Appendix A: The noise in an integrated HI map

This Appendix explains how the noise in a total HI map can be calculated. A total HI map is usually constructed from a 3 dimensional datacube containing a number of so called channelmaps. Each channelmap shows an HI image of the galaxy at a certain velocity. A total HI map is made by adding those channelmaps which contain the HI signal. Before adding the channel maps the signal in each channelmap should be isolated. When the signal is not isolated one merely adds noise to the total HI map because the location of the signal in a channel map varies with velocity due to the galactic rotation. The signals can be isolated interactively by blotting away the surrounding noise or in a more objective way by taking a certain contour level in the smoothed maps as a mask. As a consequence of adding channel maps with isolated regions, the noise in the total HI map is not constant but varies from pixel to pixel. The noise at a certain pixel in the total HI map depends on the number N of non-blank pixels at the same position in the individual channel maps that were added.

In case the data cube was obtained with an uniform taper during the observation, the noise $\sigma^{u}$ in two channelmaps will be independent. The noise equivalent bandwidth B^u in a uniform tapered spectrum is equal to the channel separation b. When adding Nuniform tapered channelmaps at a certain pixel the noise $\sigma^{u}_{N}$ at the same pixel position in the total HI map will be increased by a factor $\sqrt{N}$:

$$\sigma^{u}_{N} = \sqrt{N} \sigma^{u}.$$

Usually, the observations are made using a hanning taper in which case the noise in two adjacent channelmaps is no longer independent. A hanning taper effectively smooths the data in velocity by convolving the velocity profile at each pixel. If U_i is the pixel value in the ith uniform tapered channel map, the value H_i in the ith hanning tapered channelmap is given by

$$H_{i} ={\textstyle\frac{1}{4} U_{i-1}} + {\textstyle\frac{1}{2}} U_{i} + {\textstyle\frac{1}{4}} U_{i+1}.$$

Since the $\sigma^{u}_{i}$'s are independent and all equal to $\sigma^{u}$, the noise $\sigma^{h}_{i}$ in the ith hanning tapered channelmap can be calculated according to

$$\begin{eqnarray*}\sigma^{h}_{i} & = & \left[ ( {\textstyle\frac{1}{4}} \sigma^{u... ...= & {\textstyle\frac{\sqrt{6}}{4}} \sigma^{u} = 0.61 \sigma^{u}. \end{eqnarray*}$$

In this case the noise equivalent bandwidth B^h for a hanning tapered spectrum is given by

$$B^{h} = {\textstyle\frac{16}{6}} B^{u} = 2.67 B^{u} .$$

As a consequence, the noise in two hanning tapered channelmaps may be correlated depending on their separation. Two channelmaps separated by one channelmap are correlated because both contain a quarter of the flux from the channel map between them. Only channel maps separated by more than one channel are independent. This will be shown in the following three cases in which two hanning tapered channel maps at different separations will be added.

1.

Adding two adjacent hanning tapered channelmaps i and (i+1) gives a signal H_(i)+(i+1) of

$$\begin{eqnarray*}H_{(i)+(i+1)} & = & H_{i} + H_{i+1} \\ & = & \left( {\textstyl... ...textstyle\frac{3}{4}} U_{i+1} + {\textstyle\frac{1}{4}} U_{i+2} \end{eqnarray*}$$

and the noise $\sigma^{h}_{(i)+(i+1)}$ in that map will be

$$\begin{eqnarray*}\sigma^{h}_{(i)+(i+1)} & = & [ ( {\textstyle\frac{1}{4}} \sigma... ...sqrt{ 3 {\textstyle\frac{1}{3}} } \sigma^{h} = 1.83 \sigma^{h}. \end{eqnarray*}$$

2.

Adding the hanning tapered channels i and (i+2) gives

$$\begin{eqnarray*}H_{(i)+(i+2)} & = & H_{i} + H_{i+2} \\ & = & \left( {\textstyl... ...\textstyle\frac{1}{2}} U_{i+2} + {\textstyle\frac{1}{4}} U_{i+3} \end{eqnarray*}$$

and the noise becomes

$$\begin{eqnarray*}\sigma^{h}_{(i)+(i+2)} & = & [ ( {\textstyle\frac{1}{4}} \sigma... ...sqrt{ 2 {\textstyle\frac{1}{3}} } \sigma^{h} = 1.53 \sigma^{h}. \end{eqnarray*}$$

3.

Adding the hanning tapered channels i and (i+3) gives

$$\begin{eqnarray*}H_{(i)+(i+3)} & = & H_{i} + H_{i+3} \\ & = & \left( {\textstyl... ...e\frac{1}{2}} U_{i+3} + {\textstyle\frac{1}{4}} U_{i+4} \right) \end{eqnarray*}$$

with a resulting noise of

$$\begin{eqnarray*}\sigma^{h}_{(i)+(i+3)} & = & [ ( {\textstyle\frac{1}{4}} \sigma... ... }} \sigma^{h} \\ & = & \sqrt{2} \sigma^{h} = 1.41 \sigma^{h}. \end{eqnarray*}$$

So, channelmaps i and (i+3) are independent.

Because the noise is correlated, adding N adjacent hanning tapered channelmaps does not give an increase of the noise with a factor $\sqrt{N}$ but with a factor $\sqrt{ N - \frac{3}{4} } \cdot \frac{4}{\sqrt{6}}$ as is shown below. First the total signal H_N is calculated.

Channel	Ui-1	Ui	Ui+1	$\cdots$	Ui+N-2	Ui+N-1	Ui+N
i	1/4	1/2	1/4
i+1		1/4	1/2	$\cdots$
i+2			1/4	$\cdots$
$\cdots$				$\cdots$
i+N-3				$\cdots$	1/4
i+N-2				$\cdots$	1/2	1/4
i+N-1					1/4	1/2	1/4	+
	$\frac{1}{4}U_{i-1}$	$\frac{3}{4}U_i$	Ui+1	$\cdots$	Ui+N-2	$\frac{3}{4}U_{i+N-1}$	$\frac{1}{4}U_{i+N}$

and thus

$$\begin{eqnarray*}H_N = \frac{1}{4}U_{i-1}+ \frac{3}{4}U_i + U_{i+1}+ \cdots +U_{i+N-2} + \frac{3}{4}U_{i+N-1}+\frac{1}{4}U_{i+N}. \end{eqnarray*}$$

From this it follows that the noise $\sigma^h_N$ is given by

$$\begin{eqnarray*}\sigma^h_N & = & \left[ ({\textstyle\frac{1}{4}})^2 + ({\textst... ...\\ & = & \sqrt{ (N - {\textstyle\frac{3}{4}} ){B}^h } \sigma^h. \end{eqnarray*}$$

However, before the hanning tapered channelmaps are added to form a total HI map, the continuum must be subtracted. This operation introduces extra noise in the channelmaps which doesn't behave like a hanning tapered correlation. Here, it will be assumed that the average continuum map is formed by averaging N₁ line free channels at the low velocity end of the datacube and N₂ channels at the high velocity end which gives

$$C_{\rm low} = \frac{1}{N_1} \sum\limits_{j=1}^{N_1}H_j \mbox... ...cm} } C_{\rm high} = \frac{1}{N_2} \sum\limits_{j=1}^{N_2}H_j.$$

Since all channels are hanning tapered the noise in these maps can be calculated according to

$$\sigma_{C_{\rm low}} = \frac{1}{N_1} \sqrt{ \left( N_1 - {\textstyle\frac{3}{4}} \right) }\sigma^u$$

and

$$\sigma_{C_{\rm high}} = \frac{1}{N_2} \sqrt{ \left( N_2 - {\textstyle\frac{3}{4}} \right) } \sigma^u.$$

The average continuum map to be subtracted is then formed by

$$<C>~=~{\textstyle\frac{1}{2}} ( C_{\rm low} + C_{\rm high} ) .$$

Since $\sigma_{C_{\rm low}}$ and $\sigma_{C_{\rm high}}$ are independent it follows that the noise $\sigma_{<C>}$ in the finally averaged continuum map is given by

$$\begin{eqnarray*}\sigma_{<C>} & = & {\textstyle\frac{1}{2}} \sqrt{ \sigma^2_{C_{... ...ac{3}{4} }{ 4N^2_2 } \right) } \\ & \equiv & \sigma^u {\cal N}. \end{eqnarray*}$$

After subtraction of the continuum the channelmaps only contain signal from the HI emission line. The signal in the channelmaps containing the line emission is now given by

$$\begin{eqnarray*}L_i & = & {H_i}~-~<C> \\ & = & {\textstyle\frac{1}{4}} U_{i-1}+{\textstyle\frac{1}{2}}U_{i}+{\textstyle\frac{1}{4}}U_{i+1}-<C>. \end{eqnarray*}$$

Because $\sigma_{<C>}$ is independent from $\sigma^{u}_{i}$ in the velocity range which is not used to form the averaged continuum map, it can be written

$$\begin{eqnarray*}\sigma^l_i & = & \left[ ( {\textstyle\frac{1}{4}} \sigma^{u}_{i... ...left( {\textstyle\frac{3}{8}} + {\cal N}^2 \right) } \sigma^{u}. \end{eqnarray*}$$

When adding N adjacent hanning tapered and continuum subtracted channelmaps containing the line emission, the signal L_N will be

$$\begin{eqnarray*}L_N = {\textstyle\frac{1}{4}}U_{i-1} + {\textstyle\frac{3}{4}}U... ...}{4}}U_{i+N-1} + {\textstyle\frac{1}{4}}U_{i+N} - N \cdot <C>. \end{eqnarray*}$$

The noise $\sigma^l_N$ at each pixel in the final map can be derived analogous to the calculation of $\sigma^h_N$ and is given by

$$\begin{eqnarray*}\sigma^l_N & = & \left[ (N - {\textstyle\frac{3}{4}} ) + N^2{\c... ... ( N - \frac{3}{4} ) + N^2{\cal N}^2 }{B^h} \right) } \sigma^h . \end{eqnarray*}$$