Simulation and Sky Reconstruction for Tianlai Array Kai-Feng Yu - - PowerPoint PPT Presentation

Simulation and Sky Reconstruction for Tianlai Array

Kai-Feng Yu NAOC Tianlai Collaboration Meeting, Pingtang, 2018

Formalism

• m-mode (J.R.Shaw, et al. 2014, 2015, J.Zhang el al,

2016a, 2016b)

• Spherical Harmonics tools and the periodicity of the

signal

• A linear description: V ij

m =

X

`

Bij

`ma`m + nij m

v = Ba + n

Simulation Flow

Note:

• Artificial noise source:

calibration

• Receiver temperature: auto-

correlation

• Gain: gaussian, 1/f (ToDo)
• RFI: not included

Sky model

TOD(visibility) Add RFI Array Model Add basic receiver Temperature Apply Gain Add sample noise Final TOD Add Noise Source

Tianlai Cylinder Pathfinder

• 96 (31, 32, 33) dual

polarization receivers

• Irregular

configuration: to reduce grating lobes

…… Num_feed=31 40m 15m Num_feed=32 Num_feed=33 N …… 12.4m ……

Regular Spacing: (30, 30, 30), 0.4m Regular Spacing: (30, 30, 30), 0.3m

Grating lobes

Sky Map Reconstruction

• Invert to solve for SH coefficients .
• Beam Transfer Matrix is not full-rank, not invertible.
• Several methods to reconstruct sky at individual

frequency.

B

ˆ a

v = Ba + n

Moore-Penrose pseudoinverse

• Gaussian noise with

covariance:

• ˆ

a = ⇣ N − 1

2 B

⌘+ N − 1

2 v

N = ⌦ nnT ↵

Dirty Map

• ˆ

a = B∗N −1v

Tikhonov Regularization

BT B

(B∗B)−1

ˆ a = ⇣ N − 1

2 B

⌘+ N − 1

2 v

=

• B∗N −1B

−1 B∗N −1v

ˆ a =

• B∗N −1B + εI

−1 B∗N −1v

• if exists,
• not invertible

ε = 0.001 ε = 0.01 ε = 1

Tikhonov Regularization

Different epsilon value

ε = 10−5

Tikhonov Regularization

Point Spread Function(PSF)

aP SF (θ, φ) = (B∗B + εI)−1 B∗BaP S(θ, φ)

ε = 10−6 ε = 0.01 ε = 0.1 ε = 1

Tikhonov Regularization

Add corrections

a = ˆ a + ∆a − N = ˆ a + ∆ (ˆ a + ∆a) − N = ˆ a + ∆ˆ a + ∆2a − N = ˆ a + ∆ˆ a + ∆2ˆ a + ∆3ˆ a + · · · − N

ˆ a = (B∗B + εI)−1 B∗v = (B∗B + εI)−1 B∗ (Ba + n) = (B∗B + εI)−1 (B∗B + εI) a − (B∗B + εI)−1 εIa + (B∗B + εI)−1 B∗n

∆=ε(B∗B+εI)−1, N=(B∗B+εI)−1B∗n

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = a − ∆a + N

1 2 3 4 10

ε = 0.01

2 4 1 3 10

ε = 0.1

Summary

• Simulation Tools, Map-Making
• To-Do: Power Spectrum, Machine Learning? …

Thank you!