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

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 • One-dimensional fitting • Measuring velocity/curvature • Bonus slides ⋆ dynamic spectrum residuals ⋆ phase retrieval ⋆ deconvolution ⋆ mapping to the sky O. Wucknitz 2019 2/29

Interstellar scattering: geometric delay D s scattering screen observer pulsar θ D d D ds c τ = 1 D = D s D d 2 θ 2 D D ds O. Wucknitz 2019 3/29

Scattering field • subimages (or pixels) at θ j with complex fields V j • reference direction (direct path) θ 0 , maybe moving φ j = π D ν ( θ j − θ 0 ) 2 • geometric phase at observer c φ j = π D ν ( θ 2 or j − 2 θ j · θ 0 ) c V j e i φ j ∑ • total field j 2 � � = ∑ V j e i φ j V j V k e i( φ j − φ k ) � ∑ • total intensity � � � (dynamic spectrum) j j , k O. Wucknitz 2019 4/29

Secondary spectrum φ j = π D ν θ 2 � � • phase difference φ j − φ k j − 2 θ j · θ 0 c φ j = π D θ 2 ν − 2 π D ( θ j · ˙ θ 0 ) j • linear motion: θ 0 = ˙ θ 0 t ν t c c • 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 5/29

Wavelength instead of frequency? ν − 2 π D ( θ · ˙ φ = π D θ 2 θ 0 ) ν t c c − 2 π D ( θ · ˙ = π D θ 2 θ 0 ) t λ λ = π D θ ′ 2 λ − 2 π D ( θ ′ · ˙ θ 0 ) t θ ′ = θ λ scaling with λ : all stays on main parabola, but shifts along it O. Wucknitz 2019 6/29

Dynamic and secondary spectra B1133+16 at 1450, 432 and 327 MHz [ Stinebring et al. (2018), ApJ 870, 82 ] O. Wucknitz 2019 7/29

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 θ 2 1 − θ 2 2 ∗ Doppler θ 1 − θ 2 O. Wucknitz 2019 8/29

Mapping between delay/Doppler and image positions (1) • one-dimensional, modulo constants τ = θ 2 1 − θ 2 • delay 2 • Doppler p = θ 1 − θ 2 � τ � θ 1 = 1 • image positions p + p 2 θ 2 = 1 � τ � p − p 2 • either re-map the secondary spectrum, or directly O. Wucknitz 2019 9/29

Mapping between delay/Doppler and image positions (2) 4 20 2 10 0 0 2 10 2 20 4 =const 1 =const p =const 2 =const 10.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0 4 2 0 2 4 p 1 O. Wucknitz 2019 10/29

1-d simulations: Dynamic spectrum (no noise) 312.50 8 312.25 7 312.00 6 311.75 5 freq [MHz] 311.50 4 311.25 3 311.00 2 310.75 1 310.50 0 50 100 150 200 250 300 350 'time' O. Wucknitz 2019 11/29

1-d simulations: secondary and pos/pos spectrum secondary spectrum pos/pos spectrum 40 6 4 20 2 delay [micro-sec] theta2 [mas] 0 0 2 20 4 40 6 4000 2000 0 2000 4000 6 4 2 0 2 4 6 'Doppler' [arbitrary] theta1 [mas] O. Wucknitz 2019 12/29

Real data: B0834+06 400 20 300 15 200 10 100 5 delay [microsec] 2 [mas] 0 0 5 100 10 200 15 300 20 400 20000 15000 10000 5000 0 5000 10000 15000 20000 20 15 10 5 0 5 10 15 20 Doppler [arbitrary] 1 [mas] [ data from Walter Brisken, Dana Simard ] O. Wucknitz 2019 13/29

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 14/29

1-d fitting: noisy simulation 312.50 12.5 312.25 10.0 312.00 7.5 5.0 311.75 freq [MHz] 2.5 311.50 0.0 311.25 2.5 311.00 5.0 310.75 7.5 310.50 0 50 100 150 200 250 300 350 'time' O. Wucknitz 2019 15/29

1-d simulations: secondary and pos/pos spectrum (noisy) secondary spectrum pos/pos spectrum 40 6 4 20 2 delay [micro-sec] theta2 [mas] 0 0 2 20 4 40 6 4000 2000 0 2000 4000 6 4 2 0 2 4 6 'Doppler' [arbitrary] theta1 [mas] O. Wucknitz 2019 16/29

1-d fitting: scattering field fit true 0.2 0.1 real 0.0 0.1 0.2 15 10 5 0 5 10 15 fit 0.5 true 0.4 0.3 imag 0.2 0.1 0.0 0.1 15 10 5 0 5 10 15 scatter pos [mas] O. Wucknitz 2019 17/29

1-d fitting: recovered dynamic spectrum 312.50 8 312.25 7 312.00 6 311.75 5 freq [MHz] 311.50 4 311.25 3 311.00 2 310.75 1 310.50 0 50 100 150 200 250 300 350 'time' O. Wucknitz 2019 18/29

1-d fitting: input without noise 312.50 8 312.25 7 312.00 6 311.75 5 freq [MHz] 311.50 4 311.25 3 311.00 2 310.75 1 310.50 0 50 100 150 200 250 300 350 'time' O. Wucknitz 2019 19/29

B0834+06: velocity fit in blocks (1/4 of bands) 2.0 2.0 2.0 2.0 2.0 1.5 1.5 1.5 1.5 1.5 1.0 1.0 1.0 1.0 1.0 0.5 0.5 0.5 0.5 0.5 velocity deviation [%] 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.5 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.0 2.0 0 200 400 0 200 400 0 200 400 0 200 400 0 200 400 freq blocks freq blocks freq blocks freq blocks freq blocks O. Wucknitz 2019 20/29

B0834+06: velocity fit per time block 500 2 250 0 0.15 0.10 0.05 0.00 0.05 0.10 0.15 1000 500 2 0 0.15 0.10 0.05 0.00 0.05 0.10 0.15 500 2 250 0 0.15 0.10 0.05 0.00 0.05 0.10 0.15 1000 500 2 0 0.15 0.10 0.05 0.00 0.05 0.10 0.15 400 200 2 0 0.15 0.10 0.05 0.00 0.05 0.10 0.15 velocity deviation [%] O. Wucknitz 2019 21/29

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 22/29

Bonus pages: B0834+64 fits (with derivatives, small part of data) O. Wucknitz 2019 23/29

Observed dynamic spectrum O. Wucknitz 2019 24/29

Fitted spectrum amplitudes O. Wucknitz 2019 25/29

Fitted spectrum amplitude residuals O. Wucknitz 2019 26/29

Fitted spectrum phases O. Wucknitz 2019 27/29

Secondary spectrum of complex model, deconvolved O. Wucknitz 2019 28/29

Mapped to sky O. Wucknitz 2019 29/29