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\section{Scientific Justification}  The Herschel observations of nearby molecular clouds find ubiquitous filamentary morphology of dust emission projected on the plane of sky. Although the ubiquity suggests that the filamentary morphology of projected emission is the result of filamentary structures in the 3D space, the spatial structure in the line of sight dimension of a single filament is rarely examined. While a direct measurement is impossible, {\bf the line of sight ``thickness'' can be deduced from the dependence of molecular line emission on the volume density} and {\bf the dependence of the Spectral Correlation Function on the spatial scales of self-similarity}. Here \emph{we propose to observe the ^{13}CO (2-1)/C^{18}O (2-1) and the N_2H^+ (3-2) molecular line emission in the filament FN1 in the Serpens Main molecular cloud}, which will allow us to test and compare these two methods in measuring the line of sight ``thickness.''  \subsection{Line \subsection{The 1st method: line  of sight ``thickness'' measured by cyano-molecules} Molecular transitions of cyano-molecules are sensitive to the local volume density. The ratio of emission from a higher transition to that from a lower transition is characterized by a sharp transition as a function of volume density (Fig. ?; Green \& Chapman 1978, Wernli et al. 2007). By measuring multiple transition of cyanoacetylene (HC_3N), Avery et al. (1982) and Schloerb et al. (1983) were able to derive a volume density for the TMC-1 region, and thus the line of sight ``thickness'' of the region by comparing to the column density measurements. Li et al. (2012) applied the same method to the Taurus B213 filament (Li et al. 2012) with HC_3N (4-3) and (10-9) transitions. The result in B213 successfully re-confirms that B213 is a cylindrical filament (Hacar et al. 2013) and has a ``thickness'' of 0.12 pc in the line of sight direction. This result also conforms with the width in the plane of sky of \~ 0.1 pc, as derived from the density profile across the filament (Pelmeirim et al. 2013).  N_2H^+ has a similar property. By observing line emission from multiple transitions of N_2H^+.... Here we propose to observe the N_2H^+ (3-2) transition line emission in one of the filaments in the Serpens Main region, as identified with the N_2H^+ (1-0) line emission from the CLASSy project (FN1 in Lee et al. 2014). The filament has a mild velocity gradient and a sub-/trans-sonic velocity dispersion. The filament also shows up in the HCO^+ (1-0) emission.  \subsection{Line \subsection{The 2nd method: line  of sight ``thickness'' measured by the Spectral Correlation Function} The Spectral Correlation Function (SPF) measures the degree of similarity between two spectra, and is proposed to be applied on analysis of spectral maps (Rosolowsky et al. 1999). Padoan et al. (2001a) further conclude that there is a dependence of the SPF on the ``spatial lag'' between the two spectra that the SPF takes into account. This dependence of the SPF on the spatial lag shows a power-law relation, and the spatial scales where this power-law relation exists characterize the spatial scales of self-similarity of turbulence (which is assumed to  dominates the spectra; Fig. ?). By computing the self-similar scales characterized by the SPF of HI spectra and assuming that the self-similarity of turbulence is confined to the shortest dimension in the 3D space, Padoan et al. (2001b) measured the ``depth'' (scale height) to be \~ 180 pc in the line of sight direction of LMC, which has a face-on disk structure and thus the shortest dimension along the line of sight. \subsection{Spatial and velocity structures of filaments}  Based largely on the Herschel observations, Andr\'e et al. (2013) proposed a paradigm of star formation through filament accretion and fragmentation, and the later has been directly observed in the Perseus B5 region by Pineda et al. (\emph{Nature}, 2014). Understanding the spatial and the velocity structures of filaments is then key to understanding the star formation process, and has been done largely through molecular line observations. However, the structures as seen through molecular line observations in the position-position-velocity (PPV) space does not fully correspond to structures in the position-position-position (PPP) space. Beaumont et al. (2013) measures \emph{the discrepancy of structures identified in the PPV space and in the PPP space in up-to-date star formation simulations to be \~ 40 \%}. This discrepancy further causes the virial parameters measured in the PPV space to diverge from those measured in the PPP space, typically by a factor of 2. This uncertainty makes it difficult to judge whether a region is dominated by gravitational or kinematic energy through virial parameter analysis.  \subsection{Serpens Main and the filament FN1: filaments in a hub}  Serpens Main is classified as a hub of filaments (Myers et al. 2009). The Herschel observations of dust emission and the CLASSy survey of N_2H^+, HCO^+ and HCN line emission further reveal the filamentary nature of the region (Andr\'e (Fig. ?; Andr\'e  et al. 2010; Lee et al. 2014). The CLASSy results provide quantitative estimates of the kinematics in these filaments. The filament FN1 is identified based on  the dendrogram analysis and its continuity in the position-position-velocity (PPV) space  (Lee et al. 2014). It is selected as our target to test the two methods above because of its large velocity gradient (3.2 km/s), which makes it well distinguishable in the PPV space. Also, FN1 is not associated with any YSOs, and thus does not suffer from kinematics and chemistry complications around YSOs. It is bright in the N_2H^+ (1-0) emission, which suggests the detection of N_2H^+ (3-2) and makes it an ideal target for testing the 1st method. The large velocity dispersion of FN1 indicates that trans-/super-sonic turbulence dominates the region, which follows the assumption of the 2nd method.  \subsection{Proposed observation}