Deal with referee report

[will-henney]

We now have the referee report on the paper, which had only one comment that needs to be addressed:

will be happy to recommend the manuscript for publication after some clarification of some details:

• To study the Finite box effects, as well as that of seeing the authors do an interesting exercise with idealized data (as fractional Brownian motion -fBm- 2D data modified to include a correlation taper). This approach is very clear, but there is one issue that requires attention, and that is that of the projection from a 3D field to a 2D field. In reality the density/velocity fields are three-dimensional but some information is lost in the projection to a 2D map.

For the exercise presented in appendix A, the authors use a 2D map that was modified to include the proper spectral index (n_2D = m +2) and a correlation length r_0, in reality one would have a 3D map with a spectral index (n_3D = m +3), which if integrated along the line of sight would change the spectral index from n_3D to n_2D (and within the transition some inertial range could be 'lost') .

In other words in the current version of the paper the r_0 is entirely in the plane of the sky, while in reality should correspond to the 3D field, and at the same time the transition from a 3D to a 2D map could curve more the structure functions. I believe that it would be beneficial to obtain the 2D map from collapsing a 3D fBm (with an index m+3) and the same exponential taper used and verify that the main results hold, or discuss this an additional limitation.

[will-henney]

In order to deal with this, we need to generate a fake 3d velocity cube:

• [x] Modify make_3dfield() to have ellipticity and also the tapered power law
• [x] Use make_3dfield() to generate a 3d velocity field
• [x] Use this velocity field, together with a uniform density field to generate a PPV cube using make_ppv()
• [x] Apply the seeing to each velocity channel of the PPV cube
• [x] Calculate the mean velocity map from the blurred PPV cube

Then we should look at the velocity maps to make sure they are OK. Up to here can be in one notebook. We should save the velocity maps for each value of the seeing.

From then on, the calculation of the structure function is the same as we had previously.

[will-henney]

Javier has finished an initial exploration of this in Fake-Maps/fake-3d-tapered-maps.ipynb

The results look promising. I just have a few queries:

• [x] The smoothed channel maps have some problem at the edges, for instance: But the smoothed moment1 maps do not seem to have this problem, so I am not sure what is going on.
• [x] We need to check that the structure function slope is equal to m in the unsmoothed case.
• [x] It looks like the effect of seeing on the structure function is a bit less than what we found previously. It will be interesting to see if this is still true when we do the non-periodic maps

[will-henney]

Then the following steps will be:

• [x] Repeat for the non-periodic case (512 cube, divided into 4). I guess there is no problem with it still being periodic in the z axis
• [x] We should also try the case of having a fluctuating emissivity (equivalent to density) field as well as velocity field. I would suggest doing this by generating a log E field in exactly the same way as the velocity field, and then taking the exponential of it to get the emissivity field. The fractional std dev of the emissivity should be 2-3 times larger than the fractional std dev of the surface brightness (see Fig 15), so about 2. When we do this, we need to check carefully what the slope of the structure function is, since the projection smoothing will be less according to the arguments that we give in the discussion.

[will-henney]

Comment on the edge effects: we should make sure to use boundary="fill" in convolve_fft()

[will-henney]

Comment on the power law slopes: I think we should maybe be using 2 + m instead of 3 + m for the slope of the power spectrum

But @JavGVastro should check this in the literature

[JavGVastro]

Emissivity fluctuations case

• In this case the index used for the 3D velocity and density fields were k = 3 + 0.55 to recover a slope of m_2D = 1.
• The fluctuations in emissivity were obtained as E = e^(sig_E*density) where sig_E = 2 and sig_dens = 1.
• The Figs. below are the 0th moment map and 1st moment map, and the fit to the respective B(r) (no seeing/periodic case).
• On the bottom there is an image of the seeing/non-periodic analysis.
• In this case the seeing model adjust better to the reduction in B(r) than for the constant density case.
• For this analysis the seeing model parameter a = 0.75.

Finite maps results

• For the 3D maps with emissivity fluctuations the results using finite maps are similar to the 2D maps in the empirical case with a valid r_0 as long as L / r_0 > 10.
• Using the fit there is a difference with previous results (2D and constant density) having something like L / r_0 > 5 for a condition to recover r_0.

[JavGVastro]

Here is a summary of the case that is going to be used to update the paper. Also, there are the images files locations so everything can be easily found, tracked and reviewed.

Emissivity fluctuations case: $\sigma_E = 1$

• In this case the index used for the 3D velocity and density ($\rho$) fields was $n = 3 + 0.3$ ( to recover a slope of $m_{2D} = 1$, see below) and a $r_0 = 32$ pix.

• The fluctuations in emissivity were obtained as $E = \exp ( \sigma_E * \rho )$ where $\sigmaE = 1$ and $\sigma{\rho} = 1$ (for the paper revision).

• The Figs. below are the 0th moment map and 1st moment map (tapered case).

Image file: _PhD.Paper/Fake-Maps/fake-3d-maps/sf-fake-3d-maps-emissivity-fluct_m1_sigE1.ipynb

-Comparison of the Initial structure functions of the 1st moment map.

Image file: _PhD.Paper/Fake-Maps/fake-3d-maps/sf-fake-3d-maps-emissivity-fluct_m1_sigE1.ipynb

• Below is the fit to the respective $B(r)$ (non-seeing/periodic case) and confidence interval analysis to confirm that the spectrum from the previous point recovers a slope of $m_{2D} = 1$.

Image file: _PhD.Paper/Fake-Maps/fake-3d-maps/ci-fake-3d-maps-emissivity-fluct_m1_sigE1.ipynb

• Finite box analysis.

Conclusion: The behaviour is shared as in the 2D case in the reduction of $\sigma$ and $r_0$ following the finite box analysis where the condition to recover $r_0$ is $(L > 10 r_0)$. The main difference is that the condition $(L > 3 r_0)$ to recover $r_0$ changes to $(L > 5 r_0)$ when applyting a fit to the results. The dashes line still is described by $1 - e^{L / x r_0}$ but with $x$ equal to $4.0$ instead of $3.6$ (2D maps).

Images File: _PhD.Paper/Fake-Maps/fake-3d-maps/fake-3d-maps-finite-ems-fluct-SigE1.ipynb

• Effects of seeing.

1. Create maps with their respective seeing:

Image file: _PhD.Paper/Fake-Maps/fake-3d-maps/create-fake-3d-maps-seeing-nonp-ems-fluct-sigE1.ipynb

2. Compute structure functions.

File: _PhD.Paper/Fake-Maps/fake-3d-maps/sfs-fake-3d-maps-seeing-nonp-ems-fluc-sigE1.ipynb

3. Analysis on the reduction of $B(r)$ with respect differents s0.

The model changed from:

$S(r; s_0, r_0) = \frac{e^{-s_0 / r_0}}{1+(2s_0 / r)^{2a}}$ to

$S(r; s_0, r_0) = \left [ \left (1+\frac{1.25 s_0}{r_0} \right) \left(1+ \left (\frac{2.6 s_0}{r} \right) ^{2a} \right) \right ]^{-1}$

with $a = 0.75$

Image file: _PhD.Paper/Fake-Maps/fake-3d-maps/fake-3d-maps-structure-function-analysis-ems-fluc_m4-sig_E1.ipynb

Note: The analysis of the new model is here: _PhD.Paper/Fake-Maps/fake-3d-maps/ratio_seeingnewmodel.ipynb

[JavGVastro]

$S(r; s_0, r_0) = \left [ \left (1+a_1\frac{ s_0}{r_0} \right) \left(1+ \left (a_2 \frac{s_0}{r} \right) ^{a_3} \right) \right ]^{-1}$

• $\sigma_E = 0$ $a_1 = 1.35$, $a_2 =2.6 $, $a_3 = 1.5 $

• $\sigma_E = 1$ $a_1 = 1.25$, $a_2 = 2.6$, $a_3 = 1.5$

• $\sigma_E = 2$ $a_1 = 1.00$, $a_2 = 2.44$, $a_3 =1.7$

Fake-Maps/fake-3d-maps/ratio_seeing_newmodel-global.ipynb