Low-Frequency Gravitational Wave Searches Using Spacecraft Doppler Tracking:

Low-Frequency Gravitational Wave Searches Using Spacecraft Doppler Tracking Cassini Radio Science GW Group*
* J.W. Armstrong, R. Ambrosini,
B. Bertotti, L. Iess, P. Tortora, H.D. Wahlquist

Low-Frequency Gravitational Wave Searches Using Spacecraft Doppler Tracking:

Low-Frequency Gravitational Wave Searches Using Spacecraft Doppler Tracking The Doppler technique
Signal processing approaches + current sensitivity
Bursts
Periodic and quasi-periodic waves
Backgrounds
Data analysis ideas (which probably won’t work for ULF observations)
Data analysis ideas (which could well work for ULF observations)

DSS25 and Cassini:

DSS25 and Cassini

Three-Pulse GW Response:

Three-Pulse GW Response

Frequency/Timing Glitch:

Frequency/Timing Glitch

Antenna Mechanical Event:

Antenna Mechanical Event

Plasma Events:

Plasma Events

Slide9:

Noises at = 1000 sec
Red: plasma at S, X, and Ka-band
Blue: (hatched) uncalibrated troposphere at Goldstone
Blue: (solid) after AMC/WVR calibration
Green: antenna mechanical noise
Asmar et al. Radio Science 40,
RS2001 doi:10.1029/2004RS003101 (2005)

Spectrum of Fractional Frequency Fluctuations:

Spectrum of Fractional Frequency Fluctuations Armstrong et al. ApJ, 599, 806 (2003)

Cartoon of Signal Phase-Space:

Cartoon of Signal Phase-Space

Doppler Tracking and Pulsar Timing:

Doppler Tracking and Pulsar Timing s/c tracking pulsar timing
Tracking mode: 2-way one-way
GW coupling: 3-pulse 2-pulse
Noise coupling: 1- and 2-pulse 1-pulse
Characteristic time: T, TWLT T
Noise sources: FTS FTS
s/c buffetting PSR stability
antenna mech station location
plasma (solar wind) plasma (ISM)
troposphere troposphere

Signal Processing for Bursts:

Signal Processing for Bursts If you know the waveform and the noise power spectrum, then matched filter
Subtlety: bogus tails of distribution of matched filter outputs caused by nonstationarity of the noise, even in absence of signal
Fix with local estimation of noise spectrum + histograms of SNR vs raw matched filter output
E.g. Iess & Armstrong in Gravitational Waves: Sources and Detectors, Ciufolini, ed., World Scientific, 1997; Armstrong (2002) http://cajagwr.caltech.edu/scripts/armstrong.ram
If you don’t know the waveform, try projecting data onto mathematical basis which has burst-like properties
“Burst-like”: localized in time; perhaps approx. localized in freq.
Wavelets (many flavors)
Empirical orthonormal functions?

Signal Processing for Bursts (cont.):

Signal Processing for Bursts (cont.) In Doppler tracking, you may not know the waveforms but you do know the signal and noise transfer functions
Use two-pulse noise transfer functions to characterize data intervals as “noise-like” (with a specific noise source)
Use three-pulse signal transfer functions to characterize data intervals as “candidate signal-like”, then follow up with detailed analysis
“Data sorting”, based only on noise and signal transfer functions, as a preprocessor for burst search
True GW burst must be internally consistent across multiple data sets (e.g., Cassini has multiple simultaneous data sets, but with different sensitivities)

All-Sky Burst Sensitivity:

All-Sky Burst Sensitivity Armstrong et al. ApJ, 599, 806 (2003)

Directional Sensitivity for Mid-Band Burst:

Directional Sensitivity for Mid-Band Burst

Signal Processing for Periodic and Quasi-Periodic Waves:

Signal Processing for Periodic and Quasi-Periodic Waves If sinusoid:
spectral analysis
E.g. Anderson et al. Nature 308, 158 (1984) Armstrong, Estabrook & Wahlquist ApJ 318, 536 (1987) Bertotti et al. A&A 296, 13 (1995)
If chirp:
dechirp with exp( i t2) followed by spectral analysis [arrow of time introduced]
E.g. Anderson et al. ApJ 408 287 (1993) Iess et al. in Gravitational Waves: Sources and Detectors, Ciufolini, ed., World Scientific, 323 (1997)

Signal Processing for Periodic and Quasi-Periodic Waves (cont.):

Signal Processing for Periodic and Quasi-Periodic Waves (cont.) If periodic non-sinusoidal signal (e.g. nonrelativistic binary):
Harmonic summing/data folding
E.g. Groth ApJ Supp. Series 29, 285 (1975)
If binary system near coalescence:
Complicated time evolution of signal
May be helpful to do suboptimum pilot analysis by resampling based on assumed time-evolution of the phase
E.g. Bertotti, Vecchio, & Iess Phys. Rev. D. 59, 082001 (1999) Vecchio, Bertotti, & Iess gr-qc/9708033 Smith Phys. Rev. D36 2901 (1987)

All-Sky Sinusoidal Sensitivity:

All-Sky Sinusoidal Sensitivity

Eccentric Nonrelativistic Binary Waveform:

Eccentric Nonrelativistic Binary Waveform • Waveforms can be complicated
• This example for Doppler tracking:
- Stellar mass object in orbit about BH at galactic center
- Cassini 2003 tracking geometry
E.g. Wahlquist GRG 19 1101 (1987) Freitag ApJ 583 L21 (2003)

Signal Processing for Stochastic Background:

Signal Processing for Stochastic Background Isotropic BG limits can be derived from smoothed power spectrum of single s/c Doppler time series, since average transfer function to the Doppler is known
E.g. Estabrook & Wahlquist GRG 6, 439 (1975) Bertotti & Carr ApJ 236, 1000 (1980) Anderson & Mashoon ApJ 408, 287 (1984) Bertotti & Iess GRG 17, 1043 (1985) Giampieri & Vecchio CQG 27, 793 (1995)
Subtlety, related to estimation error statistics, the confidence with which the noise can be independently known, and use of the observed spectrum as an upper limit to the GW spectrum
E.g. Armstrong et al. ApJ 599, 806 (2003)

Signal Processing for Stochastic Background (cont.):

Signal Processing for Stochastic Background (cont.) Using multiple spacecraft would be good, too
E.g. Estabrook & Wahlquist GRG 6, 439 (1975) Hellings Phys Rev. Lett. 43, 470 (1978) Bertotti & Carr ApJ 236, 1000 (1980) Bertotti & Iess GRG 17, 1043 (1985)
If BG not isotropic then correct, angle-dependent signal transfer function must be used

Isotropic GW Background:

Isotropic GW Background Armstrong et al. ApJ, 599, 806 (2003)

Signal Processing (good ideas which I suspect will not be useful for ULF GW processing):

Signal Processing (good ideas which I suspect will not be useful for ULF GW processing) Empirical orthonormal functions/Karhunen-Loeve expansion
Let the data themselves determine a mathematical basis for the data and hope that most of the variance projects onto a small number of basis vectors
Attractive as “template independent” search for signals
Probably useful for signal-dominated detector
In simulations with low SNR time series (unfortunately the practical s/c case) modes found were always the noise modes e.g., Helstrom Statistical Theory of Signal Detection (Pergamon: Oxford), 1968
Dixon and Klein “On the Detection of Unknown Signals” ASP Conf. Series, 129 (1993)

Signal Processing (good ideas which I suspect will not be useful for ULF GW processing):

Signal Processing (good ideas which I suspect will not be useful for ULF GW processing) Bispectral analysis
Fourier decomposition of third moment: FT[<x(t) x(t+t1) x(t+t2)>]
Measures contribution to third moment from three Fourier components having frequencies adding to zero
Attractive theoretically as diagnostic of weak nonlinearities
Third moment may be intrinsically small
Convergence is slow e.g., Hasselmann, Munk, & MacDonald “Bispectra of Ocean Waves” in Time Series Analysis (Rosenblatt, ed.), (Weiley: New York) 1963
MacDonald Rev. Geophysics 27 449 (1989)

Signal Processing (good ideas which I suspect will be useful for ULF GW processing):

Signal Processing (good ideas which I suspect will be useful for ULF GW processing) Time-Frequency Analysis
Many ways to tile frequency-time (wavelets, chirplets, Gabor transforms); each can have special merit if you think your signal projects preferentially onto a specific mathematical basis
Template independent
Useful in Doppler tracking to characterize nonstationarities in the time series
Has been used in s/c tracking to “denoise” GLL time series by rejecting higher-frequency subbands

Signal Processing (good ideas which I suspect will be useful for ULF GW processing):

Signal Processing (good ideas which I suspect will be useful for ULF GW processing) Multi-taper spectral analysis
Very attractive theoretically: objective; synthesizes spectrum from average of spectra with the time series weighted by different windows
Achieves optimum resolution consistent with very low spectral leakage
Used successfully in geophysics on short, noisy, red time series
“Automatic” way to distinguish periodic signals in presence of steep continuum
Caveat: achieved some notoriety: outsiders found “too many signals” in space physics time series thought by insiders to be noise-only e.g. Percival and Walden Spectral Analysis for Physical Applications (Cambridge Univ. Press: Cambridge), 1993

Concluding Comments:

Concluding Comments Low-frequency (i.e. ≈10-6-0.1 Hz) spacecraft observations are two-way and have well-defined transfer functions for f > 1/T2
Noise analysis for s/c Doppler tracking in many ways similar to the ULF pulsar tracking problem:
Frequency standard noise
Plasma noise (ionosphere/solar wind for s/c; +ISM for pulsars)
“spacecraft buffeting” = intrinsic pulsar stability noise
Antenna mechanical noise (station location noise)
Tropospheric noise (wet + dry)
Signal processing and sensitivity analysis (noise/signal) similar

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