DHIGLS: DRAO H i INTERMEDIATE GALACTIC LATITUDE SURVEY

With the DRAO ST, one cannot simply carry out an all-sky survey to map Galactic H i at arcminute resolution. Scarce resources (i.e., the number of syntheses) must be focused strategically. This section presents briefly why we selected certain fields to study. The details of the "resource allocation" to the DHIGLS regions and within them (i.e., how the syntheses are placed), which relate closely to the science goals, are deferred to Section 3.2 and are summarized in column 5 of Table 1. Included in the science considerations is the availability of ancillary data (e.g., infrared imaging), which broadens the range of scientific investigations that are enabled by the DHIGLS data. To complement studies in the Galactic plane, the DHIGLS regions were selected to be at intermediate latitude and to have low to intermediate ${N}_{{\rm{H}}{\rm{I}}}$ within a variety of VCs.

SPIDER¹² is a 10° GHIGLS field at the top of the arch of the North Celestial Pole Loop (NCPL), a giant gas structure north of the Galactic Plane with a cylindrical morphology (Meyerdierks et al. 1991). The related DHIGLS mosaicked region DF within this field focuses on the highly structured diffuse LVC emission. The IVC emission is much fainter and the HVC negligible. Targeted mapping of the stunning dust emission at even higher resolution has been carried out with Herschel.

The European Large-Area ISO Survey (ELAIS) N1 field is an "extragalactic window" targeted by the Spitzer SWIRE survey (Lonsdale et al. 2003) and most recently the HerMES legacy program on Herschel (Oliver et al. 2012). Its low column density can be appreciated in Figure 1. The GHIGLS field N1 spans 5°.¹³ The DHIGLS mosaicked region EN focuses on the central lowest ${N}_{{\rm{H}}{\rm{I}}}$ region. It has little LVC emission, negligible IVC emission, but striking HVC emission. At least at the GBT resolution, the HVC gas morphology is not immediately obviously traced by dust as, quantitatively, the HVC has a low dust emissivity (Planck Collaboration et al. 2011). The higher resolution DHIGLS observations enable a deeper search for HVC-correlated dust emission.

In addition to the Galactic ISM science, another goal for the two deepest DHIGLS regions (DF and EN) is to provide higher resolution H i data to complement the characterization of foregrounds at the 5' Planck resolution.

DRACO (Herbstmeier et al. 1993) was selected because of its prominent and distinctive IVC gas, which has a clear dust signature with IRAS and Planck (Planck Collaboration et al. 2011). Targeted Herschel dust emission observations were carried out as well (Miville-Deschênes et al. 2016). The dynamics of the interaction of the IVC gas with the Galactic thick disk is fascinating, and the transition from atomic to molecular gas in dense regions near the "interaction surface" can also be studied. Of course, the new observations here cover only the H i component. Differences in dust properties between LVC and IVC gas could provide evidence of dust evolution.

G86 is a prominent degree-sized high-latitude cloud found to have a clear signature of dust correlated with the IVC gas using early DRAO observations (Martin et al. 1994). A more sensitive single-pointing DRAO ST synthesis MG has been carried out, along with targeted Herschel imaging.

The DHIGLS region UM within the GHIGLS field UMA probes clouds with LVC gas spanning the transition from the diffuse H i phase to the molecular phase. A small two-pointing subregion of UM, which we refer to as URSA, has been studied previously using early DRAO ST data (Miville-Deschênes et al. 2003a; revisited in Appendix E) and imaged with Herschel.

The small DHIGLS region PO is the brightest part of the GHIGLS field POL and has molecular emission. Part of this region has been imaged with Herschel (Miville-Deschênes et al. 2010). We studied one other single-field synthesis, MC, within POL.

Understanding the noise properties of any telescope is necessary to inform an observing strategy. Three maps related to the noise in a single synthesis are illustrated in Figure 2. The differences in each of these representations are discussed below.

Zoom In Zoom Out Reset image size

Inspection of an emission-free channel map from this single DHIGLS synthesis indicates that the typical rms noise for 0.824 km s⁻¹ channels is 18 mJy beam⁻¹ (or $3.2\sin \delta$ K). In fact, noise estimates for all syntheses compare well to the value measured for the original phase of the CGPS carried out in the previous decade (Taylor et al. 2003), $3.2\sin \delta$ K.

For astrophysical applications of a single synthesis, we need to correct each channel map by the primary beam of a 9 m antenna (the effect of its attenuation), which amplifies the noise away from the center. We note that the primary beam is not a Gaussian, but is well described by a cos⁶ function of radial offset (Taylor et al. 2003) with an FWHM of 108'. The corrected emission-free channel maps are therefore noticeably noisier around the periphery, as shown in the left panel of Figure 2.

The noise can also be quantified by finding the rms in the spectrum over many emission-free channels pixel by pixel; after correction for the primary beam, this gives us an empirical "noise map" as in the middle panel. In making a mosaic, the contribution from a single synthesis is weighted according to the inverse square of the noise. The model for the weight map is therefore the square of the primary beam, here normalized to unity at the pointing center (there is no dependence of the central value on δ as in the noise map). This map is shown in the right panel.

Each of the maps is truncated at a radius of 93', where the noise has risen by a factor 9.5 compared to the value at the pointing center and the weight has fallen to 0.011 (see Figure 2). As adopted for the CGPS, data beyond this radius are not used in assembling the mosaics (Section 4.1).

In a single synthesis, the rms noise level for a single channel varies across the field (Figure 2). The noise level can be made more uniform and/or greatly reduced by assembling a number of single syntheses as a mosaic. For example, the CGPS mosaic pattern was that of a close-packed hexagonal grid with pointing centers separated by $112^{\prime}$, a bit larger than the FWHM, a compromise between uniform sensitivity and spatial coverage at 1420 MHz. With this in mind, for the much fainter EN region, we chose a pattern of single pointings to emphasize sensitivity over spatial coverage with spacings of about $12^{\prime}$, while for the brighter DF, we opted for more extended coverage with spacings of about $40^{\prime}$. Figures 3 and 4 show the pointing centers for DF and EN, numbering 91 and 76, respectively (see Table 1). These figures are in ICRS coordinates ($\alpha ,\delta$) and the NCP projection (Calabretta & Greisen 2002) native to the DRAO ST data. The central position of each synthesis field is marked by a bullet. For scale, we show the FWHM of the primary as a dashed circle of diameter $108^{\prime}$. Also shown as the solid circle is the $93^{\prime}$ truncation radius ($186^{\prime}$ diameter) used in the production of the final mosaic.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The pointing centers for the other mosaics of DHIGLS regions are given in Figures 1 and 31.

A synthesized beam for each of the fields was produced corresponding to the visibility coverage in the Fourier domain. This beam was normalized to unity at its center. Because much of the data processing involves approximating the beam with a Gaussian, the synthesized beam was fitted in the image domain with a two-dimensional (2D) Gaussian to determine the effective beam size. The effective beam size was consistently smaller than the half-power beam-width (HPBW) of $58^{\prime\prime}$ and consistently peaked at 1.08 with a relative beam integral of 1.08 (see Figure 5). The mean Gaussian FWHM of the synthesized beam and its dispersion for the mosaic is given in Table 1 for each region.

Zoom In Zoom Out Reset image size

Two continuum maps are produced from an average of the channels at each end of the spectrum that are free of H i emission. The channel selection is performed independently for each synthesis with the median number of channels used for a continuum map being 30, separately for each end. The continuum contribution at each channel in the spectral cube is then determined by linear interpolation between these two continuum maps. Empirically, there is only a minimal frequency dependence of the continuum over the frequency range of the data cube and so, in the few exceptional cases where there are emission-free channels at only one end of the spectrum, the resulting single continuum map is subtracted directly from the spectral cube.

In a number of fields there are very bright point sources that produce correspondingly bright artifacts in the form of grating rings. Subtraction of the continuum will, in principle, remove the continuum source and any artifacts around it. However, within the spectral cube, the continuum emission could be absorbed by diffuse H i in some channels, resulting in a weaker point source and a concomitantly weaker set of artifacts in those channels. Following subtraction of the unattenuated continuum, a scaled negative imprint of the source and artifacts would remain in those absorbing channels. For weaker point sources, the artifacts are hidden in the noise, but, to mitigate such potential residual contamination for bright sources, it is useful to create a clean cube prior to producing the continuum maps by deconvolving those bright point sources with a model beam. This cleaning process follows routines developed for the DRAO ST by Willis (1999) and is performed on each channel in a given cube. It is performed only on those cubes having continuum point sources with flux density $F\gt 1$ Jy beam⁻¹ and this cleaning proceeds down to the level of 300 mJy beam⁻¹ as was done in the CGPS (Taylor et al. 2003).

An averaged continuum map was produced for each synthesis from the two continuum maps. Each point source in this map was fitted with a 2D Gaussian (with the fluxfit routine included in the DRAO Export Software Package madr, Higgs et al. 1997) to determine its sky coordinates ($\alpha ,\delta$) and flux density ${F}_{\mathrm{DRAO}}$, limiting the width of each fit to be no less than 95% of the effective beam parameters determined in Section 3.3.

The fitted values were compared with those in the NVSS¹⁴ database (Condon et al. 1998) to determine for each field a weighted average offset in sky coordinates (${\rm{\Delta }}\alpha$, ${\rm{\Delta }}\delta$) and a weighted average flux density ratio, ${F}_{\mathrm{DRAO}}/{F}_{\mathrm{NVSS}}$. Taylor et al. (2003) discuss this in detail. We find values of the flux density ratio ${F}_{\mathrm{DRAO}}/{F}_{\mathrm{NVSS}}$ in the range 1.2–1.5; the typical uncertainty for each determination is 0.03.

Because the synthesized beam is not quite Gaussian, ${F}_{\mathrm{DRAO}}$ measured by Gaussian fitting will be overestimated by a factor of about 1.08 (Section 3.3). We note that in the conversion from surface brightness to brightness temperature (see Section 4), this factor of 1.08 returns in the amplitude of the Gaussian-fitted beam and so it cancels out.

A plot of ${F}_{\mathrm{DRAO}}/{F}_{\mathrm{NVSS}}$ for all DRAO ST syntheses as a function of observation date reveals intriguing systematic variations (Figure 6). This is present in the data used in a single mosaic (e.g., regions DF and EN colored separately) and for the interleaved observations, ruling out a dependence on decl. There might be a seasonal trend, though it is not exactly the same year to year. We note that changes in the equivalent VLA to NVSS ratio cited as part of the VGPS (Stil et al. 2006), though also not fully understood, were shown to be related to the band-averaged scalar amplitude of the continuum-subtracted visibilities, which is in turn connected to the noise.

Zoom In Zoom Out Reset image size

The empirical pattern of the temporal variation is useful for confirming suspected anomalous ${F}_{\mathrm{DRAO}}/{F}_{\mathrm{NVSS}}$ scale factors. A few fields have strongly asymmetric ring artifacts in the point-source continuum images and were suspected of having inaccurate ratios, and we found that these fields (DF66, DF76, DF81, and DF87), observed in late 2007/early 2008, have the largest deviations from the trend seen in Figure 6 (each marked with a red cross). Therefore, these fields at the edge of the DF region were excluded.

As described in Section 3.2, our DRAO ST observations were designed with the intent of combining several syntheses with adjacent pointing centers into a mosaic. For each synthesis, the DRAO ST data cube (in Jy beam⁻¹) was converted to brightness temperature, ${T}_{{\rm{b}}}$, using the beam parameters determined in Section 3.3 and the scale factor determined from the comparison with NVSS (Section 3.5). The channels of each synthesized cube were then interpolated to a common velocity grid using a spline interpolation. This interpolation has very little effect on the spectra, but is required to ensure a consistent velocity grid among all fields.

The prepared data from the individual syntheses were then assembled into mosaics using the routine SUPERTILE in madr. As described by Dewdney et al. (2002) and Taylor et al. (2003), SUPERTILE reads in the position registration results for each of the fields (Section 3.5), adjusts the position of each field accordingly, divides by the DRAO ST primary beam (which amplifies the noise away from the beam center), truncates each data set at a radius of $93^{\prime}$ relative to its pointing center, and combines the data, keeping track of the beam shape and weights throughout the process.

The channel maps of the mosaicked H i spectral-line data are contained in a data cube. We can produce a noise map from emission-free channels as we did for a single synthesis in Section 3.1. An example is shown in the upper panel of Figure 7 for the DRAO ST DF-i mosaic. The corresponding weight map produced in assembling the mosaic is shown in the lower panel.

Zoom In Zoom Out Reset image size

Figure 7 shows the distinctive pattern of increased noise around the perimeter as a result of the lower coverage and effect of the primary beam there. Across most of the field, the noise is 1.0 K, clearly reduced compared to 3.0 K for a single synthesis because of the chosen dense packing of pointing centers (and the typical non-unity scale factor ${F}_{\mathrm{DRAO}}/{F}_{\mathrm{NVSS}}$ applied to the single syntheses prior to mosaicking).

We note that because of the weighting scheme used in SUPERTILE, the weight map can be used to create a theoretical noise map by multiplying its inverse square root by the typical noise for a single synthesis, $3.2\sin \delta$ K (modified by the typical ${F}_{\mathrm{DRAO}}/{F}_{\mathrm{NVSS}}$ scale factor for the syntheses in the mosaic). Therefore, in figures for other regions below, we display only the noise map.

The noise map is used to create three contours. Two contours, e.g., black outer dashed and solid contours in Figure 3, indicate where the noise of the mosaic is identical to the noise in a single synthesis at the FWHM of the primary beam and at the truncation radius, respectively. A third contour, e.g., white dashed in Figure 3, indicates where the noise is only a factor of two larger than in the central, lowest-noise region, thus selecting a low-noise portion of the larger mosaic (inside the outer black dashed contour because the central noise in the mosaic is lower than for a single synthesis). The white dashed contour is of particular significance in selecting data for analyses discussed below.

We have carried out a similar analysis for the EN-i mosaic. The noise map is shown in Figure 8. This clearly shows that the denser packing of pointing centers of the syntheses making up the mosaic (Figure 4) has significantly reduced the noise in the central region to 0.3 K, about a factor of 3 lower than in DF-i. The denser packing also reduces the relative extent of the low-noise region marked by the dashed white contour in Figure 4.

Zoom In Zoom Out Reset image size

Likewise, noise maps for the UM-i mosaic and the smaller DR-i and PO-i mosaics are presented in Figures 32, 35, and 38 in Appendix F.

Figure 9 shows a representative channel map of the H i signal in the DRAO ST DF-i mosaic near the peak of the average spectrum. Below, we analyze the power spectrum in a low-noise (and high signal-to-noise) region shown in Figure 9 as the "white rectangle." This is the rectangle of largest area that is aligned with the pixel grid and fits inside the "white dashed contour" defined in Figure 3, which is the region of least noise.

Zoom In Zoom Out Reset image size

Similar data are shown in Figure 10 for the EN-i mosaic and in Figures 32, 35, and 38 for the UM-i, DR-i, and PO-i mosaics in Appendix F.

Zoom In Zoom Out Reset image size

It is of interest to relate the signal and noise in H i maps to their manifestations in the Fourier domain (harmonic or u–v domain) and the angular power spectrum. In addition to enabling the science application discussed in Section 7, this provides a means of exploring the sensitivity of ST observations to various spatial scales and more immediately allows for direct comparisons with (and integration of) observations from single-dish telescopes, as will be described in Section 5. Facilitating such analyses is the power-spectrum model development, including the "noise template" and fitting presented in Appendix A building on earlier work described in Miville-Deschênes et al. (2007) and Martin et al. (2015).

The 2D angular power spectrum of a map $f(x,y)$ (image of the sky) is the square of the modulus of the Fourier transform $\tilde{f}({k}_{x},{k}_{y})$ (the visibility) where k is the spatial frequency (wavenumber) in the u–v domain:

$$\begin{eqnarray}&&P({k}_{x},{k}_{y})=| \tilde{f}({k}_{x},{k}_{y}){| }^{2}.\end{eqnarray} \tag{ 1 }$$

The collapsed one-dimensional (1D) power spectrum P(k) is formed by azimuthal averaging of $P({k}_{x},{k}_{y})$ in annuli of constant $k={({k}_{x}^{2}+{k}_{y}^{2})}^{1/2}$. We use this for both analysis and display purposes (see Appendix A.5 for exceptional images with directional structure).

The 2D Fourier transform reflects the sampling in the u–v domain, which is quite uniform for the DRAO ST syntheses, with noise increasing at angular frequencies with little or no coverage. Depending on the symmetry of the Fourier transform, differing amounts of this structure will remain in P(k). In the limit where the synthesized beam is circularly symmetric, there is a direct mapping of the features in the 2D power spectrum to the 1D P(k). Of the two deepest mosaics, DF-i is much closer to this limit than is EN-i.

Figure 11 shows P(k) for the LVC H i channel map in Figure 9 for DF-i. There the data have been fit to the 1D power spectrum formed after deconvolution in 2D using the asymmetrical synthesized beam. The fit to the power-law model of Equation (11) has power-law exponent $\gamma =-2.99\pm 0.04$ and scales the noise template by $\eta =1.59\pm 0.03$. We also show an alternative representation in which the modeling is carried out without deconvolution, requiring accounting for the lowering of power by the effective beam at high spatial frequencies (see Equation (12)). These parameters are consistent with those found above: $\gamma =-3.03\pm 0.03$ and $\eta =1.60\pm 0.03$.

Zoom In Zoom Out Reset image size

To assess the level of noise achieved and the ability to investigate the variety of spatial structure using the DRAO ST H i data, we have modeled the power spectra in the other mosaicked regions in Figure 12 for the single-channel maps shown in Figures 10, 32, 35, and 38.

Zoom In Zoom Out Reset image size

As in DF-i, the denser coverage in EN-i also lowers the power spectrum of the noise, so that, as seen in Figure 12, the power spectrum of the signal can still be analyzed over a considerable range in k, even though the H i signal in this channel is much weaker than in the channel selected for DF-i. Here, the noise template is scaled by $\eta =1.22\pm 0.09$. Generally, η decreases toward 1.0 as the H i signal diminishes, and vice versa.

The power law for this HVC channel is significantly shallower ($\gamma =-2.21\pm 0.12$) than that for the LVC channel in DF-i ($\gamma =-2.99\pm 0.04$).

For the remaining UM-i, DR-i, and PO-i power spectra in Figure 12, at high k where the noise dominates, the different levels are in accord with expectation based on the noise maps presented in Section 4.2 and Appendix F. Similarly, the values of η show the expected differences with H i signal strength, being 1.40 ± 0.03, 1.04 ± 0.05, and 1.27 ± 0.05 for the selected channels in UM-i, DR-i, and PO-i, respectively. Despite the different noise levels and signal strengths, the power spectrum of the signal can still be analyzed over a significant range of k.

Extending the comparison made between DF-i and EN-i, the power-spectrum falloff for the UM-i LVC channel is similar to that for DF-i, but visibly steeper than for the HVC channel in EN-i; this is quantified in the power-law exponent, −2.93 ± 0.08. On the other hand, the data for the IVC channels in DR-i and PO-i appear more compatible with the behavior in EN-i. Their exponents are −2.27 ± 0.14 and −2.4 ± 0.2, respectively.

Thus, the data are of sufficiently high quality that some differences can be discerned. However, because these power spectra are for individual channels and are thus responding to a mixture of velocity and density fluctuations (Lazarian & Pogosyan 2000), the exponents cannot be linked simply to the structure of the atomic gas. This will be followed up in Section 7.

We adopted observations made with the GBT as part of GHIGLS (Martin et al. 2015). On-the-fly observations of the fields produced Nyquist-sampled maps with an effective GHIGLS beam slightly elongated in the scanning direction. The GHIGLS beam can be modeled accurately in 2D and, when approximated by a 2D Gaussian, is $9\buildrel{\,\prime}\over{.} 55\times 9\buildrel{\,\prime}\over{.} 24$ FWHM (Martin et al. 2015). The data reduction pipeline for calibration of these H i measurements, the correction for stray radiation, and baseline removal is described in Boothroyd et al. (2011) and Martin et al. (2015). The spectrometer channel spacing in the final GHIGLS data cubes is 0.80 km s⁻¹ with velocity coverage well over ±200 km s⁻¹, i.e., beyond that covered by the DRAO ST. For use with the higher velocity resolution data in the PO-i mosaic and the MC and MG single syntheses from the DRAO ST (Table 1), we reprocessed the archived GBT raw data to create POLFINE and G86FINE data cubes with channel spacing 0.32 km s⁻¹.

The large SPIDER field was observed in 2° × 2° segments and then assembled into a final 10° × 10° cube. The nine central sections of SPIDER were measured twice. Other relevant GHIGLS fields (Table 1) are N1 (measured two times), G86 (3), DRACO (3), POL(1), and UMA (1). The typical end-channel (emission-free) rms noise is 105 mK per observation, reducing to 75 mK for two observations (appropriate to the central region of SPIDER). For the reprocessed G86FINE and POLFINE fields, the noise is typically 96 and 170 mK, respectively. Note that for the DRAO ST UM-i mosaic, we actually used the GHIGLS NCPL combined field to achieve slightly greater spatial coverage.

Each synthesis pointing used in the assembly of a given DRAO ST mosaic has been registered and scaled to the NVSS flux scale as described in Section 3.5. However, here, we are interested in having a consistent calibration between the H i data from the DRAO ST mosaic and GHIGLS. To this end, as described in Appendix C, we determined a cross-calibration scale factor for the brightnesses, ${f}_{\mathrm{cc}}\equiv {I}_{\mathrm{GBT}}/{I}_{\mathrm{DRAO}}=1.12\pm 0.01\pm 0.03$, that is to be applied to the DRAO ST mosaicked H i data to bring them to the GHIGLS scale.

As described below, we combined the two data sets by weighting their Fourier transforms. We refer to the DRAO ST + GHIGLS product as the "combined" (or "merged") image. This combined product is the one contained in the DHIGLS data cube, and the spectra therein are on the GHIGLS scale.

Our approach to weighting is close to that of the AIPS task IMERG.¹⁵ Using the madr software, the transformed data below some ${k}_{\min }$ are taken from the GHIGLS data and above some ${k}_{\max }$ from the DRAO ST data, and within this "image combination overlap range" the data are weighted by complementary functions summing to unity. We used a cubic polynomial in k with zero slope at the inner and outer edges of the range. Note that the overlap ranges for the cross calibration and the image combination need not be the same. However, because many of the same considerations apply (see Appendix C.2), we did adopt the same range. We note again that this ${k}_{\max }$ is well below the range affected by the DRAO ST synthesized beam. Refer to Figure 28 in Appendix B.

This approach is different than the CASA task feather ¹⁶ and IMMERGE (Stanimirovic 1999 and by Sault & Killeen¹⁷ ) in two ways. First, these use ${k}_{\min }$ = 0 arcmin⁻¹ and so, despite the weighting function applied, the interferometric data are used below the range actually sampled in the u–v domain. Second, the weighting function applied to the deconvolved single-dish data is simply the Fourier transform of the single-dish beam and so, with no restriction on ${k}_{\max }$ other than implied by the decreasing weight, data strongly affected by the noise are incorporated. Our adopted limits to the range address both of these issues.

In the combined image, we are interested in preserving the high-resolution information in the DRAO ST mosaic and so, we use that mosaic, scaled by ${f}_{\mathrm{cc}}$, in its original ICRS coordinate system. To reduce edge effects from the conversion to the Fourier domain, the image is apodized with a taper beginning 109 from the edge and ending at the outer contour of the mosaic. For details on the apodization of non-rectangular fields, refer to Appendix A.4. The apodized and zero-padded image was Fourier transformed, the weighting function applied in 2D, and then the data were back transformed, producing a "high-resolution" filtered image. An example for a single channel of DF-i is shown in the upper panel of Figure 13. Pixels exterior to the inner edge of the apodized region (see contour in this panel) were flagged for exclusion in the combined cube.

Zoom In Zoom Out Reset image size

The original GHIGLS data were interpolated to the velocity channels of the DRAO ST data and initially to a 15 ICRS grid centered on that of the DRAO ST mosaic. It is sufficient to have a rectangular region somewhat larger than the mosaic. This image was median-subtracted, apodized, Fourier transformed, and then deconvolved with a 2D Gaussian approximation of the GHIGLS beam. Subsequently, the weighting function was applied and the data back transformed to produce a second "low-resolution" filtered image, which was finally regridded to the finer grid of the DRAO ST mosaic. An example corresponding to the same channel is shown in the middle panel of Figure 13. As done for the DRAO ST data, pixels affected by the apodization of the GHIGLS data are flagged for exclusion.

The high- and low-resolution filtered data were combined (added) in the image plane. To complete the example for the single channel in the DF region, this final product is presented in the lower panel of Figure 13. Pixels in hatched area (affected by the apodization of the low-resolution filtered image) and beyond are encoded as zero in a modification of the weight map provided as an extension to the DHIGLS cube (Section 9).

This procedure was carried out for each velocity channel to produce the DHIGLS cube. The final data products are cropped in the x- and y-directions to exclude the hatched region and the remaining blank and hatched region pixels are filled with the regridded original GHIGLS data for context. Before any analysis, these can be removed (masked) using the modified weight map supplied with the data cube. DHIGLS data cubes are available on a public archive, as described at the end of Section 9. For any processing of these data cubes (e.g., convolving to the native resolution of another instrument such as Planck), take note of the synthesized beam parameters in Table 1.

H i emission has structure on all scales for which a power spectrum in the u–v domain provides a useful statistical quantification. The power spectrum of the combined channel map in Figure 13, for the data within the white dashed contour, is given in Figure 14. Given how the combined map was constructed, the power spectrum for values of k higher than the overlap range (in blue) is identical to the power spectrum of the scaled interferometric data alone (shown in red, but for clarity, only for $k\lt 0.05$ arcmin⁻¹). Because of the broad overlap in u–v coverage the latter power spectrum actually agrees quite well through most of the overlap region, before eventually falling off at low k, as in Figure 11. With the addition of the GHIGLS data, the power-law rise in the power spectrum of the combined map extends to lower k (with no discontinuity as in Figure 29). The power-law exponent found within the white dashed contour, −2.94 ± 0.02, is consistent with that found for the white rectangle (−2.95 ± 0.03) and with those in Figures 11 and 29.

Zoom In Zoom Out Reset image size

The power spectrum shows that spatial fluctuations in the structure of the H i emission are smaller on small angular scales (high k) compared to those on large scales. Thus, when the data are filtered to contain only those small scales, the amplitudes in the image domain are relatively small (and both positive and negative) as shown in the upper panel of Figure 13. In contrast to this, when the data are filtered to contain only large scales, the amplitudes in the image domain are relatively large as shown in the middle panel of Figure 13 (note the larger range on the colorbar).

We note that the high-resolution filtered image has less dynamic range than the channel map of the DF-i mosaic in Figure 9, because the weighting applied to the u–v data from the interferometer (Section 5.2) cuts off the power quickly across the overlap region toward lower k (see Figure 28), thus removing the brighter fluctuations on the larger scales. Those fluctuations are captured in the low-resolution filtered image. Likewise, the low-resolution filtered image lacks the finer spatial detail in the original GHIGLS data, because the weighting applied cuts off faster than the effect of primary beam of the GBT (Figure 28). A corollary is that the overlap in u–v coverage of the DRAO ST and the GBT is so good that the spatial structure at scales covering the central part of the overlap range could equally well be represented by data of high quality derived from either instrument.

From the comparison in Figure 13, it can be appreciated that short-spacing "correction" is actually a contribution of fundamental importance, laying out the basic structure of the map, which the interferometric data adjusts to reveal the finer detail.

Within the cube, distinct velocity structures can be identified. To quantify the division of the cube into VCs, we computed the mean, median, and standard deviation about the mean of each channel (see Figure 15), following the method in Planck Collaboration et al. (2011) and Martin et al. (2015). Components often appear blended in the mean and median spectra, but become more distinct in the standard deviation spectrum, which depends on the fluctuations across the field, not just the presence of a signal, and so it takes advantage of the rich structure within the cube. Peaks in the standard deviation spectrum were noted and each was taken to indicate a separate velocity component. For DF, these are at about 3, −30, and −50 km s⁻¹. Minima at about −15 km s⁻¹ and −40 km s⁻¹ suggest velocities at which the cube can be divided into components. For exploratory purposes, we divided the IVC range into two separate components, which we label IVC1 and IVC2. The separation here can be compared with the velocity cuts made in Planck Collaboration et al. (2011) for the GHIGLS data in the SPIDER field (of which DF is a subset): LVC/IVC separation at −14.9 km s⁻¹ and IVC/HVC separation at −88.1 km s⁻¹ (see Table 2). Here, there is no HVC component readily identifiable, unlike for the GHIGLS SPIDER field, where the lower resolution provides higher sensitivity. Also, the coverage in DF does not fully overlap with the HVC seen in the GHIGLS SPIDER data, and where it does is along the noisy edge of the mosaicked region.

Zoom In Zoom Out Reset image size

A second example is given in Figure 16 for EN. Results are tabulated in Table 2, along with those for the other DHIGLS regions and the corresponding GHIGLS fields identified in Table 1.

Zoom In Zoom Out Reset image size

The data cubes collapsed along their velocity axis produce maps of integrated emission, ${W}_{{\rm{H}}{\rm{I}}}$ $\,=\,{\sum }_{i}{T}_{{\rm{b}}}^{i}{\rm{\Delta }}v$. The integrated emission is limited to channels where there is a measurable signal. An example of such a map is shown in Figure 17 for DF with emission integrated between −80 and +30 km s⁻¹, including both IVC and LVC (Figure 15).

Zoom In Zoom Out Reset image size

The quantity ${W}_{{\rm{H}}{\rm{I}}}$ is of particular interest because, in the optically thin regime, the column density ${N}_{{\rm{H}}{\rm{I}}}$ = C ${W}_{{\rm{H}}{\rm{I}}}$ cm⁻², where $C=0.1823\times {10}^{19}$ cm⁻² (K km s⁻¹)⁻¹. Although of some significance for the GHIGLS data, an optical depth correction might seem less important for the DHIGLS data presented here because they are noisier. Nonetheless, the correction is a systematic one and for consistent treatment of the data in both surveys an optical depth correction was made assuming a single spin temperature ${T}_{{\rm{s}}}=80$ K (Martin et al. 2015) in Equation (13). The related issue of the dependence of optical depth corrections on the angular resolution of the observations is explored in Appendix D.1.

Some resulting ${N}_{{\rm{H}}{\rm{I}}}$ maps produced from the DHIGLS H i cubes are shown in Figures 18 and 19. Others for the remaining regions are included in Appendix F. These ${N}_{{\rm{H}}{\rm{I}}}$ products are available for download from the DHIGLS archive (see end of Section 9).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

As presented in Appendix C.1 and discussed further in Section 7.2, the optimal S/N in an ${N}_{{\rm{H}}{\rm{I}}}$ map is obtained by integrating only a few channels near the peak emission, about 10 in the case of DF. However, for the above ${N}_{{\rm{H}}{\rm{I}}}$ maps, a full VC integration is carried out over many more channels to cover the range in which emission is seen anywhere in the map. Therefore, the ${N}_{{\rm{H}}{\rm{I}}}$ maps in Figures 18 and 33 clearly appear noisier than color maps based on single channels with strong emission (see Figures 24 and 25 below). The spectrum at each pixel could be integrated separately, keeping only data above some noise threshold, but, unlike for the centroid velocity, this would bias ${N}_{{\rm{H}}{\rm{I}}}$. Fitting Gaussian components and then integrating them over velocity is an alternative that accounts implicitly for noisy empty channels.

On the basis of work on fBm simulations, Miville-Deschênes et al. (2003b) suggested a direct mapping between the power spectrum of the centroid velocity map of a VC and the 3D velocity field. This motivated our computation of centroid velocity maps for all of the VCs in the DHIGLS regions. The DHIGLS cubes have not been filtered and so noise is a factor in determining a reliable centroid velocity power spectrum. In order to reduce some of this uncertainty, each spectrum in the cube is first filtered using a five-channel-wide Hanning window (i.e., a Hanning function with five non-zero values). The centroid velocity field, v_c, is then determined using only those channels in the given component's velocity range with signal above twice the emission-free end-channel noise, ${\sigma }_{\mathrm{ef}}$:

$$\begin{eqnarray}&&{v}_{c}=\displaystyle \frac{{\displaystyle \sum }_{i}\,{v}^{i}{T}_{{\rm{b}}}^{i}{\rm{\Delta }}v}{{\displaystyle \sum }_{i}\,{T}_{{\rm{b}}}^{i}{\rm{\Delta }}v};{T}_{{\rm{b}}}^{i}\gt 2{\sigma }_{\mathrm{ef}}.\end{eqnarray} \tag{ 2 }$$

Centroid velocity maps for DF, EN, and UM are shown in Figure 20, 21, and 34, respectively. These images show structures on all scales and sometimes systematic gradients, most visible at large scales (e.g., the HVC in EN in Figure 21). The centroid velocity maps look somewhat less noisy than the ${N}_{{\rm{H}}{\rm{I}}}$ maps because of the spectral smoothing and clipping performed.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

As described in Appendix A.2 we deconvolved the data with the appropriate 2D synthesized beam (Table 1) before computing the 1D power spectrum and fitting it with the model in Equation (11). As in Appendices C.1 and A.4, we selected data within the non-rectangular white dashed contour. The model has three parameters to be fitted: the power-law exponent, the amplitude normalization, and the scale factor of a noise template.

The power spectra of the ${N}_{{\rm{H}}{\rm{I}}}$ maps of three VCs in DF (Figure 18) are shown in the upper panel of Figure 22, where it is readily apparent that the spectra for IVC1 and IVC2 are consistently shallower than for the LVC, indicative of relatively more power on smaller scales. This is quantified by the power-law fits, also shown in the figure. The power spectrum of the ${N}_{{\rm{H}}{\rm{I}}}$ map of the combined IVC (Table 2) is similarly shallower. The exponents for LVC and IVC tabulated in Table 3 are significantly different. Fits to data within the more restricted white rectangle give consistent results within the uncertainties. We also note that while there are systematic uncertainties ($\lt 0.1$) in the derivation of the exponents, assessment of their relative differences is not impacted (Appendix A.3).

Zoom In Zoom Out Reset image size

Table 3.  Power-law Model Results for DF, EN, and UM

Map	DF	EN	UM
LVC ${N}_{{\rm{H}}{\rm{I}}}$	−3.00 ± 0.03	−2.45 ± 0.17	−2.89 ± 0.04
va	−2.78 ± 0.08	−2.43 ± 0.16	−3.03 ± 0.09
IVC ${N}_{{\rm{H}}{\rm{I}}}$	−2.60 ± 0.04	...	−2.48 ± 0.06
v	−2.17 ± 0.10	...	−2.59 ± 0.15
HVC ${N}_{{\rm{H}}{\rm{I}}}$	...	−2.85 ± 0.07	...
v	...	−2.78 ± 0.30	...
H i, LVC+IVCb	−3.00 ± 0.04	−2.45 ± 0.17	−2.85 ± 0.04
$E{(B-V)}_{\mathrm{xgal}}$ c,d	−2.69 ± 0.07	−1.55 ± 0.11	−2.66 ± 0.08
${\tau }_{353}$ c	−2.39 ± 0.08	−1.46 ± 0.15	−2.43 ± 0.08
857 GHzc,d	−2.64 ± 0.07	−1.19 ± 0.17	−2.52 ± 0.07
857 GHzc	−2.53 ± 0.07	−1.10 ± 0.19	−2.48 ± 0.08

Notes.

^aCentroid velocity map. ^bProxy for dust emission: combined ${N}_{{\rm{H}}{\rm{I}}}$ components weighted by ${\tau }_{\mathrm{dust}}$/${N}_{{\rm{H}}{\rm{I}}}$ of the correlated dust components from Planck Collaboration et al. (2011). ^cFrom Planck Collaboration et al. (2014a), 5' resolution. ^dNo point sources.

Download table as:  ASCIITypeset image

EN has very little IVC emission. Power spectra for the other two VCs (Figure 22, middle) indicate that the HVC spectrum is marginally steeper than that for the LVC. Additionally, the HVC exponent in Table 3 can be compared to the exponent for the rectangular field in the single EN-i HVC channel at −117.7 km s⁻¹ used in Section 4.4, −2.21 ± 0.12, or more appropriately here, the exponent for the same channel in the DHIGLS cube within the white dashed contour, −2.30 ± 0.09. This suggests a steepening of the power spectrum with the inclusion of additional channels (see the discussion in Section 7.2).

The UM data are not as deep as in DF or EN, but, nevertheless, the H i signal in the power spectrum is well defined. For the ${N}_{{\rm{H}}{\rm{I}}}$ maps of the LVC and IVC components (Figure 33), we again found a steeper exponent for the LVC component (see Figure 22, lower panel, and Table 3). The value of the LVC VC is comparable to that for the single LVC channel value, −2.93 ± 0.08, from Section 4.4 for the white rectangle region in Figure 32. UM is immediately adjacent to and overlaps slightly DF (Figures 1 and 24) for which we found a similar pair of exponents for the LVC and IVC components (Table 3). We note that the UM-i mosaic contains a deeper observation of the URSA subregion for which a steeper power spectrum (exponent −3.6 ± 0.2) was found in previous work (Miville-Deschênes et al. 2003a). This is revisited in Appendix E.

Results for DR and PO are presented in Appendix F.

7.1.1. Comparison with Power Spectra of Thermal Dust Emission

Angular power spectra of the ISM gas (and dust) such as discussed above reflect a self-similarity that has long been associated with a turbulent cascade (Chandrasekhar 1949). It is not necessarily straightforward to quantify all aspects of this turbulence directly from observations. For example, as seen in Figure 24, the CNM structure in the channel maps changes quickly with velocity, but a channel map does not necessarily represent a slice of space/distance. Likewise, a spatial slice need not exhibit the same velocity throughout the map. In general, the pattern of emission in individual channels in the H i velocity cube can reflect a combination of both the density and the velocity structure of the ISM. Attempts have been made to disentangle the effects of these two fields.

For optically thin gas, Lazarian & Pogosyan (2000) predict that for typical velocity field exponents the power-law exponent of the integrated H i emission should decrease (the spectrum steepens) as the spatial thickness of the sampled region increases, approaching the power-law exponent of the underlying 3D density field. The important condition is that the physical depth ($\delta d$) associated with the VC must exceed the largest transverse scale s of the map, which can be obtained from Equation (4) below. The intermediate Galactic latitude fields DF, UM, and EN span about 7°, 5°, and 2°, respectively. Adopting a distance of about 100 pc for the LVC in these fields, then, from Equation (4) below, the transverse extent s of the entire field would be of order 12, 8.5, and 3.5 pc, respectively.

In general, the LVC is composed of WNM, CNM, and thermally unstable components (Saury et al. 2014). The scale height of the WNM, H_z ∼ 400 pc (Kalberla 2003), supports the view that the range $\delta d$ of gas contributing to the integrated column density significantly exceeds s for intermediate-latitude LVC gas over regions of the size observed in DHIGLS. Furthermore, CNM is thought to form within the WNM via a thermal phase transition. There is a pressure threshold for this transition and so $\delta d$ might be somewhat smaller for the CNM. Nevertheless, the spatially thick condition $\delta d\gt s$ seems likely to be met here. This, in turn, suggests that the measured LVC power laws (−3.00 ± 0.03, −2.45 ± 0.17, −2.89 ± 0.04 in row 1 of Table 3) represent the 3D density power spectrum of the H i gas.

Using fBm simulations, Miville-Deschênes et al. (2003b) found that increasing the number of LVC channels being integrated does indeed steepen the power law as the relative contribution of power on the smallest scales is reduced. They find that the exponent decreases by order unity and they interpret this phenomenon as a measure of the change from 2D to 3D in the captured topology.

The large velocity range needed to capture a "complete" VC could often be associated with a large thickness, although, as discussed, velocity does not map uniquely to distance. To explore such potential variations in the power spectra of the data, we started with the channel map with peak emission in the standard deviation spectrum (e.g., Figure 15) and added channels around these peaks, on alternating sides, eventually building up component maps as in Figures 18 and 19. At each step we used the model in Equation (11) to determine the power-law exponent. As the number of channels in the slice integrated increases, the S/N first increases and then decreases. Estimating the relative S/N as a function of the slice width by ${P}_{0}/\eta$ with k₀ = 0.01 arcmin⁻¹, we found the width at which the S/N was at a maximum.

The trend lines in Figure 23 show the results for DF for the three velocity ranges identified in Figure 15. The exponents for IVC1 and IVC2 are similar to one another, but different than for LVC. We also marked the exponent for the slice width at which the S/N was maximum and plotted the exponents found in the power-spectrum analysis of ${N}_{{\rm{H}}{\rm{I}}}$ of the complete VCs (Figure 22).

Zoom In Zoom Out Reset image size

Our main result is that there is no significant decrease in the exponent with width of the slice (a possible surrogate for spatial thickness). This is the case for EN and UM as well and is in contrast to a decrease of the exponent with slice width by 0.45 found by Miville-Deschênes et al. (2003a) in URSA (see also Appendix E). However, for the HVC in EN, there appears to be a steepening of the power spectrum with the inclusion of additional channels, from exponent −2.30 ± 0.09 for a single channel to −2.85 ± 0.07 for the full VC, which is in the sense expected from theory and simulations.

Given the angular extent of DF, UM, and EN, the power-spectrum samples well down to about k of 0.005, 0.007, and 0.02 arcmin⁻¹, respectively. At the other extreme, the S/N is sufficient in each underlying mosaic to probe the power spectrum readily up to k of about 0.2 arcmin⁻¹, corresponding to a scale of 5', somewhat larger than the synthesized beam.

For an adopted distance of LVC gas of 100 pc, the DHIGLS data probe down to scales of 0.14 pc and up to 10 pc in the case of DF and UM.

The IVC, on the other hand, might lie in the more distant Galactic halo, say 1 kpc. The power spectrum in that case would be probing transverse scales about 10 times larger, perhaps up to 100 pc in the case of DF and UM. Noting that the transition from probing 3D to probing 2D turbulence occurs when $\delta d\lt s$ (here, $s\sim 100$ pc), the relatively shallow spectra seen for the IVC VCs might indicate that these data are probing such a 2D topology, in which case the observed exponent would depend on the exponent of the velocity field as well (also relatively shallow, see Section 7.4). However, not much is known about the depth of the IVC. A typical IVC column density is 3 × 10¹⁹ cm⁻² and so volume density ${n}_{{\rm{H}}{\rm{I}}}=0.1$ cm⁻³ would translate to depths of 100 pc, using Equation (3) below. In Section 8, we present evidence for significant CNM in the IVC that would indicate higher ${n}_{{\rm{H}}{\rm{I}}}$ and lower depths within these structures. However, the relevant $\delta d$ is the amount by which gas across the region is spread out along the line of sight, not the depth of individual features.

The Draco nebula, estimated to be at a distance of about 500 pc (Gladders et al. 1998), is an interesting IVC because it shows an interaction with the embedding Galactic halo (Section 8.1, Miville-Deschênes et al. 2016). In the interaction region, the gas has become dense enough to be molecular, and so, the depth there might be of order a parsec, smaller than the angular extent of 15 pc. The power spectrum is again relatively shallow (−2.68 ± 0.07, Appendix F.2), perhaps suggestive of a 2D topology, but $\delta d$ across the region is not readily known. Alternatively, a shallower behavior compared to typical LVC might simply reflect different conditions in the driving of the turbulence in the two components.

The HVC gas in N1 is part of Complex C at an estimated distance of $\sim 10\pm 0.3\,\mathrm{kpc}$ (Thom et al. 2008). Adopting this distance, the power spectrum would be probing transverse scales up to 300 pc for the 2° EN field. The power-spectrum exponent is not atypical, and so, if revealing a 3D topology, would imply $\delta d\gt 300$ pc and a mean density $\lt 0.01$ cm⁻³. From the line profiles, it appears that the HVC component could be divided into a broad (WNM-like) and narrow (CNM-like) component. The narrow line components are quite compact, some about 3' or 10 pc at the adopted distance. These individual features have ${N}_{{\rm{H}}{\rm{I}}}$ about 7 × 10¹⁹ cm⁻² and so a typical ${n}_{{\rm{H}}{\rm{I}}}$ of 3 cm⁻³, perhaps greater locally if clumpy below the beam scale. This is much larger than the average density, but does not necessarily imply $\delta d\lt s$, because, again, the relevant $\delta d$ is the amount by which the gas, including the condensed structures (filaments, nuggets), are spread out along the line of sight. Furthermore, the turbulent conditions in this infalling gas could be quite different from Galactic LVC gas, making interpretation more challenging.

We also analyzed the centroid velocity maps for all VCs in DF, EN, and UM (Figures 20, 21, and 34, respectively), with the goal of determining the power spectrum of the 3D velocity field (Miville-Deschênes et al. 2003b). To quantify the power spectrum, the centroid velocity data were deconvolved by the DRAO ST beam and modeled with a power law. Interestingly, the noise component at high k is similar to the noise template used for the ${N}_{{\rm{H}}{\rm{I}}}$ power-spectrum models up to $k\sim 0.8;$ however, the shape is not as consistent, and so we chose to fit a power-law model without a noise template. This precludes the use of high-k power in the model fit, limiting the range to $k\lt 0.06$ arcmin⁻¹. An associated systematic uncertainty in the exponent in comparison to analysis of ${N}_{{\rm{H}}{\rm{I}}}$ maps of the same VC could amount to 0.1.

As summarized in Table 3, the power-law exponents for the centroid velocity maps are for the most part the same as those found for the ${N}_{{\rm{H}}{\rm{I}}}$ maps, within the uncertainties. This interesting empirical result is not a general expectation of theory and implies that the 3D velocity field and the 3D density field (see discussion in Section 7.2) have rather similar statistical properties. An exception might be the centroid velocity map of DF IVC, which, being a combination of IVC1 and IVC2 with distinct velocities and spatial structure, is more complex.

Examination of the H i data cubes shows that the structure in the channel maps changes dramatically over even small changes of velocity. At the spatial positions of these distinctive structures, the line profiles have significant narrow components. To illustrate this we made RGB color images of the H i emission using three distinct channels offset by 2.47 km s⁻¹. Examples are given for a variety of environments: LVC gas in the overlapping DF and UM regions in the NCPL (Figure 24); IVC gas in these same regions (Figure 25) and in DR (the Draco nebula, Figure 26); and HVC gas in EN (Figure 27).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For each figure, the intensity ranges for the channels represented by different primary colors are identical. Therefore, when the line profiles are sufficiently narrow and the emission at different velocities does not overlap spatially, the image appears red, green, or blue. With partial overlap at similar intensity, the image appears yellow or cyan (other blending possibilities resulting in magenta or white are rare in these examples).

The channel offset used, 2.47 km s⁻¹, corresponds to the FWHM of a thermally broadened H i line at a kinetic temperature of about 130 K, even without allowance for instrumental and turbulent broadening. The fine structure that appears in a single color in this visualization comes from narrow line components of CNM gas, as can be corroborated by independent estimates of the density and spin temperature of the gas (Sections 8.2 and 8.3).

The LVC H i seen in Figure 24 has a turbulent and filamentary nature; angular power spectra of single-channel maps are illustrated in Figures 11, 12, and 14.

In the IVC range, the channel maps of DF and UM near −50 km s⁻¹ (Figure 25) have a streaky appearance, suggestive of a preferred orientation for the structures (approximately along constant Galactic latitude). Another distinguishing feature, compared with the appearance in the LVC range, is higher contrast frothy small-scale structure, also seen in PO near −21 km s⁻¹  (Figure 38). These maps have slightly flatter power spectra (see Figure 12 and also Figure 22 and Table 3). power-spectrum analysis for images with such exceptional directional structure is discussed in Appendix A.5.

The color image of the Draco nebula, the IVC in DR (Figure 26), reveals some particularly interesting features. Relative to a diagonal running northeast to southwest (upper left to lower right), there is IVC gas to the upper right and a void, at those velocities, to the lower left. It has been suggested that infalling higher velocity gas is interacting with more tenuous gas at high altitude in the Galactic disk (Goerigk et al. 1983). The DHIGLS observations show the complexity of this interaction. Nevertheless, there are some important systematics that will be important to model. For example, in the central portion along this boundary there is a change in the characteristic velocity of the narrow line emission across the "leading edge," from red to green (in radial motion approaching more quickly) in Figure 26 on an arcminute (0.15 pc) scale, as one passes from lower left to upper right. Along this edge, the density is enhanced locally to the extent that CO is present and detectable in the Planck CO maps (Planck Collaboration et al. 2014b). The corrugated structure of the edge might result from hydrodynamic instability in the interaction region (Miville-Deschênes et al. 2016).

In EN, the LVC has no prominent small-scale structure, except for a filament at −7 km s⁻¹ at ${16}^{{\rm{h}}}{10}^{{\rm{m}}},+54^\circ 10$'; this filament was studied earlier with the GBT (Miville-Deschênes & Martin 2007) and is seen much more clearly in our DHIGLS data with the DRAO ST resolution.

By contrast, the HVC image in Figure 27 reveals an intricate pattern of coherent narrow ribbons of emission at each particular velocity, with the structure in this velocity range shifting systematically with increasingly negative velocity to the lower right, along the general direction of the ribbons. The HVC gas in EN is part of Complex C at an estimated distance of $\sim 10\pm 2.5\,\mathrm{kpc}$ (Thom et al. 2008).

It is useful to check that the volume density in these structures is relatively high, as would be expected in the CNM as compared to the WNM.

If a column density ${N}_{{\rm{H}}{\rm{I}}}$ arises in a physical depth $\delta d$, then the volume density ${n}_{{\rm{H}}{\rm{I}}}$ is given by¹⁹

$$\begin{eqnarray}&&{n}_{{\rm{H}}{\rm{I}}}=3.2\,\displaystyle \frac{{N}_{{\rm{H}}{\rm{I}}}}{[{10}^{19}\,{\mathrm{cm}}^{-2}]}\,\displaystyle \frac{[1\,\mathrm{pc}]}{\delta d}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 3 }$$

A structure of angular size ψ at distance d has a transverse size

$$\begin{eqnarray}&&s=0.29\,\displaystyle \frac{\psi }{[10^{\prime} ]}\,\displaystyle \frac{d}{[100\,\mathrm{pc}]}\,\mathrm{pc}.\end{eqnarray} \tag{ 4 }$$

Supposing that $\delta d$ can be estimated from the narrowest dimension of the structure, ${\psi }_{{\rm{n}}}$, then

$$\begin{eqnarray}&&{n}_{{\rm{H}}{\rm{I}}}=11\,\displaystyle \frac{{N}_{{\rm{H}}{\rm{I}}}}{[{10}^{19}\,{\mathrm{cm}}^{-2}]}\,\displaystyle \frac{[10^{\prime} ]}{{\psi }_{{\rm{n}}}}\,\displaystyle \frac{[100\,\mathrm{pc}]}{d}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 5 }$$

For an optically thin H i line component approximated by a Gaussian specified by its peak brightness temperature ${T}_{{\rm{p}}}$ and FWHM,

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{{\rm{H}}{\rm{I}}}}{[{10}^{19}\,{\mathrm{cm}}^{-2}]}=16\,\displaystyle \frac{{T}_{{\rm{p}}}}{[20\,{\rm{K}}]}\,\displaystyle \frac{\mathrm{FWHM}}{[4\,\mathrm{km}\,{{\rm{s}}}^{-1}]},\end{eqnarray} \tag{ 6 }$$

and

$$\begin{eqnarray}&&{n}_{{\rm{H}}{\rm{I}}}=174\,\displaystyle \frac{{T}_{{\rm{p}}}}{[20\,{\rm{K}}]}\,\displaystyle \frac{\mathrm{FWHM}}{[4\,\mathrm{km}\,{{\rm{s}}}^{-1}]}\,\displaystyle \frac{[10^{\prime} ]}{{\psi }_{{\rm{n}}}}\,\displaystyle \frac{[100\,\mathrm{pc}]}{d}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 7 }$$

Parameters and derived volume densities for a few worked examples are presented in Table 4. Consider first the green "face" and filaments in DF (Figure 24, upper panel), which we tabulate as DFa. From Equation (7), we find a volume density in the range ${n}_{{\rm{H}}{\rm{I}}}\sim 30\mbox{--}120\,{\mathrm{cm}}^{-3}$, somewhat denser than the conditions for CNM gas modeled in Wolfire et al. (2003).

Table 4.  Parameters of Select Structures and Derived Volume Density

Feature	Distance	FWHM	Tp	ψ	${n}_{{\rm{H}}{\rm{I}}}$
	(pc)	(km s−1)	(K)	(')	(${\mathrm{cm}}^{-3}$)
DFa	100–200a	4	20	15–30	30–120
DFb	100–200a	3	4	4	40
ENa	150b	4	10	6	100
ENb	10000c	4	15	3	4

Notes.

^aNCPL LVC distance, Meyerdierks et al. (1991). ^bLVC distance, Miville-Deschênes & Martin (2007). ^cHVC distance, Thom et al. (2008).

Download table as:  ASCIITypeset image

DF also hosts some remarkably long narrow structures. Among these, in the left-hand side of channel maps near 9 km s⁻¹, is one (DFb) that runs over 3° along $\delta \sim +73\buildrel{\circ}\over{.} 5$, but is barely resolved by the DRAO ST synthesized beam, indicating an axial ratio of over 100. From the parameters in Table 4, ${n}_{{\rm{H}}{\rm{I}}}\sim 40\,{\mathrm{cm}}^{-3}$, if it is indeed a filament in 3D.

Likewise, for the above-mentioned LVC filament in EN (ENa), we find ${n}_{{\rm{H}}{\rm{I}}}\sim 100\,{\mathrm{cm}}^{-3}$.

The HVC ribbons seen in EN are neutral condensed structures with a density much higher than the estimated average, ${n}_{{\rm{H}}{\rm{I}}}\sim 0.03\,{\mathrm{cm}}^{-3}$ (Thom et al. 2008), with a low covering factor. Furthermore, there is clear substructure along the ribbons. The brightest such nugget (ENb) at ${16}^{{\rm{h}}}{20}^{{\rm{m}}}{37}^{{\rm{s}}},+54^\circ 55^{\prime} 25^{\prime\prime}$ in the −125 km s⁻¹ channel map, has ${n}_{{\rm{H}}{\rm{I}}}\sim 4\,{\mathrm{cm}}^{-3}$. The integrated column density of even this brightest nugget is still quite low, ${N}_{{\rm{H}}{\rm{I}}}$ $\sim \,{10}^{20}\,{\mathrm{cm}}^{-2}$ from Equation (6), requiring the angular resolution and sensitivity of the DHIGLS product to bring out the fine detail across the region.

H i absorption of bright background radio sources (e.g., Roy et al. 2013; Murray et al. 2015) can be found in DHIGLS spectra. The absorption coefficient of H i varies inversely with spin temperature ${T}_{{\rm{s}}}$, favoring the detection of cold gas.

Thanks to the relatively small size of the synthesized beam (${\theta }_{\mathrm{sb}}$ in arc seconds in Table 1) some radio point sources are detected with high continuum brightness temperatures ${T}_{{\rm{c}}}$. For a source of flux density ${S}_{\nu }$,

$$\begin{eqnarray}&&{T}_{{\rm{c}}}=170{\left(\displaystyle \frac{[60^{\prime\prime} ]}{{\theta }_{\mathrm{sb}}}\right)}^{2}\,\displaystyle \frac{{S}_{\nu }}{[1\mathrm{Jy}]}\,{\rm{K}}.\end{eqnarray} \tag{ 8 }$$

H i absorption reduces this background emission to ${T}_{{\rm{c}}}\exp (-\tau )$. Because of continuum removal in the processing of the spectra, this results in a negative-going feature in the spectrum, of depth $\tau {T}_{{\rm{c}}}$ for small optical depth.

When τ is small, the net temperature ${T}_{{\rm{n}}}$ recorded in the DHIGLS spectrum is therefore

$$\begin{eqnarray}&&{T}_{{\rm{n}}}={T}_{{\rm{b}}}(1-{T}_{{\rm{c}}}/{T}_{{\rm{s}}}),\end{eqnarray} \tag{ 9 }$$

which becomes negative when ${T}_{{\rm{c}}}\gt {T}_{{\rm{s}}}$ or ${S}_{\nu }\gt 0.4$ Jy for ${T}_{{\rm{s}}}=80$ K. From the NVSS catalog browser²⁰ we find about 0.25 such sources per square degree and although this is small we have found instances of favorable alignments producing the expected narrow absorption features.²¹

In DF a prominent example is 4C + 71.09 (NVSS J101132+712440) which can be seen as a dark dot against emission at 5.12 km s⁻¹ (green) in Figure 24 (upper panel). For this source, ${T}_{{\rm{c}}}=380$ K. The spectrum is most negative at 5.94 km s⁻¹ with ${T}_{{\rm{n}}}=-70$ K. The FWHM of the absorption feature is about 3.5 km s⁻¹. ${T}_{{\rm{b}}}$ interpolated spatially in the channel map is 25 K, with about 20 K attributable to narrow emission of FWHM 4 km s⁻¹, slightly broader than the absorption. Rearranging Equation (9), the spin temperature can be estimated from

$$\begin{eqnarray}&&{T}_{{\rm{s}}}={T}_{{\rm{c}}}{T}_{{\rm{b}}}/({T}_{{\rm{b}}}-{T}_{{\rm{n}}}).\end{eqnarray} \tag{ 10 }$$

For this example, using the numbers collected in Table 5, this evaluates to ${T}_{{\rm{s}}}=84$ K and confirms that the optical depth is appropriately small ($\lt 0.25$) and that this is CNM gas as suggested above on the basis of ${n}_{{\rm{H}}{\rm{I}}}$. That the line width is slightly broader than thermal is usually attributed to turbulence, which also appears to play a role in the pressure distribution (Wolfire 2015).

Table 5.  Parameters^a of Select Absorption Features and Derived ${T}_{{\rm{s}}}$

Region	NVSS Source	${T}_{{\rm{c}}}$	${v}_{\mathrm{LSR}}$	${T}_{{\rm{n}}}$	${T}_{{\rm{b}}}$	${T}_{{\rm{s}}}$
DF	J101132 + 712440	380	+5.1	−70	25	84
UM	J094912 + 661459	444	+2.6	−106	25	85
UM	J094912 + 661459	444	−53.0	−58	13	80
DR	J164829 + 600722	41	−23.7	+5	18	55

Note.

^aVelocity in km s⁻¹ and temperatures in K.

Download table as:  ASCIITypeset image

There are many more examples of absorption indicating cold CNM gas in the LVC range. In UM, 4C + 66.09 (NVSS J094912+661459) can be seen deeply absorbed at 2.64 km s⁻¹ (blue in Figure 24, lower panel) with FWHM about 3.8 km s⁻¹ as for nearby emission. Using the values in Table 5, ${T}_{{\rm{s}}}=85$ K.

Although lines of sight to bright radio sources rarely intersect sufficiently bright IVC gas, which has a lower covering factor than the LVC gas, there are a few examples in the DHIGLS data. Toward 4C + 66.09 in UM the deepest absorption in the IVC range is at −53.0 km s⁻¹ (cyan in Figure 25, lower panel), with FWHM about 3 km s⁻¹ (absorption and emission). In this IVC feature, ${T}_{{\rm{s}}}=80\,{\rm{K}}$. Using ${\psi }_{{\rm{n}}}\sim 10^{\prime}$ and taking $d\lt 500$ pc, we find ${n}_{{\rm{H}}{\rm{I}}}\gt 20\,{\mathrm{cm}}^{-3}$. Thus, the IVC gas that reveals the frothy and streaky structure across the whole region is dense, like a CNM gas.

In DR, absorption of the weaker source NVSS ${\rm{J}}164829\,+600722$ still produces a relative deficit visible in the −23.7 km s⁻¹ emission (cyan) in Figure 26 and we find ${T}_{{\rm{s}}}\sim 55$ K. This gas is part of the prominent higher column density region in the south, which is molecular (Herbstmeier et al. 1993).

For the HVC range, because of factors like insufficient spectral coverage, faintness of emission, and low covering factor, there are no examples of absorption in DHIGLS data.

The noise in a mosaic made using the interferometric data from the DRAO ST is quite complex. The spectral noise is not uniform over the map (see example in upper panel of Figure 7). The power spectrum of an emission-free channel is not flat (see example in Figure 11), but it has a consistent shape and scale for a given mosaic. For example, at values of k corresponding to antenna spacings of $34L$ to $38L$ ($D\sim 146$–163 m or $k\sim 0.20$–0.22 arcmin⁻¹), the noise is slightly reduced because the strategy for the separation of the moveable antennas and their discrete movements results in twice as much coverage at these antenna spacings. This can be seen clearly in the power spectrum of the emission-free channel in Figure 11. At higher values of k, the desired H i signal is being reduced by the synthesized beam and this example shows how, even in relatively bright intermediate Galactic latitude regions like DF, the noise dominates at large k. As described in Section 3, a Gaussian taper was applied in the Fourier domain, apodizing the visibilities at high k. In Figure 11 it can be seen how this taper has affected the level and shape of the power spectrum of the noise above k = 0.4 arcmin⁻¹.

We exploit the consistent shape to calculate a median of the 2D power spectra for a set of emission-free channels for each mosaic. We call this the "noise template." After deconvolution by the relevant beam, we form the 1D equivalent, N_d(k). At high k this is greatly amplified compared to the version without deconvolution, N(k). Because the noise is always larger when there is an H i signal (e.g., Martin et al. 2015), this template needs to be scaled by a fitting factor $\eta \gt 1$, as in Equations (12) and (11) below.

The 1D power spectrum of deconvolved data can be described with a parameterized model consisting of a power-law representation of the signal plus a noise component:

$$\begin{eqnarray}&&{P}_{\mathrm{model},{\rm{d}}}{(k)={P}_{0}(k/{k}_{0})}^{\gamma }+\eta {N}_{{\rm{d}}}(k),\end{eqnarray} \tag{ 11 }$$

where P₀ is the amplitude of the power law at some representative scale k₀ and $\gamma$ is the scaling exponent. The exponent is alternatively called the spectral index or the slope (in a log−log representation).

Without deconvolution, the original data are modified by the effective beam decreased at high k through ${\tilde{\phi }}^{2}(k)$ so that the model would be

$$\begin{eqnarray}&&{P}_{\mathrm{model}}{(k)={\tilde{\phi }}^{2}(k){P}_{0}(k/{k}_{0})}^{\gamma }+\eta N(k).\end{eqnarray} \tag{ 12 }$$

For application in the above 1D model the effective ${\tilde{\phi }}^{2}(k)$ can be estimated directly from the 1D power spectrum formed from the 2D power spectrum of a representative synthesized beam at the center of the DRAO ST mosaic. As a check, fitting such a 1D power spectrum with a Gaussian results in equivalent 1D Gaussian beams for DF and EN that have quite similar FWHM, $56\buildrel{\prime\prime}\over{.} 8\pm 0\buildrel{\prime\prime}\over{.} 7$ and $58\buildrel{\prime\prime}\over{.} 9\pm 0\buildrel{\prime\prime}\over{.} 9$, respectively, both reasonably consistent with the 2D Gaussian beam statistics as summarized in Table 1.

Example power spectra of signal and noise are shown in Figure 11 for the DF-i mosaic. Generally, a noise template is calculated from many emission-free channels (Appendix A.1). In this example, for simplicity, the noise template is from a single representative emission-free channel (v = 39.7 km s⁻¹, as also used in Figure 2, left).

We adopt the model represented by Equation (11) to probe the power spectrum of the signal to smaller spatial scales, while properly accounting for the noise.

We also mitigated against further uncertainty in model fits by excluding data above a value ${k}_{\max }$; the results below are not sensitive to the precise value, so long as the noise can be adequately assessed. We adopted 0.5 arcmin⁻¹ (which corresponds to baselines of 361 m).

In the DRAO ST observations there are no data corresponding to spacings shorter than 13 m and so there is little power at $k\,\lt$ 0.018 arcmin⁻¹, except from foreshortened baselines. This is reflected in the power spectrum of the signal in Figure 11, where there is a precipitous departure below the rising power law at 0.018 arcmin⁻¹. The remedy for this is to add short-spacing data acquired with a large single antenna (Section 5). For the power-spectrum fits to the DRAO ST data alone, we excluded data below a value ${k}_{\min }$, here taken to be 0.018 arcmin⁻¹.

The three best-fit model parameters were found using the IDL routine mpfit.pro (Markwardt 2009), with the described uncertainties as weights. The weights were further modified at high k using the prescription in Martin et al. (2015), specifically substituting fractional error b = 0.07 into their Equation (4) to account for the uncertainty of the effect of the beam.

The fits to the models for DF-i data are shown in Figure 11 in Section 4.4. The exponent is $\gamma =-3.03\pm 0.03$. The uncertainties that we cite are the formal 1σ errors from the fits. Alternative choices in the fitting analysis could lead to systematic uncertainties in the exponent, which we estimate to be no larger than 0.1. We also found that relative differences between the exponents for different channels or VCs (Section 7.1) are robust against the systematic effects.

Apodizing a (median-subtracted) map prior to computing the Fourier transform has the benefit of reducing edge effects. Normally maps can be made rectangular and so only a simple rectangular taper is required. The unique shapes of the DHIGLS regions are distinctly non-rectangular. One alternative is to define an inscribed rectangle, as we have done, but this inevitably leaves out high signal-to-noise data. This led us to an alternative approach aimed at incorporating as much spatial information from each mosaic as possible, in this case using data within the white dashed contour (e.g., Figure 3).

We modified the shape of the taper so that rather than following a straight edge of the map it followed a contour of constant weight (or noise) in the DRAO ST mosaic. The median-subtracted maps were tapered to zero orthogonal to the chosen contour. The 35' taper was centered on the contour. These tapered maps were then cropped to a rectangular map aligned with the zeroed edges of the map. Because of the unique shapes of the DHIGLS regions, information is missing between the zeroed edges of the map and the boundary of the rectangle; this was padded with zeros. In practice, the 2D taper was obtained by convolving a unit mask defined by the contour with a circularly symmetric tapering function of the desired shape and width and then cropping.

The drawback of any tapering (apodization) of the map is that the resulting Fourier transform has also been modified: the true Fourier transform has been convolved with the Fourier transform of the taper. The Fourier transforms of the various non-rectangular zero-padded apodization functions obtained as described above are all centrally peaked, suggesting that a convolution in the Fourier domain will not have a significant effect on the measured power-law exponent of the original DHIGLS map. (The amplitude is affected, but that is not of interest here.)

We have tested this conjecture using simulations; fBm maps were created with a given power-law exponent (Miville-Deschênes et al. 2003b). We apodized these maps with the custom-shaped DHIGLS taper, Fourier transformed them, and computed the power spectrum. Each power spectrum was then modeled to find the power-law exponent. We found that the results were consistent with those used to make the fBm maps within the model uncertainties.

Implicit in the construction of the power spectrum in 1D is that the 2D power spectrum is azimuthally symmetric. For exceptional images with a pronounced streaky appearance, the directionality affects the 2D power spectrum. For example, for the single channel of IVC emission in the UM region at −51.76 km s⁻¹, seen in green in Figure 25, lower panel, the directionality of the structure is roughly on the upper left to lower right diagonal, and in the image of the 2D power spectrum there is more power in the u–v plane orthogonal to this, roughly in two opposed quadrants. This can be brought out most effectively in the image by removing the radial dependence, i.e., by dividing the 2D power spectrum by the model fit to the 1D power spectrum, which has exponent −2.35 ± 0.03.

To quantify what is seen visually, as a simple approximation we made a 1D power spectrum by azimuthally averaging within these opposed quadrants, and another 1D power spectrum for the other two complementary quadrants. The power-law exponents were quite similar (−2.28 ± 0.04 and −2.27 ± 0.05, respectively). Thus, the systematic effect is apparently not large. However, as expected, the amplitudes were different, in this case by a factor of about two. Given the similar exponents, it is not surprising that the amplitude for the overall 1D power spectrum is close to the mean of the amplitudes for the two separate 1D power spectra. We are often interested in power spectra of images of integrated emission, ${N}_{{\rm{H}}{\rm{I}}}$ (Section 7.1). Because the H i structure changes over many channels, the directionality is less pronounced in ${N}_{{\rm{H}}{\rm{I}}}$ (e.g., compare Figures 25 and 33), so that forming the 1D power spectrum is more suitable and less prone to systematic effects. For the case above, recall that the exponent for the somewhat noisier ${N}_{{\rm{H}}{\rm{I}}}$ image of the full IVC VC in UM was −2.48 ± 0.06 (Table 3).

Also, in comparing the power spectra of images of two independent tracers, for example ${N}_{{\rm{H}}{\rm{I}}}$ and thermal dust (Section 7.1.1), it seems likely that any systematic effects will be similar given a common directionality.

CGPS (Figure 28, upper panel, red). Short-spacing data were added for each separate synthesis before mosaicking. For the single-dish data, the 26 m is quite undersized for the task. The lower bound adopted is lower than the minimum baseline (8.6 m versus 12.9 m; see Higgs et al. 2005). Even so, the single-dish data needed in the overlap range are well down in the 26 m beam.

SGPS (lower panel, red). The Parkes 64 m beam is also cutting off data significantly over the entire overlap range. This was mitigated somewhat by mosaicking of the ATCA data (McClure-Griffiths et al. 2005), which reduces the effective minimum baseline by $D/2$, where D is the diameter of the antennas making up the interferometer. Still, the feathering implies an extrapolation of the interferometric data to low k. The reduction by the synthesized beam is barely relevant.

VGPS (lower panel, blue). The minimum baseline of the VLA is quite large (36 m), so that the overlap range is affected by the beam of the GBT despite its 100 m aperture. The lower and upper bounds adopted for cross calibration (not given in Stil et al. 2006) were 36 and 90 m (J. Stil 2016, private communication).

DHIGLS (upper panel, blue). The mosaicking by SUPERTILE was not designed to achieve the $D/2$ reduction in minimum baseline. Still, relative to the facilities combined in other surveys, there is a potentially much broader u–v overlap range because the minimum baseline is small, the GBT is large, and the synthesized beam is small. Relative noise is also a consideration. Details are given in Section 5.

The cross-calibration comparison was done at the lower resolution of the GHIGLS data and in the u–v domain. The evaluation of ${f}_{\mathrm{cc}}$ was carried out optimally using the full 2D information.

We used the data for the DF region to refine our approach, as illustrated below. The GHIGLS SPIDER data are in the Galactic GLS projection whereas the DF-i mosaic is in the ICRS NCP projection. To minimize reprocessing of the GHIGLS data and to ensure the same spatial coverage for the cross calibration, the DF-i cube was reprojected to the same Galactic GLS projection as the GHIGLS data, adopting the same 18'' pixel grid spacing as in the original DF-i cube. We also processed the weight map in the same way, to track the region in which the two data sets could be compared, and similarly the synthesized beam map. The velocity channels were interpolated to correspond to those in the SPIDER cube.

For the cross calibration the signal needs to stand out from the noise. With regard to the noise, we compared the data in the region where the noise level in the DF-i mosaic is no more than a factor of two larger than the noise minimum (see the white dashed contour in Figure 3). With regard to the signal, we compared the data near the peak in the LVC H i spectrum. Specifically, we used a map of ${W}_{{\rm{H}}{\rm{I}}}$, the integral of the H i spectra, including only ${N}_{\mathrm{ch}}$ channels near the peak of the H i standard deviation spectrum (see Figure 15). We estimated the relative signal-to-noise ratio (S/N) as a function of ${N}_{\mathrm{ch}}$ using ${W}_{\mathrm{sd}}/{\sqrt{N}}_{\mathrm{ch}}$, where ${W}_{\mathrm{sd}}$ is the standard deviation about the mean (alternatively, it can be estimated via analysis of the power spectrum). We adopted the range 1.6 km s⁻¹ $\lt \,{v}_{\mathrm{LSR}}\lt 6.5$ km s⁻¹ to create the optimal ${W}_{{\rm{H}}{\rm{I}}}$ map, which results in an S/N about twice that of a single channel.

As in Appendix A.4, starting with the original GHIGLS data in the image plane, we first removed the median value of the data (in this case ${W}_{{\rm{H}}{\rm{I}}}$) within the white dashed contour, then apodized with a 35' taper centered on the contour to reduce the data to zero, and beyond that in the encompassing square image padded with zeros. We then computed the 2D Fourier transform $\tilde{f}({k}_{x},{k}_{y})$ of this modified image. The tapering and padding with zeros, applied equally to two fields to be compared, will not affect the derived scale factor. As discussed in Appendix A.4, there is no significant effect on the power-law exponent either. We note that this scheme is similar to that used by Bertincourt et al. (2016) for power spectra of non-rectangular regions in the determination of a cross-calibration factor for Herschel/SPIRE and Planck/HFI data.

To produce the corresponding product for the DRAO ST data, starting with ${W}_{{\rm{H}}{\rm{I}}}$ from the finely gridded reprojected DF-i data described above, we deconvolved with the synthesized beam, convolved with the 2D GHIGLS beam, and regridded to the coarser GHIGLS image grid. We then selected the region, removed the median, apodized, and zero-padded in the same way as for the GHIGLS data, and finally computed the 2D Fourier transform.

Solely for visualizing the relative calibration of the data we also produced another closely related product from the DRAO ST image data, omitting the steps of convolving with the 2D GHIGLS beam and regridding to the coarser GHIGLS image grid. As in Appendix A, we computed the 1D power spectrum from these 2D data and from those for GHIGLS. Figure 29 shows the power spectra of the two data products and the power-law fits calculated for the different but overlapping k ranges appropriate to the two data sets. The power-spectrum exponent for the GHIGLS data, −3.00 ± 0.05, is very close to that for the DRAO ST data, −3.06 ± 0.03. This is an important check that has not been possible for the other H i surveys mentioned. Furthermore, it is clear that a vertical shift is required to align the DRAO ST data with those from the GBT data. The amount of the shift is ${f}_{\mathrm{cc}}^{2}$.

Zoom In Zoom Out Reset image size

Next we consider the "overlap range" in k, or, in practice, an annulus in the u–v domain, in which the data are to be compared.

On the low side, this range is constrained by the minimum baseline of the interferometer. As above for the DRAO ST, we chose 13 m ($0.018\,{\mathrm{arcmin}}^{-1}$). This is a conservatively higher choice relative to the minimum baseline, compared to other H i surveys (Figure 28). Nevertheless, it is still not far into the range of k that is affected by the beam of the (short-spacing) single-dish data, as was necessary for the VGPS (larger minimum baseline), CGPS (smaller single dishes), or SGPS (combination of the two).

The choice for the high-k side depends on properties of the GBT data. It needs to take into account the relative noise of the two data sets, which can be assessed using power-spectrum model fits according to Equation (11). As is apparent from the results shown in Figure 29, because of the effect of the GHIGLS beam, as we move to higher values of k in the power spectrum, the GBT noise becomes larger relative to not only the DRAO ST noise, but also the H i signal. Another consideration is that data in the entire range of k being considered are affected by the GHIGLS beam, and any uncertainties in the beam would produce more pronounced effects at higher values of k (the synthesized beam is not a consideration). To be conservative, we adopted 35 m ($0.0485\,{\mathrm{arcmin}}^{-1}$).

Despite these conservative choices, the overlap range is relatively broad and much less impacted by the beam of the single-dish data as compared to the other surveys (Figure 28); this provides the opportunity for a more critical assessment of the cross-calibration determination.

There are a number of different methods for determining the cross-calibration factor . The one we adopted, as in IMMERGE, was to correlate the complex 2D Fourier-transformed data (keeping real and imaginary parts distinct) within the overlap-range annulus in the u–v domain. To mitigate against any residual cross, we masked two rectangular regions each with a width of three pixels centered along each axis (i.e., along u = 0 and v = 0). The intercept from this correlation is consistent with zero within its error (Figure 30) and subsequently in fitting the data we enforced this by adding data at $(0,0)$ explicitly.

Zoom In Zoom Out Reset image size

As judged from the noise power spectra, the GHIGLS data have greater uncertainty than those from the DRAO ST. This is confirmed by the standard deviation of the values of the modulus in rings of constant k. Therefore, we performed the usual "y on x" regression implied by the format of Figure 30. As in IMMERGE, for the linear fit, we minimized the absolute deviation (L1-norm). For the DF field, we found ${f}_{\mathrm{cc}}=1.12\pm 0.01$. We also used ordinary least squares fitting (i.e., minimizing the L2-norm), finding 1.10 ± 0.01. Furthermore, the bisector slope is also the same within 1σ. These checks reinforce that the result is robust.

In the same way, we found independent values of ${f}_{\mathrm{cc}}$ for four other regions, EN, UM, DR, and PO (see Table 6). The DRAO ST syntheses included in these mosaics had different values of ${F}_{\mathrm{DRAO}}/{F}_{\mathrm{NVSS}}$ applied, indicating the efficacy of that part of the calibration.

Table 6.  Values of Cross-calibration Scale Factor ${f}_{\mathrm{cc}}={I}_{\mathrm{GBT}}/{I}_{\mathrm{DRAO}}$

Region	${f}_{\mathrm{cc}}$
DF	1.12 ± 0.01
EN	1.10 ± 0.02
UM	1.09 ± 0.01
DR	1.12 ± 0.02
PO	1.17 ± 0.02

Download table as:  ASCIITypeset image

We have investigated the systematic errors that might arise. One is from the choice of overlap range. For example, the values of ${f}_{\mathrm{cc}}$ using data from the independent annuli between 13 and 26.4 m and between 26.4 and 35 m are 1.09 ± 0.01 and 1.16 ± 0.01, respectively.

Deconvolving the GHIGLS data and not convolving the DRAO ST data with the GHIGLS beam changes the effective weighting going into the slope. We found ${f}_{\mathrm{cc}}=1.13\pm 0.01$.

We investigated correlating values of the Fourier amplitude (modulus of the visibility) in 2D. This again changes the weighting. We also note that at higher k the amplitudes are smaller, but there are more independent data samples. We found ${f}_{\mathrm{cc}}=1.14\pm 0.01$.

We also correlated power-spectrum moduli in 2D, finding ${f}_{\mathrm{cc}}$ from the square root of the slope. This is akin to fitting a common power law to the two sets of data. Again, the weighting changes and we found ${f}_{\mathrm{cc}}=1.12\pm 0.01$. We note that correlating power-spectrum moduli in 1D (a weighted fit is indicated) is less satisfactory, because of the compression of the data in k and azimuthal coverage before the correlation (though it should make no difference for noise-free data).

From these investigations, we estimate that the systematic error in ${f}_{\mathrm{cc}}$ is 0.03.

We also looked at the cross-calibration factor synthesis by synthesis. The DRAO ST data have to be corrected for the effect of the primary beam, which limits the spatial coverage to about 90' and makes the data noisier at the low end of the k range. An annulus between 25 and 50 m ($0.034\lt k\lt 0.069$ arcmin⁻¹) was selected. We found a mean cross-calibration factor 1.13 ± 0.01, consistent with the above. As another check, we used DRAO ST data that were not scaled to NVSS and found that the values of ${I}_{\mathrm{GBT}}/{I}_{\mathrm{DRAO}}$ were well correlated with the inverse of ${F}_{\mathrm{DRAO}}/{F}_{\mathrm{NVSS}}$ (Section 3.5). This indicates that if cross-calibration synthesis by synthesis were sufficiently precise, there would be no point in first doing the synthesis by synthesis scaling to NVSS.

We adopted a common cross-calibration factor of $1.12\,\pm 0.01\pm 0.03$ for all seven DHIGLS regions. A correction of the same order was found by Pidopryhora et al. (2015) for combining GBT and VLA data. This is mentioned not for direct comparison to our ${f}_{\mathrm{cc}}$, but to point out that a non-unity cross-calibration factor is not unprecedented.

The correction in Equation (13) clearly depends on the contrast of the observed brightness ${T}_{{\rm{b}}}$ relative to ${T}_{{\rm{s}}}$. Because peaks in the intrinsic H i emission on the sky are diminished if the observing beam is larger than the spatial structure, the estimated optical depth correction is reduced at lower resolution.

We estimated the impact on GHIGLS ${N}_{{\rm{H}}{\rm{I}}}$ maps by comparing results from data at two differing resolutions, assuming ${T}_{{\rm{s}}}=80$ K. For the first map, ${N}_{{\rm{H}}{\rm{I}}}$ was created from the DHIGLS cube using Equation (13) and then convolved to the GHIGLS resolution and regridded to 35 pixels. We call this ${N}_{{\rm{H}}{\rm{I}}}$(DHIGLS). For the second map, we convolved the DHIGLS cube to the GHIGLS resolution, regridded, and then computed ${N}_{{\rm{H}}{\rm{I}}}$. We call this ${N}_{{\rm{H}}{\rm{I}}}$(GHIGLS). The resulting difference, expressed as ${\rm{\Delta }}={N}_{{\rm{H}}{\rm{I}}}(\mathrm{DHIGLS})/{\rm{C}}\,-\,{{\rm{N}}}_{{\rm{H}}{\rm{I}}}(\mathrm{GHIGLS})/{\rm{C}}$, is positive.

Within the high signal-to-noise region bounded by the white dashed contour in DF, and for the LVC component, we found $\mathrm{log}({\rm{\Delta }})=0.02\pm 0.2$ for Δ in units of K km s⁻¹, meaning that at the lower resolution of actual GHIGLS, ${N}_{{\rm{H}}{\rm{I}}}$/C is underestimated on the order of 0–2 K km s⁻¹ for this region. From Table 3 in Martin et al. (2015), the uncertainty for the LVC component of (GHIGLS field) SPIDER, ${\sigma }_{{N}_{{\rm{H}}{\rm{I}}}}=0.3\times {10}^{19}$ cm⁻², or ${\sigma }_{{N}_{{\rm{H}}{\rm{I}}}}/C\sim 2$ K km s⁻¹, comparable to the maximum additional Δ correction. Thus, the uncertainty of ${N}_{{\rm{H}}{\rm{I}}}$ cited in Martin et al. (2015) might be somewhat underestimated in this systematic way by not accounting for the resolution dependence of optical depth effects. Further discussion of the effect of the optical depth correction and the dependence on ${T}_{{\rm{s}}}$ can be found in the analysis of the GHIGLS data in Martin et al. (2015).

Whether there is a significant effect on the estimated optical depth correction for the DHIGLS data would depend on whether the bright spatial structures are still unresolved at the resolution of the DRAO ST. There is the further uncertainty in the correction because the value of ${T}_{{\rm{s}}}$ is not known precisely and is unlikely to be constant.

The separation in the irregular grid of pointing centers (Figure 31) for the UM-i mosaic in the GHIGLS UMA field is somewhat smaller than the FWHM of the DRAO ST primary beam. The upper panel of Figure 32 shows the resulting noise map of UM-i. The coverage is relatively sparse compared to EN-i and DF-i, and so the noise level is higher, though at 2 K is still lower than at the center of a single synthesis. This is slightly better than achieved in the CGPS, but the signal in this region is of course much lower than in the Galactic plane. An example signal channel from the UM-i mosaic is shown in the lower panel of Figure 32.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The ${N}_{{\rm{H}}{\rm{I}}}$ maps produced from the UM data cube are shown in Figure 33, assuming ${T}_{{\rm{s}}}=80$ K. The corresponding centroid velocity maps (Section 7.4) are shown in Figure 34. The power spectra are discussed in Section 7.1 using data within the white dashed contour in Figure 31.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Color maps made from the DHIGLS cube are presented in the lower panels of Figures 24 and 25. Narrow emission and absorption lines in both LVC and IVC gas are discussed in Section 8.

The five pointing centers for the DR-i mosaic in the GHIGLS DRACO field can be seen in Figure 1 and the noise map for DR-i is given in Figure 35. The minimum noise level, 1.5 K, is only slightly better than at the center of a single synthesis, reflecting the relatively sparse coverage. Also given in Figure 35 is a channel map from DR-i representative of IVC gas with significant H i signal.

Zoom In Zoom Out Reset image size

The strong IVC emission that defines the Draco nebula can also be seen in the ${N}_{{\rm{H}}{\rm{I}}}$ map in the lower part of Figure 36. For data within the white dashed contour, the power-law exponent is −2.68 ± 0.07 (Figure 37). For comparison, the exponent evaluated for the smaller rectangular region is −2.57 ± 0.18 and that for the single channel of DR-i analyzed in Section 4.4 is −2.27 ± 0.14 (Figure 12). On the other hand the LVC is very weak and so the exponent is less well determined: −2.87 ± 0.24.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

A color map made from the DHIGLS cube in the IVC range is presented in Figure 26. Narrow emission and absorption lines in the IVC gas are discussed in Section 8.

The three pointing centers for the PO-i mosaic in the GHIGLS POL field can be seen in Figure 1 and the noise map of PO-i is given in Figure 38. The noise level is similar to that at the center of a single synthesis (Figure 2). The lower panel of Figure 38 shows a representative channel map of PO-i in the IVC range.

Zoom In Zoom Out Reset image size

The ${N}_{{\rm{H}}{\rm{I}}}$ maps produced from the PO data cube follow in Figure 39. The power-law exponent for the IVC data within the white dashed contour is −2.82 ± 0.08 (Figure 40). For comparison, the exponent evaluated for the smaller rectangular region is −2.80 ± 0.15 and that for the single channel of PO-i analyzed in Section 4.4 is −2.4 ± 0.2 (Figure 12). The LVC power spectrum has a somewhat small amplitude and a similar exponent: −3.17 ± 0.16.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The noise maps for the two single syntheses, MG in the GHIGLS G86 field and MC in the GHIGLS POL field, are like that shown for DF20 in Figure 2, with slightly different values of $\sin \delta$.

In Figure 41, it can be seen that compared to the LVC in MG, the IVC has a higher column density and more spatial structure. The ${N}_{{\rm{H}}{\rm{I}}}$ map for LVC in MC follows in Figure 42 (IVC is much less significant).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In this paper we are focusing on the mosaics from H i spectral-line data. However, we note that, as was the case for the CGPS, survey products derived from observations using the DRAO ST also include complementary continuum mosaics at 408 MHz and 1420 MHz and polarization mosaics in four bands near 1420 MHz. Such data enable many studies (Kothes et al. 2010). Point-source catalogs can be constructed with flux density, spectral index, and polarization information. The polarization properties of radio galaxies have been analyzed using the deepest continuum data then available, in EN (Taylor et al. 2007; Grant et al. 2010). Rotation measures of point sources can be used to probe the magnetic field (Brown et al. 2003), in this case at higher latitude including the Galactic halo (Mao et al. 2012). Diffuse emission at 408 MHz and at 1420 MHz in polarization can be mapped (Landecker et al. 2010), now at intermediate latitude. The potential correlation with H i structures can also be explored, but all of these studies are beyond the scope of this paper.