogate for H2, we determined the abundance of the OH molecule to be X(OH) = 1.09 × 10−7 with respect to H2. The radial distribution of CH column densities along the sightlines probed in this study, excluding SgrB2(M), showcase a dual peaked distribution peaking between 5 and 7 kpc. The similarity between the correspondingly derived column density profile of H2 with that of the CO-dark H2 gas traced by the cold neutral medium component of [CII] 158 μm emission across the Galactic plane, further emphasises the use of CH as a tracer for H2.


Introduction
Hyperfine structure (HFS) splitting is a commonly observed phenomenon in spectroscopy that arises from interactions between unfilled electron shells and the nuclear spin magnetic moment of atoms or molecules. It leaves a unique imprint on the spectrum of a given chemical species and transition, much like fingerprints. Over the years, knowledge of HFS in atoms and molecules has served both atomic-, molecular-, and astro-physicists alike in interpreting observations. The effects of such splitting, despite being small, leads to energy differences that are typically of the order of ∼ 0.03 K in terms of temperature. Since HFS interactions broaden, shift, and even alter the shape of spectra (Booth & Blackwell 1983), not taking them into consideration might lead to erroneous interpretations of astrophysical quantities. Overall, taking HFS splitting into account makes it possible to precisely model and resolve the underlying spectral line shapes, since the observed integral line profile is a superposition of the various HFS components. In recent years, the onset of improved observational techniques with greater sensitivity and higher spectral resolution have paved the way for rigorous spec-The reduced spectra (Fig. 3) are only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ troscopic studies and abundance measurements, in particular in the sub-millimetre and far-infrared (FIR) regions. Observations made between 2010 and 2013 with the Herschel-High Frequency Instrument for the Far-Infrared (HIFI, de Graauw et al. 2010), later followed by the German REceiver at Terahertz frequencies (GREAT, Heyminck et al. 2012) on board of the Stratospheric Observatory for Infrared Astronomy (SOFIA, Young et al. 2012) have resulted in high resolution (down to 10 −7 ) absorption spectra of a plethora of light hydride molecules of which the study is important for various reasons (for an overview see Gerin et al. 2016). In many cases, the background sources reside far away in the inner Galaxy. Thus the differential Galactic rotation causes complex velocity crowding, which is further complicated by the HFS of many of these molecules, including CH and OH, which are studied here. All this requires advanced techniques of analysis of the observed line profiles. Several different approaches have previously been used to fit the HFS of various spectral lines through techniques ranging from simulated annealing algorithms (Wiesemeyer et al. 2016) and Monte Carlo approaches that use genetic algorithms (Estalella 2017) to direct deconvolution (Gerin et al. 2010), and multi-Gaussian fitting techniques (Neufeld et al. 2015;Persson et al. 2016). In this paper, we introduce a technique that decomposes the contributions of HFS from astrophysical spectra, based on the theory developed by Wiener (1949). A linear and time-invariant method, the Wiener filter theory has found its application in practical signal reconstruction scenarios. It stands out from other filtering techniques because it provides the maximum a posteriori estimate of the unknown signal through a simple least squares analysis. Consequently, the Wiener filter has also found its application in astronomy as a tool that aids in cleaning contaminated images retrieved by aperture synthesis techniques (Caprari et al. 2000). In cosmological studies, deconvolution using the Wiener filter has been employed to accurately decompose the observed polarization of the cosmic microwave background (CMB) radiation into maps of pure polarization for its two components: E and B (Bunn & Wandelt 2017). Given the proven capability of the Wiener filter deconvolution as a method of choice for decomposition, we decided to adapt it for a novel application, the deconvolution of HFS from astrophysical spectra. Concentrating for most of this paper on the transitions of a single molecule, we illustrate the working of the Wiener filter algorithm on spectra that are of astrophysical importance, through the spectroscopy of the N, J = 2, 3/2 → 1, 1/2 ground state rotational transitions of CH, the methylidene radical. CH was amongst the first molecules detected in the interstellar medium (ISM) by Dunham (1937), and identified by Swings & Rosenfeld (1937). Being the simplest organic carbyne and one of the lightest hydrides, CH initiates the formation of a large fraction of the molecules present in the ISM, thereby playing a pivotal role in both the physics and chemistry of interstellar gas clouds. CH has been widely studied in various spectral regimes to infer the thermal, chemical and dynamical evolution of diffuse and translucent cloud regions (Rydbeck et al. 1973;Turner & Zuckerman 1974;Stacey et al. 1987). Moreover, the advent of space-and air-borne telescopes like Herschel and SOFIA have established the use of CH alongside other interstellar hydrides as tracers for H 2 in the diffuse regions of the ISM (Gerin et al. 2010;Wiesemeyer et al. 2018). Figure 1 shows the low-lying energy level structure of the ground electronic state of the CH molecule, abiding by Hund's case b. The total angular momentum states labelled J splits into a pair of Λ-doublet levels of opposing parity (±Λ). Each of these are further split into hyperfine levels given by F = J ± I, where I is the nuclear spin of the H atom, I(H) = 1/2. The outline of this paper is as follows: In Sect. 2.1, we develop the theory behind the working of the Wiener filter, and in Sect. 2.2, we detail our procedure for spectral line analysis: the Wiener filter algorithm. Having motivated our astronomical case study on the N, J = 2, 3/2 → 1, 1/2 HFS transitions of CH, in Sect. 3, we describe our observations. In Sect. 4, we present the implementation of the Wiener filter on the absorption spectra and its analysis, while the results obtained are discussed in Sect. 5, in which we also compare the distribution of CH with that of OH, and explore the usefulness of CH as a tracer for H 2 . We conclude this work in Sect.6.

Theory
Convolution is a formal mathematical operation used to describe the interaction between an input signal, f (ν), and an impulse response function, h(ν), to produce an output signal, g(ν), in a linear system. In spectroscopic terms, the input and output signals are analogous to the original and observed spectra while the impulse response in this case arises from the HFS splitting. For a discrete sample, such a convolution model can be mathematically formulated as follows: With astrophysical observations, one is posed with the inverse problem of convolution, which is deconvolution, wherein the output spectrum is known, while its main constituent, the original spectrum, remains unknown. The convolution theorem renders this task trivial, as it reduces the operations of both convolution and deconvolution into those of simple multiplication and division, in Fourier space, given a priori knowledge of g(ν) and h(ν). Hence, in Fourier space, Eq. 1 becomes where G(ν), F(ν) and H(ν) represent the Fourier transforms (FTs) of g(ν), f (ν) and h(ν), respectively. However, such a pointwise division of the Fourier transforms is sufficient to obtain an estimate of f (ν), only as long as the transform of the response function remains non-zero over all frequencies. Moreover, realistic systems are influenced by noise, n(ν), which further degrades the observed spectrum, d(ν). This additive noise is often amplified when directly deconvolved, because it acts as a lowpass filter and gives rise to faux features in the reconstructed signal. Such issues of sensitivity are often tackled by using filters, and in the following sections, we introduce and employ one such filter, namely the Wiener filter.

Wiener filter deconvolution
Wiener's theory formulated a linear tool, the Wiener filter (hereafter WF) for additive noise reduction aimed to resolve the signal restoration problems (singularities) faced by direct deconvolution. The WF model assumes all signals to be stationary 1 , and modelled by linear stochastic processes with a signalindependent noise. The WF output in the Fourier or inverse frequency domain,F(ν) is the product of the (noise) degraded observed spectrum, D(ν), and the frequency response of the filter, W(ν): The filter further models the input signal by implementing a minimum mean square error (MSE) 2 constraint on the deconvolution: Expanding the quadratic term in Eq. 6, We assume that N(ν) is independent of F(ν), therefore (F(ν).N(ν) * ) = (N(ν).F(ν) * ) = 0. This reduces Eq. 7 to, The derivative of the MSE, 2 with respect to W(ν) yields using ∂(zz * ) ∂z = 2z * (property of conjugates), where the asterisk represents the complex conjugate. Minimising ∂ 2 /∂W(ν) by equating Eq. 9 to zero and solving for W(ν) results in the general form of the Wiener filter, Since the exact forms of both N(ν) and F(ν) are modeldependent, they remain as unknowns for most practical systems. However, for additive white noise that is independent of the signal, the F(ν) N(ν) ratio can be approximated by the signal-to-noise ratio (S/N), or some form of normalized noise variance, σν, characterised by the system noise. Rearranging the terms to fit the inverse filter formulation, the Wiener filter and restored spectra are given as: (10) Figure 2 illustrates the idea behind the WF deconvolution algorithm as expressed in Eq. 11. Therefore, a reliable estimation of the "original", underlying spectrum using the WF deconvolution is dependent upon the assumptions made in determining the kernel term, W(ν), of the filter (Eq. 11). The kernel term is dependent on the noise of the system and on the response function, which in this case is tailored to address the effects of HFS splitting. The following section details the kernel estimation, as well as the adopted algorithm.

WF algorithm
This section features a brief description of our WF fitting and deconvolution algorithm written in python, based on the mathematical formulation discussed in Sect. 2.1. The WF algorithm uses the radiative transfer equation to express the observed line profile, T l , of a given molecular transition containing n HFS hyperfine components in terms of optical depths, τ. This is done under the assumption of local thermodynamic equilibrium (LTE) and unsaturated absorption. The formal solution of the radiative transfer equation for a constant excitation temperature reads: where the excitation temperature, T ex and the background continuum, T c are expressed as Rayleigh-Jeans equivalent temperatures, and Φ l and Φ c are the beam filling factors of the line and continuum emission, respectively. For the specific case of absorption spectroscopy of far-infrared ground state transitions in low-excitation gas (hν » kT ex ), we can neglect the first term and recast the radiative transfer equation displayed in Eq. 12 as T l = Φ c T c e −τ . In cases where we cannot neglect the emission term, particularly in the immediate environment of the continuum source, a realistic description needs to account for an excitation gradient, and therefore requires a full non-LTE solution, which is outside the scope of this paper. The next step of the algorithm determines the kernel term, which constitutes the hyperfine response and the noise term. Hence, the quality of the restored spectrum relies both on the impulse response of the HFS and an accurate representation of the system noise. The effects of HFS splitting are characterised by using weighted impulses in the frequency domain. The impulse response of each hyperfine component is represented by a Dirac-δ function, weighted by the relevant spectroscopic parameters. The weighting function, Ω (HFS) , accounts for the fraction of the specified species in the upper energy level, E u (in Kelvin) of a given transition that has an upper-level degeneracy, g u and Einstein A coefficient, A E as follows: For a given hyperfine transition, all of the above spectroscopic terms remain constant, except for the partition function, Q, which itself is a function of rotation temperature, T rot , which in our case (LTE) is equal to the excitation temperature, T ex . When deriving the formulation of the WF, in Sect. 2.1, the noise term was approximated to be the inverse of the S/N. While this serves as an acceptable choice for the noise contribution, it can be further refined and expressed explicitly as a function of the receiver system, which we will not discuss here in this paper.
The WF kernel is then determined following Eq. 10, and the algorithm then fits the spectrum. The standard deviation of the residual distribution is used to measure the quality of the spectral fit, and based on this, the algorithm carries out deconvolution following Eq. 11. In the last step, the WF algorithm applies an inverse FT to convert the modelled spectrumF(ν) intof (ν) in frequency space. The deconvolved optical depth signature, τ decon is then converted to molecular column density values using for each velocity channel, i, since, the backend provides the frequencies (and the velocities) as a discrete sequence of data, which was further smoothed by us to a useful spectral resolution as explained in Sect. 3 and where Ω represents the mean HFS weight. Having established the general scheme of the WF algorithm, the following sections detail its functioning on spectra that are of astrophysical importance.

GREAT observations and data reduction
The observations presented here were performed using up-GREAT 2 /LFA (Risacher et al. 2018), on board SOFIA over its cycle 5 and 6 campaigns, in a number of the observatory's flights (on 26 June and 6 July in 2017, and 20 and 22 June in 2018). Our source selection consists of several far-infrared bright hot cores from the ATLASGAL 3 survey (Csengeri et al. 2017) and SgrB2(M). The observed sources, whose strong far-infrared dust continuum emission serves as background for our absorption measurements, are listed in Table. 1. The receiver configuration is comprised of the (7+7) pixel low frequency array (LFA) receiver on upGREAT, in dual polarization. Only one set of HFS transitions near 2006 GHz (Table. 2) was observed, because the atmospheric transmission along some sight-lines was affected by telluric ozone absorption lines. The wavelengths of the ozone features at 149.1558 µm and 149.7208 µm originating from the signal band coincide with the CH hyperfine transitions at 2010 GHz. These ozone lines, despite having relatively narrow spectral features, contaminate the CH absorption measurements made toward the strong continuum sources. Therefore, the receivers were tuned to 2006.7 GHz in the upper side band mode. Both atmospheric as well as systematic fluctuations arising from the instrumental set-up were removed by using a double beam chop-nod mode. The secondary mirror chopped at a rate of 2.5 Hz with a chop amplitude of 60 and 105 for the observations carried out in 2017 and 2018. An advanced version of the MPIfR-built Fast Fourier transform spectrometer (dFFTS4G) (Klein 2017) was used as the backend to carry out the spectral analysis of the procured data. A velocity resolution of 0.036 km s −1 was achieved by using a 4 GHz bandwidth per pixel, with a channel spacing of 244 kHz over 16384 channels for almost all the 2006 GHz spectra. The spectra were further calibrated using the KALIBRATE program (Guan et al. 2012) with an underlying forward efficiency of 0.97 and a mainbeam efficiency of 0.68. The fully calibrated spectra were sub-

WF deconvolution
In Fig. 3, we display how the WF fits to the CH spectra observed towards the different lines of sight (LOS), following the algorithm described in Sect. 2.2. In Appendix A, we illustrate how the observed degraded spectrum is the convolution product of the deconvolved spectrum and the hyperfine weights. For two sources, we compare the line profiles of the observed spectrum with those of the deconvolved one in Fig. 4. The column densities determined from the former will be systematically larger over velocity intervals corresponding to the molecular clouds due to the line broadening effect of the HFS splitting, as evidenced from the differences between the two line profiles. While the extent of these variations are sightline-dependent, we find the column densities of CH determined from the observed spectra (prior to deconvolution) to be on average 24.8% larger than those determined post deconvolution, for the molecular cloud features. The quality of the WF fit toward the varied sightlines are assessed prior to deconvolution by means of the fit residuals (see Appendix B). Furthermore, we discuss the efficiency of the WF fit and deconvolution in Appendix C.
The column densities are derived following Eq. 14 by assuming an excitation temperature of ∼ 3.1 K, which is close to the temperature of the cosmological blackbody radiation (2.738 K), and was found to represent contributions of the Galactic interstellar radiation field (Gerin et al. 2010), and integrated over velocity intervals thought to be characteristic of spiral-arm and inter-arm features, based on the spiral-arm structure of the Milky Way presented by Reid et al. (2014), and consistent with those used by Godard et al. (2012) and Wiesemeyer et al. (2016). The ground state occupation assumption at LTE (T ex = T rot = 3.1 K) is valid for the physical conditions that prevail over diffuse and translucent regions, and is questionable only within the molecular cloud components themselves. The dense envelopes of these    (Müller et al. 2005). The frequencies were determined by Davidson et al. (2001). ( * ) -indicate the transitions that are not observed in this experiment. molecular clouds can result in higher values of T ex as a consequence of collisional excitation of CH. The consequences of determining column densities by assuming a single excitation temperature component along each LOS is briefly discussed in Appendix D. Upon modelling the absorption features associated with the dense molecular environments of our continuum sources, we find the derived column densities of CH increase by ∼ 15% at T ex = 10 K, and up to < 50% at T ex = 30 K. This further indicates that the uncertainties in our column density estimates at velocity intervals corresponding to those of the molecular cloud components predominantly arise from our assumption of a single excitation temperature. Hence, the column densities derived over their associated velocity components represent a lower limit on the N(CH) values. The derived column densities for each sightline are summarised in Table. 3. The CH column densities per velocity component are found to vary between ∼ 10 12 and 1.2 × 10 14 cm −2 along inter-arm and spiral-arm clouds along the different sightlines. We inspect the reliability of the column densities derived using the WF by comparing them with other techniques in Sect. 4.2.
As the optical depths are computed from the line-tocontinuum ratio, uncertainties in the continuum level give rise to systematic errors in the derived column densities. The uncertainties are partly due to the receiver-gain ratio between the signal and the image band, possibly deviating from unity. For the mixer technology (hot-electron bolometers) used by upGREAT, this quantity is difficult to measure without re-tuning the receiver. For the HIFI bands employing the same technology and clos-est to our tuning, a dedicated study (Kester et al. 2017) provides typical deviations of 1 to 4%. At this level, given the available sensitivity and typical optical depths ranging from 0.1 to 1, the impact of the continuum uncertainty on the derived column densities is difficult to disentangle from that of the thermal noise in the absorption profile. However, following the DSB error estimation presented in Wiesemeyer et al. (2018), we account for the uncertainties present in the continuum level that will subsequently lead to errors in the optical depths and derived column densities. Further, the errors in the WF computed column densities are determined by sampling a posterior distribution of the deconvolved optical depths 5 . We sampled 5000 artificial spectra, each generated by adding a pseudo-random noise contribution with the same standard deviation as the line free part of the continuum, to the absorption spectra prior to applying the WF fit and deconvolution, over each iteration. The deconvolved optical depths and subsequently derived column densities (per velocity interval) sample a point in the channel-wise distribution of the column densities across all spectra. The empirical spread of these distributions yield the asymmetric errors in the computed column densities. However, the column densities are not always normally distributed, and often showcase skewed distributions, as normality is not imposed. For this reason, the profiles of the distributions are best described by using the median and inter-5 The error in the derived column densities (per velocity interval) scale with the deconvolved optical depths by a constant value. This constant is a function of the spectroscopic parameters that govern the transition and the excitation temperature. quartile range from which the sample mean and standard deviation are derived following Wan et al. (2014). This is done so as to remove any biases introduced in the determination of the mean due to the asymmetry of the distribution. The new mean and standard deviation are then used to determine the asymmetric errors through positive and negative deviations. While errors in the deep absorption features corresponding to spiral-arm velocities arise from imperfections in the fit, the errors in the interarm gas features are dominated by uncertainties in the true continuum.

Comparison of WF algorithm with other procedures
In this section, we briefly compare column densities of the CH molecule inferred using the WF algorithm with those obtained from other commonly used techniques. To carry out an unbiased analysis, we performed the WF algorithm on the N = 1, J = 3/2 → 1/2 HFS spectra of CH near 532 GHz, obtained using Herschel/HIFI toward SgrB2(M) as a part of the HEXOS 6 survey toward SgrB2(M). The spectroscopic parameters of these HFS transitions are summarised in Table.   A comparison between the CH column densities derived using the WF algorithm and the XCLASS fits is displayed in Fig. 5 for all absorption features along the LOS except, for the strong absorption from the envelope of the SgrB2(M) molecular cloud, which traces denser gas. The weighted regression at lower column densities systematically follows a one-to-one correlation while at larger column densities the values derived using the WF algorithm are almost always smaller than those computed using XCLASS, with an overall ratio of N(CH) WFD /N(CH) XCLASS = 0.76 ± 0.06. The deviations may arise due to the intrinsically different means by which the two methods account for HFS splitting, an effect that is particularly prominent at the larger column density values. Additionally, inconsistencies in the postprocessing of the Herschel/HIFI data (baseline fitting, etc) also contribute to the observed deviations. Nonetheless, the CH abun-Article number, page 7 of 15 A&A proofs: manuscript no. WF_paper v lsr +Θ 0 sin(l)cos(b) , with R 0 = 8.3 kpc, Θ 0 = 240 km s −1 and assuming a flat Galactic rotation curve, i.e, Θ(R G ) = Θ 0 . Note that we do not account for the non-circular motion of the 3 kpc arm (Dame & Thaddeus 2008). (b) The positions of the four main spiral arms and the local arms have been inferred from the spiral arm characteristics presented in Reid et al. (2014)  dances determined using both methods are in agreement within the systematic uncertainties.
In addition, we also compare the column densities obtained using the WF deconvolution with those obtained by Wiesemeyer et al. (2018) when using simulated annealing (SA) to the N, J = 2, 3/2 → 1, 1/2 hyperfine transitions of CH toward W49 N. The column densities derived using both methods have a tight relation with a slope of, N(CH) WFD /N(CH) SA = 1.08±0.08. The column densities derived using the two methods deviate in value only within the inter-arm gas components and the overall concordances between the two methods further validates the use of the non-iterative WF deconvolution scheme.

Results
In Sect. 5.1, we investigate the correlation between the column densities of CH with values determined for another widely studied light hydride, hydroxyl (OH), and evaluate its abundance using CH as a proxy for H 2 . To further appreciate the global characteristics of the CH molecule as a surrogate for H 2 within the Milky Way, in Sect. 5.2, we study its radial distribution across the Galactic plane.

CH versus OH column density
For quite some time, CH has been used as a tracer for H 2 in the diffuse regions of the ISM based on its observed correlation with H 2 (Federman & Willson 1982;Mattila 1986;Sheffer et al. 2008). As a precursor to the formation of CO, OH has also gained recognition as a promising tracer for the "COdark" component (Grenier et al. 2005) of molecular gas in the ISM (Tang et al. 2017;Li et al. 2018;Engelke & Allen 2018;Rugel et al. 2018). In Table 3, we present column densities of H 2 estimated using the N(CH)/N(H 2 )∼ 3.5 +2.1 −1.4 × 10 −8 relation derived by Sheffer et al. (2008). This correlation was established using optical observations of the A 2 ∆ -X 2 Π system of CH at 4300 Å and the (2-0), (3-0), and (4-0) bands of the Lyman B-X transitions of H 2 toward 37 bright stars. Akin to the CH observations presented in our study, the optical CH and UV H 2 lines presented by Sheffer et al. (2008) are seen in absorption toward sightlines that probe molecular clouds in the local diffuse ISM (10 19 < N(H 2 ) < 10 21 cm −2 ).
In order to carry out a comparison between N(OH) and N(CH), we use data published in Wiesemeyer et al. (2016) of the 2 Π 3/2 , J = 5/2 → 3/2 ground-state transitions of OH, procured using GREAT/SOFIA as a part of the observatory's cycle 1 campaign. The spectroscopic parameters of these OH transitions are summarised in Table. E.1. For this preliminary search, observations were carried out toward all the sightlines discussed in this paper, except for SgrB2(M). The OH spectra are saturated at velocities corresponding to the SFR-related molecular cloud components (that provide the background continuum emission) themselves, as shown in Fig. 6. The saturated absorption features of the OH spectra arising in the envelopes of the hot cores of these star-forming regions are indicative of the fact that a complete ground-state occupation assumption is no longer valid (Csengeri et al. 2012). Moreover, from Fig. 6, it can be seen that most of the observed OH absorption features have corresponding features in the CH spectrum with notable differences at velocity intervals between −13 and −21 km s −1 , and from −36 to −42 km s −1 , in this example. The WF fit and deconvolution is applied to the OH spectra with a T ex = 3.1 K (for consistency), and integrated over the same velocity intervals as those used for CH. Our assumption of a complete ground-state occupation for OH is broken down for spectral features associated with the dense molecular clouds. The WF faces singularities when T l tends to zero and thus avoids the OH features arising from these regions because they showcase saturated absorption. Therefore, the use of a lower excitation temperature does not affect our analysis.
The derived CH and OH column densities are presented in Fig. 7. The used data correspond to column densities based on values computed within the velocity intervals used in Table. 3. The plotted data points represent values re-sampled to 1 km s −1 wide velocity bins (except towards the velocity intervals corresponding to the molecular clouds, as well as those contaminated by outflows). The N(CH)-N(OH) correlation plots for the individual sources are presented in Appendix F. Across the different sightlines, we obtained a range of N(OH)/N(CH) values between one and 10. Moreover, almost all the sources, except AGAL351.581−00.352 and AGAL332.826−00.549, have significant correlation coefficients between 0.46 and 0.75 with false-alarm probabilities below 5% (summarised in Table. F.1) within the spiral arm data, while there is no remarkable corre- lation observed in the inter-arm regions. The lack of, or weak correlations present in the above mentioned sources can be attributed to the fact that the OH spectra toward these two sources showcase more saturated features along the LOS than in the remainder of the sources. Fitting a linear regression to the combination of all the sightlines yields N v (OH)/N v (CH) = (3.85 ± 0.15), with a Pearson's correlation coefficient of 0.83 (false-alarm probability below 1%) for velocity intervals of 1 km s −1 . The functional form of the regression was chosen after comparing the standard error of a regression fit through the origin with that of an ordinary least square regression, both of which yielded best fit slopes that are consistent with one another (within a 5% error). From Fig. 7, we find that this relation spans roughly two orders of magnitude between 0.1 and 10, with a majority of the ratios lying between one and 10. We noted that those data points with N(OH)/N(CH) > 10 correspond to the broad line wings of saturated OH features, while those with N(OH)/N(CH) < 1 have larger errors in the determined N(OH) arising from their continuum-level uncertainties. Since the spectral features arising from star-forming regions are unsaturated in the case of the CH molecule, its effective broadening into the inter-arm regions is less. We further extend this analysis by combining our data set with optical observations 8 of the CH B-X transitions at 3886 and 3890 Å, and OH A-X transitions at 3078 and 3082 Å presented toward 20 bright OBtype stars at low Galactic latitudes (−17 • < b < +23 • ) from Weselak et al. (2010). Moreover, Weselak et al. (2010) show that the LOS toward these stars probe physical conditions (gas densities, n(H) = 250-600 cm −3 and temperatures, T kin = 20−30 K) that are typical for diffuse molecular clouds (Snow & McCall 2006) similar to those of our data set, and report a N(OH)/N(CH) value of 2.52±0.35. The revised abundance ratio (combining both sets of data) displayed in Fig. 8 follows N(OH)/N(CH) = (3.14 ± 0.49). Using this abundance ratio and the Sheffer et al. (2008) relation, we derive the OH abundance using CH as a surrogate for H 2 to be X OH = N(OH)/N(H 2 ) = (1.09 ± 0.27) × 10 −7 . This result is consistent with the column density ratios presented by Liszt & Lucas (2002), and is in the range of theoretical predictions presented by Albertsson et al. (2014). When using a N(CH)/N(H 2 ) ratio of ∼ 4.97 × 10 −8 as derived by Weselak (2019), we find the corresponding OH abundance to be X(OH) = 1.56 × 10 −7 .

Radial distribution of CH
The azimuthally-averaged CH column densities obtained toward all the sightlines in this study, except SgrB2(M), are plotted as a function of galactocentric distances, R G , in Fig. 9. The galactocentric distances to the different spiral arm features were computed assuming a flat rotation curve, with the distance of the Sun from the Galactic centre given by R 0 = 8.3 kpc, and an orbital velocity of Θ 0 = 240 km s −1 of the Sun, with respect to the Galactic centre . The uncertainties in the CH column densities are typically of the order of a few 10 12 cm −2 (km s −1 ) −1 , and are too small to be visible in some parts of the plot. This analysis is limited to a small number of strong background continuum sources, distributed at low Galactic latitudes, which lie close to the Galactic plane. We find that the CH column densities peak at galactocentric distances between 4 and 8 kpc. LOSs within the l = 0-180 • range dominate contributions toward the first column density peak ob- served around 5 kpc, while the second peak near 7 kpc originates from LOSs in the l = 180-360 • range. The column density peak near 5 kpc is associated with absorption arising from a combination of the Perseus and Norma spiral-arm crossings, and the 7 kpc peak may be attributed to the intersection of the Scutum-Crux spiral-arm crossing along the sightlines. The higher observed column densities peaking near these spiral arms can be attributed to the larger (mass) molecular content that is present at these galactocentric radii. Owing to the larger error bars, the significance of a third peak near 9 kpc is unclear. However, it is in line with contributions that may arise from the Scutum-and Sagittarius-arm far side crossings. Due to a lack of data points at galactocentric distances smaller than 2 kpc, the nature of the CH molecular gas distribution there remains unknown. We do not include contributions from the sightline toward SgrB2(M) in this analysis, because in the Galactic centre direction, it is difficult to assign the CH absorption to different components based on their LSR velocities. A similar dual-peaked radial distribution profile has been reported by both Pineda et al. (2013), and Velusamy & Langer (2014) for the cold neutral medium (CNM) component of [CII] emission.
[CII] is a proficient tracer that is sensitive to various evolutionary stages of the ISM, but by taking advantage of differing densities, Pineda et al. (2013) were able to decompose contributions of the [CII] emissivity from the different phases of the ISM, excluding the warm ionised medium (WIM) for sources in the Galactic plane, b = 0 • . Despite the smaller sample size, our results broadly resemble the shape of the [CII] integrated intensity distribution profile.
The framework of carbon chemistry in the diffuse regions, which is initiated by the radiative association of the ground vibrational state of H 2 with ionised carbon, explains the observed correspondence between CH and [CII] radial profiles (Gerin et al. 2016). Peaking at the same R G values as those of HI (see Fig. 12 in Pineda et al. (2013)), suggests that CH likely traces the more diffuse regions of the CNM between [CII] and the extended HI emission. Upon comparing the distribution of H 2 column densities derived from CH (using the Sheffer et al. (2008) relation) with those derived from both CO (CO-traced H 2 gas) and [CII] (CO-dark H 2 gas) by Pineda et al. (2013) (their Fig. 18) we find that: (1) at R G < 2 kpc, we cannot comment on the behaviour of the distribution due to the lack of data points, (2) 2 kpc < R G < 8 kpc, the CH-traced H 2 column density is comparable to the [CII]-traced H 2 column density, while being smaller than the CO-traced H 2 column density and, (3) R G > 8 kpc, despite attaining a constant level at these distances, similar to the COdark H 2 gas traced by [CII], the column density distribution of H 2 obtained using CH is lower than that traced by [CII].
The general agreement between the column density of the CO-dark gas traced by [CII] and the H 2 column densities traced by CH at galactocentric distances < 8 kpc, is a good indicator of the usefulness of CH as a tracer for the CO-dark component of molecular gas.

Conclusions
In this paper, we presented the analysis of SOFIA/GREAT observations of the N, J = 2, 3/2 → 1, 1/2 transitions of CH detected in absorption toward strong, FIR-bright continuum sources by using the Wiener filter algorithm. Although it has previously not been used as a conventional tool for spectral line analysis, we find the WF deconvolution algorithm to be a self-consistent concept that sufficiently alleviates biases manifested in spectra with close hyperfine spacing, provided that the transitions are optically thin. The absorption features along each LOS are broadly classified into contributions arising from different spiral arm and inter-arm regions based on the spiral arm model of the Milky Way presented by Reid et al. (2014). Using the WF as the basis for our spectral analysis, we derived a linear relationship be-tween CH and OH, N(OH)/N(CH) = 3.85±0.15 that is consistent with previous work. Upon increasing the number of sightlines in this analysis by including those studied by Weselak et al. (2010), we derived a revised relationship with a N(OH)/N(CH) = 3.14 ± 0.49. By subsequently using CH as a surrogate for tracing H 2 , we derived an OH abundance of X(OH) = (1.09 ± 0.27)×10 −7 . It is important to note that this result pivots on the validity of the N(CH)-N(H 2 ) relation used. We find the radial distribution of the CH abundances of our sample to peak between 4 and 8 kpc. In addition to the tight correlation that exists between the CH and H 2 established by Sheffer et al. (2008), the likeness of the CH abundance distribution to that of [CII] , a well known tracer, of the CO-dark H 2 gas, lends credence to its use as tracer for H 2 in these regions. Furthermore, this work may initiate follow-up observational studies, investigating a larger sample of sources covering a wide range of physical conditions at varying galactocentric distances to study the Galactic distribution of CH. This will reinforce its use as a tracer for H 2 in regions where hydrogen is molecular, and carbon may be both neutral or ionised, but not traced by CO, over Galactic scales.  advantages, as it is less time-consuming, and, additionally, aids in any subsequent Monte Carlo-based error estimates. The wavelet transform (WT) is another method used to quantify the degree of noise attenuation. Based on the quantization of a signal in the form of wavelets, the WT has the advantage that it characterises the local behaviour of a signal while maintaining all temporal information, unlike the FT. The Fourier-wavelet regularisation deconvolution (ForWARD) algorithm developed by Neelamani et al. (2004) is one such technique that combines deconvolution in the Fourier domain with noise suppression in the wavelet domain. In this approach, the WF deconvolution product is first decomposed in the wavelet domain into a scaling function, which acts as a low-pass filter and a wavelet function or a band-pass filter, which in turn constitutes each wavelet level. A second step of wavelet de-noising was added to our existing WF algorithm, in order to improve the robustness and efficiency of this method. Once again, we visualise the fit through its residuals as shown in In general, both sets of residuals are scattered randomly and centred around zero. The outliers in the WF fit are curbed when it is followed by the use of the WT, giving rise to a less variant or narrower distribution. However, the wavelet domain analysis is more suited to transient or time-varying signals, whereas our spectra are time independent. A subtle problem that this creates in a time-invariant system is that the wavelet domain is sparse (reduced number of data points) because of the wavelet decomposition of the spectral signal, and remains sparse over consecutive iterations. This makes it difficult to conclude whether the WT actually takes into account the outliers, or whether their effects are averaged out in the process of forming wavelets. While the addition of the WT to the WF deconvolution makes it more robust and rigorous, it has a longer computational time, and application to time-invariant systems makes it non-ideal.

Appendix D: Impact of using a single excitation temperature on derived column densities
The column densities presented in Table. 3 are derived using the WF deconvolution by assuming a single excitation temperature . Secondary x-axis represents H 2 column densities obtained using the CH/H 2 abundance ratio given by Sheffer et al. (2008). The corresponding Pearson's correlation coefficients are given in the lower-right corners.