Miscellaneous Information

Data Backup

Due to a limit on the capacity of the data storage, the observatory does not back up your OTF raw data and their outgrowth: they are removed in about 90 days . Therefore the observers must backup their own data. We strongly recommend to bring external HDD or laptop PCs which are capable to store the data.

Derivation of Observation Parameters and Sensitivity

Suppose a mapping area of l1 ["] times l2 ["] (l1 along the scan, l2 across the scan). Time to be taken in a scan row (on-source) is tscan [s]; scan speed on the celestial sphere is vscan ["/s] = l1/tscan; separation between the scan rows is {\mit\Delta}l ["]; number of scan rows is Nrow = l2/{\mit\Delta} l   +1 (for single-beam receiver) or Nrow = l2/(5 {\mit\Delta} l)   +1 (for BEARS); and the grid spacing of a map to be made is d ["] \times d ["]. And we call the number of on-source scans per one OFF as NscanSEQ.

In this situation, total on-source scan time becomes
t_{\rm ONtot} = N_{\rm row}t_{\rm scan} .
The total time spent to run an observe table including OFF, R-SKY, antenna slew, etc. is estimated to be
t_{\rm OBStot} = N_{\rm row} \left(t_{\rm scan} + t_{\rm OH} + \frac{t_{\rm OFF}}{N_{\rm scan}^{\rm SEQ}} \right) f_{\rm cal} ,
where tOFF [s] is an integration time for one OFF, and fcal is an overhead of R-SKY calibration (if you use 1 minute to obtain R-SKY at every 15 minutes, fcal=16/15). tOH [s] is an overhead time per one scan row, which consists of go and return to the OFF point 2\timesttranOFF, time for approach run tapp [s], and time for transit run ttran [s], thus is written as
t_{\rm OH} = \frac{2t_{\rm tran}^{\rm OFF}}{N_{\rm scan}^{\rm SEQ}} + t_{\rm app} + \frac{N_{\rm scan}^{\rm SEQ}-1}{N_{\rm scan}^{\rm SEQ}} t_{\rm tran}.
Using the distance to the OFF point dOFF [arcmin], ttranOFF is empirically written as ttranOFF = ceil( 4.4\timesdOFF0.26 ) . On the other hand, tapp and ttran can be obtained using a relation shown in Fig. 2-1.

Now the ratio of on-source time to the total time spent is
\eta_{\rm ON/OBS} = \frac{t_{\rm ONtot}}{t_{\rm OBStot}} = \frac{t_{\rm scan}}{t_{\rm scan}+t_{\rm OH}+t_{\rm OFF}/N_{\rm scan}^{\rm SEQ}} \cdot \frac{1}{f_{\rm cal}} .

The total ON-source integration time for a map grid point is a sum of time during which the beam(s) scans within the grid. Effectively a factor \eta is multiplied and it becomes
t_{\rm cell}^{\rm ON} = \frac{\eta t_{\rm scan} d^2}{l_1 {\mit\Delta}l} \mbox{ (single-beam), } \frac{5\eta t_{\rm scan} d^2}{l_1 {\mit\Delta}l} \mbox{ (BEARS)} .
The factor \eta is a constant determined by the type and parameters of the convolution function and is calculated as follows. Suppose that observed points i = 1,2,... are uniformly distributed around the grid point and each point has a spectrum of Ti(k) [k = 1,..,Nch], rms noise temperature of \sigmai , and a weight of the convolution function of wi . We assume the on-source integration time t0 and the noise temperature \sigmai = \sigma0 = Tsys/sqrt(B t0) of each point to be constant. The convolved spectrum T(k) is written as T = (\sumwiTi)/(\sumwi) , and its noise temperature \sigma becomes \sigma = sqrt(\sumwi2)/\sumwi \times \sigma0 = Tsys/sqrt(B tcellON) [here tcellON \equiv (\sumwi)2/\sum(wi2) \times t0 ]. If we take the grid spacing as the unit of spacial length and redefine t0 as the on-source integration time per unit area (1 grid cell), summations can be rewritten with integrals: tcellON = (\intw dx dy)2/\intw2dx dy \times t0 \equiv \etat0 . The values of \eta for convolution functions Bessel*Gauss, Sinc*Gauss, Gauss, Pillbox, and Spheroidal with default parameters are, respectively, 4.3, 1.2, 6.3, 1.0, and 10.2.

Assuming that the system noise temperature is Tsys [K] and the spectral resolution of a map to be made is B [Hz], the noise (of the map) due to on-source integrations is estimated to be
{\mit\Delta}T_{\rm A}^*({\rm ON}) = \frac{T_{\rm sys}}{\eta_{\rm q}\sqrt{B t_{\rm cell}^{\rm ON}}} ,
where \etaq is the quantization efficiency of the spectrometer. The digital backend (AC) for 45-m is a 2-bit system and has \etaq=0.88. On the other hand, the number of OFF points used to consist a map grid point is about 1+(d-{\mit\Delta}l)/ (NscanSEQ{\mit\Delta}l) (for single-beam receiver) or 5d/{\mit\Delta}l (for BEARS) [in fact the effective number of OFF points is larger than this value, since the convolution function spreads out of the cell. Taking this into account, the optimum OFF-point integration time becomes smaller by a factor of, roughly, \eta-1/4. The observing efficiency hardly changes] . When NscanSEQ is small, and d and {\mit\Delta}l are the same order (applicable for almost all the observations), number of OFF points used for single-beam RX can be approximated as d/{\mit\Delta}l. Thus the effective OFF integration time for a grid is
t_cell^off = t_off x d/delta l (single-beam), 5 x t_off x d/delta l (BEARS) ,
and the noise due to OFF points becomes
ΔTa*off = Tsys/eta_q*sqrt(B t_cell^on) .
Therefore the total noise level of the map is written as
delta Ta* = sqrt(delta Ta*(on)^2+ delta Ta*(off)^2) = Tsys eta_q^(-1) B^(-1/2) sqrt(t_cell(on)^(-1)+t_cell(off)^(-1)) .

The noise level of a map achieved in unit observation time {\mit\Delta}TA*(0) is written as
{\mit\Delta}T_{\rm A}^*(0) = {\mit\Delta}T_{\rm A}^* \sqrt{t_{\rm OBStot}} = \frac{T_{\rm sys}}{\eta_{\rm q}\sqrt{B}} \sqrt{\left( \frac{1}{t_{\rm cell}^{\rm ON}} + \frac{1}{t_{\rm cell}^{\rm OFF}} \right) \left(t_{\rm scan} + t_{\rm OH} + \frac{t_{\rm OFF}}{N_{\rm scan}^{\rm SEQ}} \right) N_{\rm row}f_{\rm cal}} .
This value is minimized when tOFF is optimal:
\frac{\partial}{\partial t_{\rm OFF}} {\mit\Delta}T_{\rm A}^*(0) = 0
leads to
t_{\rm OFF}^{\rm optimal} \simeq \sqrt{\left(t_{\rm scan}+t_{\rm OH} \right) \frac{\eta d t_{\rm scan}}{l_1}} \sqrt{N_{\rm scan}^{\rm SEQ}} .

Fig. 2-1: Effective beam after convolution
Fig. 2-1: Empirical relation between the antenna driving speed and tapp/ttran.

Convolution in "Make Map"

Convolution functions implemented are as follows (Fig. 1-2):

Fig. 1-2: Convolution functions
Fig. 1-2: Convolution functions implemented so far; Bessel*Gauss (a=1.55/\pi, b=2.52), Sinc*Gauss (a=1.55/\pi, b=2.52), Gauss (a=1.0), Pillbox, and Spheroidal (m=6, \alpha=1.0).

As shown in Fig. 1-2, the default convolution function, Bessel*Gauss, has an FWHM of twice of the map grid spacing d. Fig. 5-1 shows responses to a point source (i.e., effective beam) when grid spacing is set to be 0.1, 0.2, ..., 1.0 times the telescope beam FWHM. Peak intensity of a point source and effective beam width varies as Fig. 5-2. For example, when the grid spacing d is a half of the beam FWHM, peak temperature of a point source decreases by a factor of 0.7, and the effective beam width becomes 1.3 times broader.

If d is too small, effective integration time for a grid (tcellON) becomes small, which leads to a large noise level. Since the spatial resolution is limited by the telescope beam, the map is too much oversampled. On the other hand, if d is too large, the effective beam width broadens to 2d as shown here. An appropriate grid spacing should be chosen according to the scientific aim.

In principle, the 45-m dish responds to the spatial structure of the source up to the scale of lambda/45[m]. Therefore, if you intend to preserve the spatial information of the data (i.e., to avoid the aliasing effect) as much as possible, the grid spacing must be smaller than lambda/45[m]/2 (for lambda=2.6 [mm], it becomes 6.0 ["]). Please keep this in mind, in particular if the map is to be combined with interferometric data.

Fig. 5-1: Effective beam after convolution
Fig. 5-1: responses to a point source (effective beam) when grid spacing is set to be 0.1, 0.2, ..., 1.0 times the telescope beam (FWHM). The horizontal axis is the distance from the source (normalized by the beam FWHM), while the vertical axis is the observed intensity (normalized by the value without convolution). The telescope beam is assumed to be a Gaussian.

Fig. 5-2: Effective beam after convolution
Fig. 5-2: Variations of (left) peak intensity of a point source, (right) effective beam width (FWHM) when the grid spacing (the horizontal axis) varies from 0 to 1 times the beam FWHM. The vertical axes is normalized by the value without convolution.

A C program convbeam.c which calculates an effective beam (convbeam2d.c for 2-d version). The telescope beam size, grid spacing, and type of convolution function can be specified.

% cc -lm -o convbeam convbeam.c
% ./convbeam > hoge.dat
HPBW of the telescope [arcsec]:15
Grid spacing of the map [arcsec]:6
Conv function [0:B*G 1:S*G 2:G 3:PB 4:SF]:0
FWHM ~ 17.4 [arcsec]

If the telescope beam is 15 ["], the grid size is 6 ["] and the convolution funcion is the "Bessel*Gauss" type, FWHM of the effective beam is about 17 ["]. Sets of radius and response to a point source are written into the redirected standard output, hoge.dat (the FWHM is written in the standard error output).

GLS Projection in "Make Map" and WCS

When a map is made, celestial (spherical) coordinate values are projected onto an X-Y (planar) coordinate pixel values. By default, global sinusoidal projection (GLS) is applied: i.e., celestial coordinate (RA,DEC) is transferred into (X,Y) as

sin(X/2) = sin((RA-RA0)/2)*cos(DEC)
Y = DEC-DEC0

where (RA0,DEC0) is a reference position [in general, source position defined in the "Source Table"]. The Galactic coordinate (l,b) is also converted to (X,Y) in the same way.

Coordinate headers of FITS cubes written by the "Make Map" task follow the AIPS format:

CRVAL1 = RA0
CRPIX1 = 1 - XBLC/CDELT1 = 1 - (2*asin(sin((RABLC-RA0)/2)*cos(DEC)))/CDELT1
CRVAL2 = DEC0
CRPIX2 = 1 - YBLC/CDELT2 = 1 - (DECBLC-DEC0)/CDELT2

Here it should be noted that the center of projection is not (RA0,DEC0), but (RA0,0). It differs from the World Coordinate System (WCS) manners, thus you have to take care if you would like to convert the coordinate values into WCS.

Bit Number of Spectrum Data

OTF raw data has a dynamic range of 12 bits (4096 levels). Though the bit number is smaller than that of NewStar raw data (32 bit; 4.3 billion levels), it is unavoidable to reduce bit number because data production rate is quite high. Since the signal-to-noise ratio of individual spectrum taken with OTF is low due to very short (0.1 s) integration, sensitivity loss due to the quantumization is, in general, negligible. However, the data may be affected in the following situation: bandpass becomes nearly 0 (i.e., TA* diverges) in the bandwidth; or an extremely (really extremely) strong spurious signal appears.

In particular, when you use the AC in 32-MHz mode, or 512-MHz spectrometry of the output from single-beam receivers, divergence of TA* often happens because of characteristics of bandpass filters. We strongly recommend to cut off channels at band edges in the Scan Table.

Doppler Correction

In OTF observations with 45-m, Doppler correction for vrad (relative motion of the telescope with respect to the LSR/Heliocentric system) is done in the data production ("merge") process after the observations. That is, LO frequency shift according to the change of vrad is not performed during the observations. At first (ON-OFF)/(R-SKY) operation is performed channel by channel from the raw output from the spectrometer (R, SKY, OFF, and ON), and then resampling along the frequency axis according to vrad for each ON point is done. If there is emission at OFF point, the it appears like an absorption profile in the resultant spectrum. The velocity of "absorption" feature gradually shift according to vrad of ON point. This effect is hard to be corrected: it should be checked carefully enough that there is no emission at the OFF point.

If we calculate (ON-OFF)/(R-SKY) after resampling R, SKY, OFF, and ON along the frequency individually, OFF-point emission will appear as an absorption in a constant velocity. However, this procedure cannot be adopted since the "differential" of the band characteristics heavily affects the spectral baseline.

FAQ

A noise rms map has a stripe pattern ?

In order to do "Doppler Tracking", we use the software channel shift system in the data processing (see here); that is, the shift correction smaller than 1ch is carried out by interpolation of adjacent 2 channels, and this results in the noise rms pattern. Because the Vrad, the factor of the Doppler tracking, varies with the changing of spatial position, the noise rms pattern turns out to be periodic and striped.

stripe

This problem, however, would affect little or no effect to the intensity of spectra if each width of spectral lines is larger than the resolution of the spectrometer.

Blank value is written in my FITS cube. Why?
Blank is written when there are not sufficient number ("Minimum number of data" in the Map GUI) around the grid point. Caution: the map command skips spectra whose baselines are not subtracted yet.
I have mistook the IF configuration and the frequency shifted. What should I do?
The data may become usable by (1) rewriting header FRQ00 to the frequency which has been actually observed, and (2) specifying a reference frequency (rest frequency of the line) when executing the Map command. The header can be modified using the task "Modify Header" in the button panel.
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Last Update: 1st June 2008