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

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)

 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!

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!

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!

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>

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

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

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)

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

 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.

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

Observed velocity map & SF Cho 2018

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?

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

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)

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

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

We select 2,000 points 22

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

=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

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

4-point average (double-annular filter):