Calibration Response and Uncertainty

Calibration reconstructs dimensionless differential-arm strain from digital control-system signals. Because feedback suppresses the residual motion that the photodetector sees, calibration must restore both sensed residual motion and commanded test-mass motion. The resulting response is complex, frequency dependent, and slowly time dependent; its uncertainty changes signal amplitude and phase and must accompany the strain data into inference.

Required background. Probability and Inference for Gravity supplies uncertainty propagation and nuisance-parameter marginalization.

Helpful background. Gravitational Radiation and Waveforms fixes strain and waveform phase conventions. Computational Relativity supplies interpolation, convergence, and reproducible software checks.

Reconstructing free differential motion

Use a negative-feedback convention in which the uncontrolled or free differential arm motion is the sum of residual and controlled motion:

$$\Delta L_{\rm free}(f) =\Delta L_{\rm res}(f)+\Delta L_{\rm ctrl}(f).$$

Let the sensing function $C(f;t)$ map residual displacement to the digital error signal,

$$d_{\rm err}=C\,\Delta L_{\rm res},$$

let the digital filters $D(f;t)$ produce a control signal $d_{\rm ctrl}=D d_{\rm err}$, and let the actuation function $A(f;t)$ map that signal to controlled displacement,

$$\Delta L_{\rm ctrl}=A D d_{\rm err}.$$

The open-loop gain is $G=ADC$. Substitution gives

$$\Delta L_{\rm free} =\left(\frac1C+AD\right)d_{\rm err} =\frac{1+G}{C}d_{\rm err}.$$

For arm-length normalization $L$, the strain response is therefore

$$\widetilde h(f;t) =R(f;t)\widetilde d_{\rm err}(f;t), \qquad R(f;t)=\frac{1+G(f;t)}{L\,C(f;t)}.$$

Equivalent calibration implementations may combine an error stream and a control stream directly, distribute filters between real-time and offline stages, or use different sign conventions. They must reduce to the same free-motion identity and declare latency, units, channel names, and validity interval. The feedback and calibration architecture is described in LIGO Scientific Collaboration and Virgo Collaboration 2020, §6.1, PDF.

Dimensional analysis is a strong check. If $d_{\rm err}$ is in counts and $C$ in counts per metre, then $(1+G)/C$ is metres per count and $R$ is strain per count. A calibration filter with unexplained dimensions should not be trusted even when its numerical curve looks plausible.

Measuring and tracking the response

The functions $C$ and $A$ are models constrained by transfer measurements. A known displacement or force is injected, the corresponding digital channels are measured, and model parameters are fitted. In photon calibration, modulated radiation pressure provides a traceable test-mass displacement after accounting for laser power, incidence geometry, elastic deformation, timing, and actuation location. Calibration lines injected continuously can track selected time-dependent parameters between dedicated measurements.

A defensible calibration epoch records:

• the reference sensing, digital-filter, and actuation models;
• fitted pole, zero, gain, delay, and time-dependent correction parameters;
• the physical standard and injection geometry used to anchor displacement;
• the interval of validity and the rules for gaps or state transitions;
• amplitude and phase residuals relative to independent measurements;
• the algorithm and software version used to construct released strain.

The systematic-error decomposition and photon-calibrator checks used for early Advanced LIGO data are developed in Cahillane et al. 2017, §§II–VI. Those numerical results belong to the stated observing epochs; the measurement logic remains useful more generally.

Complex uncertainty and its physical effects

Let $R_0$ be the response used to reconstruct the released nominal strain and $R_{\rm true}$ the unknown true response. Define the nominal-to-true calibration factor as

$$\mathcal C(f;t) =\frac{R_0(f;t)}{R_{\rm true}(f;t)} =\bigl[1+\delta a(f;t)\bigr] e^{i\delta\phi(f;t)}.$$

For small errors,

$$\frac{R_0-R_{\rm true}}{R_{\rm true}} \simeq\delta a+i\delta\phi.$$

Amplitude error rescales the inferred signal and is therefore correlated with luminosity distance and any amplitude-dependent population quantity. Phase error alters waveform phasing and relative detector timing. A phase error that is locally linear in frequency,

$$\delta\phi(f)=+2\pi f\,\delta t,$$

is indistinguishable from an actual delay $h(t-\delta t)$ in one detector under the volume’s transform $\widetilde h(f)=\int h(t)e^{+2\pi i f t}\,dt$, until other calibration information is supplied. Frequency-dependent structure can instead mimic waveform phase corrections or polarization differences.

Calibration uncertainty is generally correlated across frequency because a small number of physical parameters or smooth residual functions generate the curve. Replacing it by independent error bars in every Fourier bin invents far too many degrees of freedom. Common representations include posterior samples of response curves, spline or Gaussian-process residuals with a covariance model, and physical calibration parameters with priors. The likelihood should use the representation released for the dataset rather than reverse-engineer one from a plotted envelope.

The dimensionless factor $\mathcal C$ is distinct from the sensing function $C$, which has units of counts per metre. If the true calibrated stream is $d_{\rm true}=h(\theta)+n_{\rm true}$, then the released nominal stream is $d_0=R_0d_{\rm err}=\mathcal C d_{\rm true}$. Both signal and noise are transformed. When the PSD is estimated directly from $d_0$, it is already expressed in nominal strain coordinates, and the model waveform in those coordinates is

$$\widetilde h_{\rm cal}(f;\theta,\lambda) =\mathcal C(f;\lambda)\widetilde h(f;\theta) =\bigl[1+\delta a(f;\lambda)\bigr] e^{i\delta\phi(f;\lambda)} \widetilde h(f;\theta),$$

and the likelihood marginalizes over calibration parameters $\lambda$. A source that instead defines $\mathcal C_{\rm alt}=R_{\rm true}/R_0$ must apply $\mathcal C_{\rm alt}^{-1}$ to the waveform. Applying a reciprocal factor to the data is equivalent only if the convention, Jacobian, and PSD treatment are transformed consistently.

Interpolation is part of the calibration model

Response data are evaluated on finite frequency and time grids. Direct interpolation of wrapped phase can send a smooth change from $179^\circ$ to $181^\circ$ through $0^\circ$ if the second angle is stored as $-179^\circ$. Real and imaginary parts have no phase-wrap discontinuity, but a coarse Cartesian interpolation can cut across a rapidly rotating complex response and underestimate its magnitude. For a smooth nonzero response, interpolation of $\log\lvert R\rvert$ and unwrapped phase is often useful; it still needs a sufficiently fine grid and explicit treatment of true zeros, poles, discontinuities, and epoch boundaries.

Interpolation must be tested where the response changes fastest. If $R_{\rm fine}$ is a high-resolution reference and $R_{\rm interp}$ the reconstructed curve, inspect

$$\epsilon_R(f) =\frac{R_{\rm interp}(f)}{R_{\rm fine}(f)}-1$$

in amplitude and phase, not only in complex absolute error. Refine the frequency grid until the maximum error is below the analysis-specific tolerance. Repeat the test for time interpolation and for representative calibration posterior samples.

Public-data boundary and computational checks

GWOSC strain products are already calibrated; their release metadata and caveats govern how they may be used. Public strain alone does not contain enough information to reconstruct the full internal sensing and actuation model. For O4 products, use the GWOSC technical details available on 2026-08-18 and pin dataset, minor version, detector, GPS interval, sample rate, quality selection, file hash, and stated calibration uncertainty.

A reproducible calibration test can be performed without claiming access to collaboration controls:

1. construct a synthetic $C$, $A$, and $D$ containing known gains, a cavity pole, an actuation pole, and a delay;
2. generate free motion, close the loop algebraically, and reconstruct it with $R=(1+ADC)/(LC)$;
3. refine sample rate and frequency grid, vary segment length and edge padding, and test interpolation of amplitude and unwrapped phase;
4. draw calibration-error curves from a declared smooth covariance, inject them into a chirp, and recover the expected amplitude and timing shifts;
5. on a declared public strain interval, propagate only the calibration uncertainty actually supplied with that release and report unavailable inputs explicitly.

The synthetic round trip should recover the injected free strain to floating-point and discretization tolerance. Failures near edges, phase wraps, response zeros, or a validity boundary must be reported rather than hidden by trimming after inspection. No collaboration calibration pipeline is asserted to have been executed for this page.

Common pitfalls

Calibrating only the residual error signal. At high loop gain, feedback suppresses residual motion; omitting controlled motion can remove most of the physical signal.

Treating uncertainty as one scalar percentage. Amplitude and phase depend on frequency and time and are correlated. Their impact depends on the signal and detector network.

Interpolating across an epoch boundary. A model change or invalid interval is not a smooth drift. Select the correct response on each side and exclude or specially treat the boundary.

Exercises

1. Recover the response. At one frequency, let $C=2.0\times10^{12}\,\mathrm{count\,m^{-1}}$, $AD=3.0\times10^{-13}\,\mathrm{m\,count^{-1}}$, and $L=4.0\times10^3\,\mathrm{m}$. Find $G$ and $R$.

Solution

The dimensionless open-loop gain is

$$G=ADC=(AD)C=(3.0\times10^{-13})(2.0\times10^{12})=0.60.$$

Therefore

$$R=\frac{1+G}{LC} =\frac{1.60}{(4.0\times10^3)(2.0\times10^{12})} =2.0\times10^{-16}\,\mathrm{count^{-1}}.$$

The same result follows from $R=L^{-1}(C^{-1}+AD)$.

2. Convert phase error to timing. A calibration phase residual is $3^\circ$ at $100\,\mathrm{Hz}$ and is interpreted locally as a pure delay. What is the corresponding timing magnitude?

Solution

Convert $3^\circ$ to radians: $\lvert\delta\phi\rvert=\pi/60$. Then

$$\lvert\delta t\rvert =\frac{\lvert\delta\phi\rvert}{2\pi f} =\frac{1}{120(100\,\mathrm{Hz})} =8.3\times10^{-5}\,\mathrm{s},$$

or $83\,\mu\mathrm{s}$. One frequency cannot establish that the error is truly a delay; a delay requires phase linear in $f$ over the relevant band.

References

• Cahillane, Craig, et al. “Calibration Uncertainty for Advanced LIGO’s First and Second Observing Runs.” Physical Review D 96 (2017): 102001. DOI. Open PDF.
• Gravitational Wave Open Science Center. “O4 Technical Details.” Version available 2026-08-18. Official documentation.
• LIGO Scientific Collaboration and Virgo Collaboration. “A Guide to LIGO–Virgo Detector Noise and Extraction of Transient Gravitational-Wave Signals.” Classical and Quantum Gravity 37 (2020): 055002. DOI. Open PDF.