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\subsection{The 1st method: Depths measured by comparing volume density and column density}  The excitation of molecular transitions is 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).   For example, by measuring multiple transitions of cyanoacetylene (HC_3N), Avery et al. (1982) and Schloerb et al.\ (1983) derived a volume density for the TMC-1 region, and thus the line of sight depths of the region by comparing to the column density measurements. Also, 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'' depth  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: Depths measured by 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). 
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%The first set of observations (N_2H^+ 3-2, at 279.511760 GHz) allows us to test the 1st method described above, which estimates the depths by the dependence of emission from N_2H^+ transitions on the volume density.   The N_2H^+ 3-2 line (at 279.511760 GHz) will be compared with the N_2H^+ 1-0 line from CLASSy, and the line ratio will be used to determine the volume density using RADEX (van der Tak et al.\ 2007), a radiative transfer code which constrains physical conditions and abundances from observations.   The depth is then estimated by comparing to column density maps based on pixel-by-pixel SED fittings from Herschel observations and the 2MASS/NICEST near-infrared extinction.  We plan to compare the volume densities based on the SMA observations with the ``averaged" column densities from a bigger beam of Herschel, and therefore we do not plan to match the angular resolution of the two observations.  The second set of observations (^{13}CO 2-1 and C^{18}O 2-1, at 220.398684 GHz and 219.560358 GHz, respectively) allows us to test the 2nd method.  %, which estimates the depth by the dependence of the Spectral Correlation Function on the spatial lag.   By calculating the Spectral Correlation Function of ^{13}CO (2-1) and C^{18}O (2-1), we will be able to produce a map of the SCF as a function of the spatial lag. By fitting to a power law, we can then derive the depth from the largest self-similar scale characterized by the power law.   The result from this method will be compared with the first method.  \section{Technical Justification}  To obtain maps that cover a sufficient region along the filament and across the filament, we propose mosaicking observations composed of ten pointings in the hexagonal arrangement to Nyquist-sample the region. At 270 GHz, this will give us a coverage of \~ 100 arcsec (\~ 0.23 pc at the diatance of Serpens Main) along the filament, and of \~ 80 arcsec (\~ 0.18 pc) across the filament. At 230 GHz, the coverage is \~ 1.3 times larger in length unit. (spectral setup...)  %\subsection{Complementary data and planned analysis}  %Data from the proposed observations are complemented by the CLASSy data taken by CARMA (Lee et al. 2014). Other complementary data include...