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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 sky (Andr$\acute{e}$ et al.\ 2010).   %Although  the ubiquity suggests that the filamentary morphology of projected emission is the result of filamentary structures in the 3D space, Constraining the ``depths" of these filamentary structures along the line of sight is very important as project effects play a critical role in the interpretation of observational results including density profiles (Juvela et al. 2012), mass-size relation (Kauffmann et al. 2010), linewidth-size relation (Shetty et al. 2010), and kinematics (Dib et al 2010).   A filament, as a projected two-dimensional structure, can be consistent with various theoretical models.  For example, a ``pancake-like" structure in the third dimension has been suggested by numerical simulations with supersonic turbulence and strong magnetic fields (Nakamura \& Li 2008) as well as those simulations with converging flows (Ballesteros-Paredes et al. 1999).  In addition, a ``cigar-like" structure may be generated from pure turbulence-driven MHD simulations (Padoan et al. 2001).  A few simulations suggest that gravitationally-driven clouds will collapse to sheet-like structures under the influence of self-gravity (Lin et al 1965).  Investigating the depths of filamentary structures observationally provides a valuable tool to discern between theoretical scenarios and thus provides insights to the formation mechanism of these structures.  However,  the spatial structure in the line of sight dimension of a single filament is rarely examined.While a direct measurement is impossible,  {\bf the The  line of sight ``thickness'' ``depths''  can be deduced derived  from the dependence of molecular line emission on the comparison between  volume density} density and column density},  and {\bf the dependence of the Spectral Correlation Function on the spatial scales of self-similarity}. Here \emph{we propose to observe the N_2H^+ (3-2) and the ^{13}CO (2-1)/C^{18}O (2-1) molecular line emission in the filament FN1 in the Serpens Main molecular cloud}, which cloud}. The proposed observations  will allow us to test independently derive  and comparethese two methods in measuring  the line depths  of sight ``thickness.'' filaments using these two methods.  \subsection{The 1st method: line of sight ``thickness'' Depths  measured by cyano-molecules} comparing volume density and column density}  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. 1; 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).  \subsection{The 2nd method: line of sight ``thickness'' Depths  measured bythe  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. 2). 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 (Fig. 2 \& Fig. 3).  \subsection{Spatial and velocity structures of filaments}