Detector Networks and Triangulation

A single interferometer measures one projection of a gravitational wave and provides only weak directional information. A network adds long baselines, differently oriented detector tensors, and independent noise realizations. Relative arrival times constrain the sky to timing rings; relative amplitudes and phases then test polarization consistency and usually select a smaller sky region. This page develops that geometry without assuming a particular observing run or event.

Required background. The Wave Equation and Characteristics supplies plane-wave phase and propagation along characteristics.

Helpful background. Gravitational Radiation and Waveforms fixes the tensor-wave and polarization conventions. Probability and Inference for Gravity supplies likelihoods, nuisance parameters, and credible regions.

A coherent network response

Let $\boldsymbol n$ point from Earth toward the source, so the wave propagates along $-\boldsymbol n$. For detector $I$ at geocentric position $\boldsymbol x_I$, define its arrival time relative to the geocentre by

$$t_I(\boldsymbol n)=-\frac{\boldsymbol n\cdot\boldsymbol x_I}{c}.$$

In the small-antenna limit, the calibrated strain in the frequency domain is

$$\widetilde d_I(f) =\sum_{A=+,\times} F_I^A(\boldsymbol n,\psi;t) \widetilde h_A(f) e^{-2\pi i f\,\boldsymbol n\cdot\boldsymbol x_I/c} +\widetilde n_I(f).$$

The sidereal time $t$ matters because Earth rotates each detector tensor relative to the sky. A change of polarization basis applies the same orthogonal spin-two rotation to the component columns $(h_+,h_\times)^T$ and $(F_I^+,F_I^\times)^T$, leaving their scalar product invariant. At frequencies where arm length is not negligible, $F_I^A$ is replaced by the complex finite-arm response. A versioned calibration response is also needed before digital detector output can be compared coherently.

It is useful to collect the two polarizations into a vector $\boldsymbol h=(h_+,h_\times)^T$ and the network data into $\boldsymbol d$. At each frequency,

$$\boldsymbol d=\mathbf F\boldsymbol h+\boldsymbol n,$$

where row $I$ of $\mathbf F$ contains the delayed antenna factors for detector $I$. The rank and conditioning of $\mathbf F$ determine how well the network can separate polarizations. Adding a detector helps most when its projected tensor is not nearly redundant with the existing rows; sensitivity alone does not determine polarization information.

The response formalism and the distinction between detector and network correlations are developed in Romano and Cornish 2017, §§5.2–5.3, PDF.

Timing rings and triangulation

For a pair of sites, let $\boldsymbol b_{IJ}=\boldsymbol x_I-\boldsymbol x_J$. Their geometric delay is

$$\Delta t_{IJ}=t_I-t_J =-\frac{\boldsymbol n\cdot\boldsymbol b_{IJ}}{c}, \qquad \lvert\Delta t_{IJ}\rvert\le\frac{\lvert\boldsymbol b_{IJ}\rvert}{c}.$$

A measured delay fixes the projection of $\boldsymbol n$ onto the baseline. On the celestial sphere this is a circle, not a point. If $\theta$ is the angle between $\boldsymbol n$ and $\boldsymbol b_{IJ}$, then

$$\cos\theta=-\frac{c\Delta t_{IJ}}{\lvert\boldsymbol b_{IJ}\rvert}.$$

Two nonparallel baselines intersect their timing circles in at most two directions in ideal noiseless geometry. Three sites always define a plane, and timing alone gives an exact reflection ambiguity across the plane spanned by their baseline vectors. The two solutions merge for a direction in that plane. Antenna amplitudes, phase evolution, or a fourth site outside the three-site plane can break the ambiguity. Real localization is a likelihood problem over sky position and source parameters, not a construction from three exact timestamps.

The timing precision of a compact signal depends on both signal-to-noise ratio and bandwidth. Define the signal-to-noise ratio and noise-weighted moments by

$$\rho^2 =4\int_0^\infty \frac{\left\lvert\widetilde h(f)\right\rvert^2}{S_n(f)}\,df, \qquad \overline{f^k} =\frac{4}{\rho^2}\int_0^\infty \frac{f^k\lvert\widetilde h(f)\rvert^2}{S_n(f)}\,df, \qquad \sigma_f^2=\overline{f^2}-\overline f^{\,2}.$$

For a known waveform shape at high signal-to-noise ratio, after allowing an unknown constant phase and amplitude, the local Fisher estimate is

$$\sigma_t\simeq\frac{1}{2\pi\rho\sigma_f}.$$

The subtraction of $\overline f^{\,2}$ is essential: an unknown phase absorbs a common phase shift, so timing is measured by the change of phase across the signal’s bandwidth. With phase known exactly, the corresponding estimate would use $\sqrt{\overline{f^2}}$ instead. Correlations with additional waveform parameters and calibration can weaken either estimate.

Thus a loud narrow-band signal can time worse than a quieter broadband signal. The associated angular thickness is roughly

$$\sigma_\theta\simeq \frac{c\,\sigma_{\Delta t}} {\lvert\boldsymbol b_{IJ}\rvert\lvert\sin\theta\rvert},$$

away from the baseline directions. This approximation diverges as $\sin\theta\to0$ because a delay changes only quadratically with angle near a baseline pole. The timing estimate and its limitations are derived in Fairhurst 2009, §3.2, Eqs. (21)–(24), PDF.

Timing is only one source of localization information

The same waveform must explain all detectors after applying their delays and responses. A coherent likelihood therefore uses four coupled clues:

Network information	What it constrains	Important degeneracy or systematic
Arrival-time differences	Projections of sky direction onto baselines	Mirror regions; clock and phase errors
Relative amplitudes	Antenna response and polarization mixture	Distance–inclination–polarization degeneracy
Relative phases	Delay, polarization, and waveform phase	Calibration phase and waveform modeling
Presence or absence in a detector	Compatibility with sensitivity and antenna nulls	Nonstationary noise; treating a weak response as a veto

A detector with low expected response may legitimately contain no visible signal. Conversely, a loud transient in only one instrument is not automatically astrophysical merely because another detector lies near an antenna null. Coherent inference compares signal and noise hypotheses using the actual sensitivity, data quality, and response of every instrument.

Coincidence and coherence should also be distinguished. A coincidence test asks whether independently identified candidates agree within broad time and parameter windows. A coherent analysis evaluates a common waveform directly in the network data. Coincidence is computationally useful, but it discards some phase and subthreshold information.

Observational inputs and checks

An event-level network calculation must record site coordinates and arm orientations, geodetic-to-celestial transformations and time standards, the calibrated strain and uncertainty for each detector, the analyzed segments and frequency band, the noise model, waveform convention, and detector participation. These inputs are event- and release-specific. Public-data interfaces and detector metadata here are frozen to GWOSC documentation available on 2026-08-18; no present detector state is inferred from this page.

Three checks catch common sign and geometry errors:

• a source on the perpendicular bisector of a baseline must give zero geometric delay;
• exchanging $I$ and $J$ must reverse $\Delta t_{IJ}$ without changing the timing ring;
• injecting one waveform into synthetic detector streams and shifting by the predicted delays must align the recovered phases across the chosen band.

For a public-data reproduction, record event identifier, data-release version, GPS interval, sample rate, calibration version, PSD estimate, waveform family, and sky-coordinate convention. Compare the recovered delay posterior with the light-travel bound and repeat after varying the PSD segment and calibration nuisance parameters. This is a reproducibility prescription, not a claim that a collaboration localization pipeline was run for this page.

Common pitfalls

Treating three detectors as three independent baselines. Three sites provide only two independent delay differences. The third pairwise delay is their algebraic combination, although all data still contribute to a noisy likelihood.

Ignoring detector orientation. Baseline geometry controls timing, while detector tensors control amplitudes and polarization response. A network map that uses only site positions is incomplete.

Mixing source and propagation directions. If $\boldsymbol n$ is changed from “toward the source” to “direction of propagation,” every delay phase changes sign. Declare the convention once and test it with a synthetic injection.

Exercises

1. Recover a timing ring. Two sites have separation $B=3000\,\mathrm{km}$, and detector $I$ receives a signal $5.0\,\mathrm{ms}$ later than detector $J$. Find $\boldsymbol n\cdot\widehat{\boldsymbol b}_{IJ}$ and check causality.

Solution

With $\Delta t_{IJ}=+5.0\,\mathrm{ms}$,

$$\boldsymbol n\cdot\widehat{\boldsymbol b}_{IJ} =-\frac{c\Delta t_{IJ}}{B} \simeq-\frac{(3.00\times10^8)(5.0\times10^{-3})}{3.00\times10^6} =-0.50.$$

The allowed directions form the circle whose normal is $\widehat{\boldsymbol b}_{IJ}$ and whose angular radius satisfies $\cos\theta=-0.50$, so $\theta=120^\circ$. The maximum delay is $B/c\simeq10\,\mathrm{ms}$; the measured delay is causal.

2. Compare timing precision. Signal A has $\rho=20$ and $\sigma_f=80\,\mathrm{Hz}$; signal B has $\rho=30$ and $\sigma_f=20\,\mathrm{Hz}$. Estimate their single-detector timing uncertainties.

Solution

The Fisher estimates are

$$\sigma_{t,A}\simeq\frac{1}{2\pi(20)(80)}=9.9\times10^{-5}\,\mathrm{s},$$

and

$$\sigma_{t,B}\simeq\frac{1}{2\pi(30)(20)}=2.7\times10^{-4}\,\mathrm{s}.$$

Signal A times about $2.7$ times more precisely despite its lower signal-to-noise ratio because its effective bandwidth is larger. Correlations with intrinsic phase, sky response, and calibration can broaden both values.

References

• Fairhurst, Stephen. “Triangulation of Gravitational Wave Sources with a Network of Detectors.” New Journal of Physics 11 (2009): 123006. DOI. Open PDF.
• Gravitational Wave Open Science Center. “API Documentation.” Version available 2026-08-18. Official documentation.
• Romano, Joseph D., and Neil J. Cornish. “Detection Methods for Stochastic Gravitational-Wave Backgrounds: A Unified Treatment.” Living Reviews in Relativity 20 (2017): 2. DOI. Open PDF.