1. Chungnam National University, Korea
Removing Large-scale Variations in regularly and irregulary spaced data
Jungyeon Cho
Chungnam National University, Korea

2. Large-scale variations are commonly observed
e.g.) Hour-glass morphology of B
 Makes it difficult to accurately measure fluctuations of small-scale B
A technique called Davis-Chandrasekhar-Fermi method requires this quantity
It can obtain |B|
Direction of B
Pattle + (+Cho) (2017)

3.  ssf=? x Answer: Fitting! x Small-scale fluctuations (SSF) only Q(x)
standard deviation
Q(x) x
SSF + LSV (large-scale variation) x
Q(x)
If LSV is linear, how can we get ssf ?
Answer: Fitting!

4. x x SSF + LSV Q(x) What if the LSV is quadratic?
We can get ssf from fitting! x
Q(x)
What if the LSV is complicated and unknown?
We may get ssf still from fitting!

5. The technique is simple: Multi-point Second-order Structure Function!
In this talk I present a simple technique that is complementary to fitting or other techniques like wavelet transformations.
The technique is simple:
Multi-point
Second-order Structure Function!

6. What is structure function (SF)?
Driving scale if it’s turbulence x
Let’s consider a quantity Qr=| Q(x+r)-Q(x) |.
 How does Qr change as r increases?
Structure Function: typical fluctuation as a function of separation r
<Qr>
Driving scale (ls)
 (2-point) 2nd-order SF:
SF22pt=<Qr2>

7. (Usual) Behavior of SF22pt
SF22pt=<|Q2pt|2>
Q2pt=Q(x)-Q(x+r) r
2Q2 x
x+r lc
SSF

8. Behavior of SF22pt in the presence of LSV
SSF + LSV (large-scale variation)
LSV x
Q(x)
SSF
2Q2 r lc
 It is very difficult to get SSF from SF22pt

9. Then how can we get SSF in the presence of a linear LSV?
SSF + LSV (large-scale variation) r
2Q2
SF23pt
Q(x)
x-r
x+r x
Answer: 3-point SF2 ( SF23pt)

10. Q(x-r)+Q(x+r)-2Q(x) 2
Q(x+r)
Q(x)
Q(x-r)+Q(x+r) 2
Q(x-r)
x-r x
x+r

11.  4-point SF2 can remove a quadratic LSV.
In general, r
2Q2
SF2n-pt
Plateau!
 4-point SF2 can remove a quadratic LSV.
5-point SF2 can remove a cubic LSV, and so on.

12. Numerical Tests We test the possibility using numerical simulations.
Turbulence data
LOS
+ LSV of a sinusoidal form
It can be any observable quantity
(e.g., column density, centroid velocity,...)
Note: Amplitude of the sine wave > typical turbulence fluctuation

13. Observed velocity map & SF
Cho 2018

14. If it’s confusing, what about this?
Let’s consider a simple question:
SSF+LSV
A sinusoidal large-scale variation (LSV) + =
Small-scale fluctuations (SSF)
Can we retrieve the map of small-scale fluctuations?

15. Of course, we can use a filter...
A smoothing filter
Traditional filters:
Gaussian, top-hat,...

16. Annular filter works perfectly only if LSV is linear
Double-annular filter works perfectly if LSV is cubic
We can derive these filters from the structure functions
Cho (2018)

17. Then what about irregularly-spaced data?
Starlight polarization
 Shows direction of B
(Kandori+ 2018)

18. xi
Q(xi)
Q(xj)
Q(x3rd) r lS
SF23pt
2s2 xj
x3rd
SF2=< | |2 >

19. We select 2,000 points
22

20. Summary Help us to remove large-scale fluctuations
Our multi-point structure function technique can
Help us to remove large-scale fluctuations
Help us to obtain small-scale maps

21. =0 How can we efficiently filter-out LSV? x x+r x-r
Suppose that LSV is linear =0 =
This average can filter-out a linear LSV

22. Our technique can help us to obtain small-scale maps
Our approach: multi-point average technique!
2-point average (=annular filter): rp rp

23. 4-point average (double-annular filter):