Spatial Transfer Functions

The majority of the past week has been dedicated to debugging; it looks like cross-scanned simulations finally work.

The plot below is a derivation of the spatial transfer function for a number of different intrinsic sky power-law power spectra.

Justifying the above plot is essential.

First, the very steep power-laws [-3 in the example below] show a recovery fraction >1. This is simply because their S/N was inadequate - the output power spectrum is nearly flat, but at a level higher than the sky.

Second, the most plausible power-laws [-1.5 in the example below] show pretty good recovery (90-95% over the relevant range):

There are some "white" power losses, particularly in the flatter power-spectra. My best guess is that this has something to do with the relative scales being offset from a mean of 1, but so far all tests to show that that is the cause have in fact shown no problems at all. What else could cause a scale-independent power loss?

Also, the flat power spectrum (and inverted) aren't quite flat because I impose a "galactic scale height" on them. Should I stop doing that?

additional problems

There are a few cases in the L=30 field that look awful. 050706_o31 probably observed the inside of the dish. 070727_ob6 shows some streaking that I can't easily explain... though it appears that there are some bad bolos that need to get flagged out. I wonder if that's systematic over 070727 observations....

[minor] Ongoing problems...

Since I have many tasks running in parallel, I need to summarize them sometimes...

1. There are many "bad" observations that haven't been placed in "bad" directories. A list of the error messages generated is here: Making Infiles wiki
   
2. There may be streaking in the BGPS ds2 images of l030 and l032. Still trying to re-run cleanly to find out
   
3. There may be unflagged high-points in l030 and l032. Also under examination
   
4. Simulations have been generating bulk-offset outputs; my suspicion was that the relative scales were being set incorrectly because astro dominated atmo, so I bumped up the atmo scale. The tests have run but I haven't examined the outputs
   
5. V1 sims have shown streaking, possibly because of the previous bullet, but certainly (in part) because cross-scans haven't worked
   

Direct comparison of column-density power spectra

I've multiplied the 13CO integrated data cube, the Herschel 500 micron*, and the BGPS v1.0 and v2.0 by the appropriate conversion factor to get the maps into units of column density assuming T=20K and some opacity for the dust maps. BGPS v1 has been multiplied by the 1.5 "correction" factor.

* The Herschel maps are arbitrarily scaled; I didn't derive an actual column conversion but just guess-and-checked once or twice until I got something pretty close.

The power spectra look pretty outstanding:

The bumps and wiggles in the 50-200" range are quite well-matched in Herschel and Bolocam. Some map edge effects are visible in the Herschel maps, resulting in multi-frequency bumps at small spatial scales. The Herschel noise floor is also quite evidently lower. Also noteworthy is that BGPS v1 (scaled up by 1.5) matches the others "pretty well" but is a worse match, in general, to the Herschel data than is the BGPS v2.

Here are some zoomed-in plots on the inverse scale:

And finally, the 13CO data is totally unrepresentative of the dust data. There is very little agreement on any scale. This may imply that the 13CO and/or dust temperature is too high, as a decreased T_D or T_ex will decrease 13CO column and increase dust column. However, it also raises a question - on what scales should dust and CO agree?

This is getting into some real science, perhaps. The shapes of the CO and dust power spectra disagree: the CO is pretty well-fit by a power law, while the dust is not. What hypotheses can explain this?

1. There is a systematic temperature difference / preference in which CO or dust is hotter on the largest scales.
   
   -Dust is probably warmer on larger scales, however CO should be less abundant / more readily dissociated on these largest, most diffuse scales. CO shouldn't exist (or at least, should be underabundant) in regions with high temperature. Maybe? This probably needs to be quantified.
   
2. There is a systematic dust opacity difference on large scales resulting in lower dust emission.
   
   -This is almost certainly true: the dust population increases in opacity with age, following OH94. Dust on the largest spatial scales should not have coagulated / collected ice, leading to a lower opacity on the largest scales
   
   -This may also be true even though CO is present: dust coagulation is less efficient than CO formation at n~10^3-10^4 (I think - again, off the cuff, but consistent with OH94)
   
3. The CO overestimates all scales, either because of incorrect bulk abundance or temperature considerations.
   
   This is problematic..... if you drop the CO values at all scales, it becomes deficient in the 50-200 arcsecond range, where the dust measurements agree quite well
   
4. There is a preferred distance in both images
   
   -It is not clear that the observed effects would occur because of this
   
   -It is also quite evident from other analyses that there IS a preferred velocity, at least, and it completely dominates all others and has the same shape as the integrated power spectrum. So a distance effect is most likely ruled out.
   

Power Spectra of 13CO and Bolocam

Examples of power-spectra in the left panel:

The BGPS power law is fitted only over the "valid" range, which is shown in black. The BGPS power spectrum agrees pretty well with the Herschel power spectrum. Both are substantially steeper than (any) of the 13CO power spectra. Unfortunately, this somewhat calls into question any comparison between the data sets - it seems most likely that 13CO and dust actually trace different mass, particularly since both Herschel and Bolocam show a "bump" centered at 6' (10^-1 image sizes) that is not evident in 13CO.

Measuring preferred angles in the Herschel data on larger scales

This is something of a repeat of yesterday's exercise, but for Herschel data on larger scales, where filamentation is expected (at different angles?) on a different scale than the Galactic plane.

First example: L030 500 microns

There is evidently a preferred direction that is correlated down to the resolution of the map, though the preference is smallest at the smallest scales. This may simply be a statement that there are more sources along the galactic plane, though, since there really isn't any particularly obvious filamentation in the image.

In L59, on the other hand, there is at most a very weak preference except at the largest scales corresponding to the Galactic Plane. This is somewhat interesting because there IS obvious filamentation in the L59 image, but it does not have a preferred direction. Unfortunately, it is not obvious whether or where filamentation shows up in fourier space if it does not have a preferred direction. There is no excess at any spatial scale that I can pick out.

Hunting for preferred directions

In our last group meeting, we discussed simulations using filamentary structures. I've been trying to determine how best to generate random/artificial filamentary structures in images. The first step in that direction is coming up with a way to measure asymmetries and therefore preferred directions within a real map. In order to do this, I've had to develop a number of new tools in agpy for azimuthally binned radial profiles and radially-averaged azimuthal profiles (radialprofile.py).

Examining real maps is necessary because a simple sinusoidal dependence of the power introduces filamentation along the same directions at ALL scales, which is not obviously (or in some cases, obviously not) the correct solution. Filaments are observed on large scales, but sometimes there can be 'kinks' in opposite directions on small scales.

So, the map I picked to examine was one with the most flagrantly obvious filamentary structure in it: the Motte DR 21 MAMBO map. It was also a choice of convenience because I already had the data on my laptop....

The preferred direction is quite obvious in this map: there is a long filament going up and down the map. Therefore, the DC component should be substantially higher in one direction than the other.

In the power-spectral-density image, it is quite clear that there is a preferred direction, though it is not obvious that the fourier transform is rotated 90 degrees from the image. My fourier intuition somewhat fails me here.... I realize that a broad, smooth profile in real space should be narrow and highly peaked in fourier space, but I don't fully understand why it effectively spreads out in the perpendicular direction, which I have confirmed that it does with a simple experiment.

I also don't know what the Shah function is, but it implies a periodic dip in the image at every 1/5th of the image, or every 50 pixels.

These are the power spectra averaged over different angles as labeled. -15 corresponds to -15 to +15, 15 corresponds to 15 to 45, etc. The highly peaked 75-105 power spectrum shows the large-scale filamentary profile. I think the difference in the DC component is actually an artifact of the azimuthal binning process: each pixel can only be assigned one angle, so the DC value isn't included in all of them... I'll need to find a workaround for that because it's quite deceptive.

The more interesting way to view the data - and perhaps to analyze maps - is to take radial averages in some range of spatial scales and plot the azimuthal dependence. There is a clear sinusoid at large scales. The legend shows "spatial frequency" in 1/pixel units. The distribution becomes more even with angle and even changes preferred direction at smaller scales (higher frequencies).

Next step is testing different approaches. I think an added, steeper-power-law component would probably be the best way to start.

Another suggestion, courtesy Bruce Elmegreen, is to attempt this sort of asymmetric power law sampling in 3 dimensions (with only 1 or 2 dimensions asymmetric) and then projecting down onto two dimensions.