Mapping and Fitting 1D Scattering Screens Olaf Wucknitz - - PowerPoint PPT Presentation

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Mapping and Fitting 1D Scattering Screens Olaf Wucknitz - - PowerPoint PPT Presentation

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Mapping and Fitting 1D Scattering Screens Olaf Wucknitz wucknitz@mpifr-bonn.mpg.de Scintillometry 2019 Bonn, 5 November 2019 Mapping and Fitting 1D Scattering Screens Screens often one-dimensional Map to position-position space

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SLIDE 1

Mapping and Fitting 1D Scattering Screens

Olaf Wucknitz

wucknitz@mpifr-bonn.mpg.de

Scintillometry 2019 Bonn, 5 November 2019

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SLIDE 2

Mapping and Fitting 1D Scattering Screens

  • Screens often one-dimensional
  • Map to position-position space
  • One-dimensional fitting
  • Measuring velocity/curvature
  • Bonus slides

⋆ dynamic spectrum residuals ⋆ phase retrieval ⋆ deconvolution ⋆ mapping to the sky

  • O. Wucknitz 2019

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SLIDE 3

Interstellar scattering: geometric delay

Ds D

ds

D

d

  • bserver

θ pulsar scattering screen

cτ = 1 2θ2D D = DsDd Dds

  • O. Wucknitz 2019

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SLIDE 4

Scattering field

  • subimages (or pixels) at θj with complex fields Vj
  • reference direction (direct path) θ0, maybe moving
  • geometric phase at observer

φj = πDν c (θj −θ0)2

  • r

φj = πDν c (θ2

j −2θj ·θ0)

  • total field

∑

j

Vj eiφj

  • total intensity
  • ∑

j

Vj eiφj

  • 2

= ∑

j,k

VjV k ei(φj−φk) (dynamic spectrum)

  • O. Wucknitz 2019

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SLIDE 5

Secondary spectrum

  • phase difference φj −φk

φj = πDν c

  • θ2

j − 2θj ·θ0

  • linear motion: θ0 = ˙

θ0t φj = πD θ2

j

c ν − 2πD (θj · ˙ θ0) c ν t

  • dynamic spectrum is function of t,ν
  • two-dimensional Fourier transform delay, Doppler
  • secondary spectrum
  • for wide band: use axes (ν,ν t) instead of (ν,t)
  • either transform and rebin, or use DFT in t
  • O. Wucknitz 2019

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SLIDE 6

Wavelength instead of frequency?

φ = πD θ2 c ν − 2πD (θ· ˙ θ0) c ν t = πD θ2 λ − 2πD (θ· ˙ θ0) λ t = πD θ′2 λ − 2πD (θ′ · ˙ θ0) t θ′ = θ λ scaling with λ: all stays on main parabola, but shifts along it

  • O. Wucknitz 2019

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SLIDE 7

Dynamic and secondary spectra

B1133+16 at 1450, 432 and 327 MHz [ Stinebring et al. (2018), ApJ 870, 82 ]

  • O. Wucknitz 2019

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SLIDE 8

One-dimensional scattering

  • secondary spectrum is autocorrelation of field FT
  • inverting is difficult, not generally unique
  • equivalent to phase retrieval of dynamic spectrum
  • well-constrained problem if one-dimensional

⋆ 2d constraints, 1d unknowns ⋆ axes of secondary spectrum: ∗ delay θ2

1 −θ2 2

∗ Doppler θ1 −θ2

  • O. Wucknitz 2019

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SLIDE 9

Mapping between delay/Doppler and image positions (1)

  • one-dimensional, modulo constants
  • delay

τ = θ2

1 −θ2 2

  • Doppler

p = θ1 −θ2

  • image positions

θ1 = 1 2 τ p +p

  • θ2 = 1

2 τ p −p

  • either re-map the secondary spectrum, or directly
  • O. Wucknitz 2019

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SLIDE 10

Mapping between delay/Doppler and image positions (2)

10.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0 p 20 10 10 20

1=const 2=const

4 2 2 4

1

4 2 2 4

2

=const p=const

  • O. Wucknitz 2019

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SLIDE 11

1-d simulations: Dynamic spectrum (no noise)

50 100 150 200 250 300 350 'time' 310.50 310.75 311.00 311.25 311.50 311.75 312.00 312.25 312.50 freq [MHz] 1 2 3 4 5 6 7 8

  • O. Wucknitz 2019

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SLIDE 12

1-d simulations: secondary and pos/pos spectrum

4000 2000 2000 4000 'Doppler' [arbitrary] 40 20 20 40 delay [micro-sec]

secondary spectrum

6 4 2 2 4 6 theta1 [mas] 6 4 2 2 4 6 theta2 [mas]

pos/pos spectrum

  • O. Wucknitz 2019

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SLIDE 13

Real data: B0834+06

20000 15000 10000 5000 5000 10000 15000 20000 Doppler [arbitrary] 400 300 200 100 100 200 300 400 delay [microsec] 20 15 10 5 5 10 15 20

1 [mas]

20 15 10 5 5 10 15 20

2 [mas]

[ data from Walter Brisken, Dana Simard ]

  • O. Wucknitz 2019

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SLIDE 14

Fitting

  • velocity/curvature from shear
  • could do eigenvector decomposition of θ-θ spectrum
  • caveat: noise properties, distortion
  • direct fit to dynamic spectrum!
  • model is 1-d complex field V (θ),

maybe derivatives

  • iterative fit of all parameters
  • outer loop for velocity/curvature (or orbit)
  • coherent fit over duration and band!
  • O. Wucknitz 2019

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SLIDE 15

1-d fitting: noisy simulation

50 100 150 200 250 300 350 'time' 310.50 310.75 311.00 311.25 311.50 311.75 312.00 312.25 312.50 freq [MHz] 7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5

  • O. Wucknitz 2019

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SLIDE 16

1-d simulations: secondary and pos/pos spectrum (noisy)

4000 2000 2000 4000 'Doppler' [arbitrary] 40 20 20 40 delay [micro-sec]

secondary spectrum

6 4 2 2 4 6 theta1 [mas] 6 4 2 2 4 6 theta2 [mas]

pos/pos spectrum

  • O. Wucknitz 2019

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SLIDE 17

1-d fitting: scattering field

15 10 5 5 10 15 0.2 0.1 0.0 0.1 0.2 real fit true 15 10 5 5 10 15 scatter pos [mas] 0.1 0.0 0.1 0.2 0.3 0.4 0.5 imag fit true

  • O. Wucknitz 2019

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SLIDE 18

1-d fitting: recovered dynamic spectrum

50 100 150 200 250 300 350 'time' 310.50 310.75 311.00 311.25 311.50 311.75 312.00 312.25 312.50 freq [MHz] 1 2 3 4 5 6 7 8

  • O. Wucknitz 2019

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SLIDE 19

1-d fitting: input without noise

50 100 150 200 250 300 350 'time' 310.50 310.75 311.00 311.25 311.50 311.75 312.00 312.25 312.50 freq [MHz] 1 2 3 4 5 6 7 8

  • O. Wucknitz 2019

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SLIDE 20

B0834+06: velocity fit in blocks (1/4 of bands)

200 400 freq blocks 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 velocity deviation [%] 200 400 freq blocks 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 200 400 freq blocks 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 200 400 freq blocks 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 200 400 freq blocks 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

  • O. Wucknitz 2019

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SLIDE 21

B0834+06: velocity fit per time block

0.15 0.10 0.05 0.00 0.05 0.10 0.15 250 500

2

0.15 0.10 0.05 0.00 0.05 0.10 0.15 500 1000

2

0.15 0.10 0.05 0.00 0.05 0.10 0.15 250 500

2

0.15 0.10 0.05 0.00 0.05 0.10 0.15 500 1000

2

0.15 0.10 0.05 0.00 0.05 0.10 0.15 velocity deviation [%] 200 400

2

  • O. Wucknitz 2019

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SLIDE 22

Summary

  • direct coherent fit should be optimal, tried 1-d
  • computationally expensive, subset(s) of data
  • may need derivatives wrt. time and freq
  • good curvature/velocity precision (formally 0.02 %)
  • need more efficiency, e.g. FFT
  • can include bandpass, intrinsic variability
  • can include other telescopes and baselines
  • very promising for orbits, Earth’s orbit
  • should go towards two-dimensional
  • what happens within a pixel?
  • O. Wucknitz 2019

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SLIDE 23

Bonus pages: B0834+64 fits (with derivatives, small part of data)

  • O. Wucknitz 2019

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SLIDE 24

Observed dynamic spectrum

  • O. Wucknitz 2019

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SLIDE 25

Fitted spectrum amplitudes

  • O. Wucknitz 2019

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SLIDE 26

Fitted spectrum amplitude residuals

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SLIDE 27

Fitted spectrum phases

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SLIDE 28

Secondary spectrum of complex model, deconvolved

  • O. Wucknitz 2019

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SLIDE 29

Mapped to sky

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