.. _how-coupling:

Coupling functions
==================

Thresholds
----------

Measuring a coupling function requires a significant increase in amplitude spectral density in both DARM and the sensor.
Each value of a coupling function is flagged based on a sensor threshold and DARM threshold.
When the sensor ASD exceeds the sensor threshold and the DARM ASD exceeds the DARM threshold, the value is a measurement.
When only the sensor ASD exceeds its thresholds, the value is an upper limit.

Calculation
-----------

A measured value is computed as

:math:`CF(f) = \sqrt{\frac{Y_{inj}^2(f) - Y_{bkg}^2(f)}{X_{inj}^2(f) - X_{bkg}^2(f)}}`

where :math:`X_{inj}(f)` and :math:`X_{bkg}(f)` are the values of the sensor ASD during injection and background,
and :math:`Y_{inj}(f)` and :math:`Y_{bkg}(f)` are the values of the DARM ASD during injection and background

An upper limit is computed as

:math:`CF(f) = \frac{Y_{bkg}(f)}{\sqrt{X_{inj}^2(f) - X_{bkg}^2(f)}}`

Below is an example plot of a coupling function for a magnetometer at LHO.

.. image:: images/coupling.png
    :width: 800

Estimated ambient
-----------------

Multiplying coupling function by a sensor background spectrum gives an estimated ambient in DARM for that sensor.

Below is an example plot of a spectrum and estimated ambient plot for a magnetometer at LHO.

.. image:: images/spectrum.png
    :width: 800