.. _output:

Solution Quantities and Diagnostics
-----------------------------------

After obtaining the refined fringe solution, the algorithm computes derived
quantities: the signal-to-noise ratio (SNR), residual phase, formal error estimates,
probability of false detection, and other quality indicators. These are given as follows.

Signal-to-Noise Ratio
~~~~~~~~~~~~~~~~~~~~~
The signal-to-noise ratio (for a broadband continuum source) is given by:

.. math::

   \mathrm{SNR} = A_{\mathrm{fringe}} \; f_2 \, f_e \, f_w \; \kappa_{\mathrm{BW}}
   \sqrt{N_{\mathrm{pol}} \cdot W_{\mathrm{tot}} \cdot \frac{\Delta t}{T_{\mathrm{samp}}}} \;

where :math:`N_{\mathrm{pol}}` is the effective number of polarizations (1 or 2),
:math:`W_{\mathrm{tot}}` is the total summed weights (proportional to used bandwidth) across all channels and APs,
:math:`\Delta t` is the integration time per AP, and :math:`T_{\mathrm{samp}}` is
the sample period. The remaining factors are hard-coded correction constants, :math:`f_2 = 0.881` (2-bit sampling [#f1]_),
:math:`f_e = 0.970` (DiFX/Mark4 empirical normalization factor), and :math:`f_w = 10^{-4}`
(amplitude unit conversion from "Whitneys"), and :math:`\kappa_{\mathrm{BW}}` is a bandwidth
correction factor accounting for notched or passband-reduced channels:

.. [#f1] This hard-coded factor needs to be modified for other bit-depth sampling schemes.

.. math::

   \kappa_{\mathrm{BW}} = \frac{1}{\sqrt{n_{\mathrm{summed\_polprod}}}} \;
   \sqrt{\frac{\sum\limits_{c,a} f_{c,a}}{\sum\limits_{c,a} 1}}

where :math:`f_{c,a}` is the used bandwidth fraction for channel *c* at AP *a*,
and :math:`n_{\mathrm{summed\_polprod}}` is the number of summed polarization products
from the weight object.

Residual Phase
~~~~~~~~~~~~~~

The residual phase is the argument of the coherently summed (weighted), counter-rotated
visibility at the peak SBD lag. Note that this is done over the visibility array after
the polarization product has been selected or a summation over pol-products has been performed :math:`(p'=0)`, thus:

.. math::

   Z_{\mathrm{res}} = \sum\limits_{c,a} W[p',c,a,0] \;
   \mathcal{S}'[p',c,a,\ell^*] \;
   \Psi(c, a, \tau_{\mathrm{MBD}}, \dot{\tau}_{\mathrm{DR}})

.. math:: \phi_{\mathrm{res}} = \arg(Z_{\mathrm{res}}) \quad \text{(radians)}

The residual phase in degrees is :math:`\phi_{\mathrm{res}}^{\circ} =
\mathrm{fmod}(\phi_{\mathrm{res}} \cdot 180/\pi,\; 360)`.

The term :math:`\Psi(c, a, \tau_{\mathrm{MBD}}, \dot{\tau}_{\mathrm{DR}})` is the fringe-rotation
correction factor, implemented by ``MHO_FringeRotation::vrot``.
It is a complex phasor that de-rotates each channel/AP sample by the phase accumulated due
to the fitted multi-band delay and delay rate, and is given by:

.. math::

   \Psi(c, a, \tau_{\mathrm{mbd}}, \dot{\tau}_{\mathrm{dr}})
   = \exp\!\Bigl(-2\pi i \bigl[
       \nu_c \, \dot{\tau}_{\mathrm{dr}} \, \Delta t_a
       + \tau_{\mathrm{mbd}} \, (\nu_c - \nu_{\mathrm{ref}})
       + \delta_{\mathrm{SB}}(\tau_{\mathrm{mbd}}, c)
   \bigr]\Bigr)

where:

- :math:`\nu_c` is the sky frequency of channel *c* (MHz)
- :math:`\dot{\tau}_{\mathrm{dr}}` is the fitted delay rate (:math:`\mu\mathrm{s}/\mathrm{s}`)
- :math:`\Delta t_a = t_a + \Delta t/2 - t_{\mathrm{FRT}}` is the time offset
  of AP *a* from the fourfit reference time (FRT), in seconds
- :math:`\tau_{\mathrm{mbd}}` is the fitted multi-band delay (us)
- :math:`\nu_{\mathrm{ref}}` is the reference frequency (MHz)
- :math:`\delta_{\mathrm{SB}}` is a sideband-dependent correction term that accounts
  for the zero-padding offset in the SBD domain (see ``Math/src/MHO_FringeRotation.cc``)

Integration Time
~~~~~~~~~~~~~~~~

The integration time is a measure of what portion of the scan's data was actually used
by the correlator and the fringe-fitter, and is given by:

.. math::

   t_{\mathrm{int}} = \frac{W_{\mathrm{tot}} \cdot \Delta t}
   {n_{\mathrm{summed\_polprod}} \cdot N_c}

Total Delays
~~~~~~~~~~~~

The a priori delay :math:`\tau_0` and a priori rate :math:`\dot{\tau}_0` are computed
from the geometric delay model (station coordinates, source position,
Earth orientation parameters). Total delays are:

.. math::

   \tau_{\mathrm{total}}^{\mathrm{SBD}} &= \tau_0 + \tau_{\mathrm{SBD}} \\
   \tau_{\mathrm{total}}^{\mathrm{MBD}} &= \tau_0 + \tau_{\mathrm{MBD}}
       + \Delta_{\mathrm{ambig}} \\
   \dot{\tau}_{\mathrm{total}} &= \dot{\tau}_0 + \dot{\tau}_{\mathrm{DR}}

where :math:`\Delta_{\mathrm{ambig}}` is an ambiguity correction applied when
``mbd_anchor = "sbd"``:

.. math::

   \Delta_{\mathrm{ambig}} = \tau_{\mathrm{amb}} \cdot
   \mathrm{round}\!\left(\frac{\tau_{\mathrm{total}}^{\mathrm{SBD}}
   - \tilde{\tau}_{\mathrm{total}}^{\mathrm{MBD}}}{\tau_{\mathrm{amb}}}\right)

where :math:`\tilde{\tau}_{\mathrm{total}}^{\mathrm{MBD}} = \tau_0 + \tau_{\mathrm{MBD}}`
is the uncorrected total multi-band delay, and :math:`\tau_{\mathrm{amb}} = 1/\delta_\nu`
is the MBD delay ambiguity (in :math:`\mu\mathrm{s}`), which is the inverse of the *frequency*
grid spacing :math:`\delta_\nu` (MHz) used in the transform from channel to MBD space.

Error Estimates
~~~~~~~~~~~~~~~

The formal error estimates are:

.. math::

   \sigma_{\mathrm{MBD}} &= \frac{1}{2\pi \, \Delta\nu_{\mathrm{spread}} \cdot \mathrm{SNR}} \\[2ex]
   \sigma_{\mathrm{SBD}} &= \frac{4\sqrt{12} \, \delta_{\mathrm{SBD}}}
   {2\pi \, \mathrm{SNR} \, (2 - |\overline{S}|)} \\[2ex]
   \sigma_{\mathrm{DR}} &= \frac{\sqrt{12}}{2\pi \, \mathrm{SNR} \cdot \nu_{\mathrm{ref}}
   \cdot N_a \cdot \Delta t} \\[2ex]
   \sigma_{\phi} &= \frac{180}{\pi} \cdot \frac{\sqrt{1 + 3\overline{S}^2}}{\mathrm{SNR}}
   \quad \text{(degrees)}

where :math:`\overline{S}` is the SBD averaging factor from ``calculate_sbavg``,
:math:`\Delta\nu_{\mathrm{spread}}` is the frequency spread across channels,
:math:`\delta_{\mathrm{SBD}}` is the SBD separation, :math:`\nu_{\mathrm{ref}}` is
the reference frequency, :math:`N_a` is the number of APs, and :math:`\Delta t` is
the AP period. The mean net-sideband indicator, :math:`\overline{S}`, computed by ``calculate_sbavg``
is given by the average over all channels of: (+1 per USB channel, -1 per LSB channel,
and 0 per DSB channel pair). In HOPS related output files and plots, the above delay errors are quoted
in units of :math:`\mu s`, rate errors in :math:`\mu s/s`, and phase errors in degrees.
Note that when ionospheric phase correction is active, the MBD error is recomputed from the 
ionospheric covariance matrix, not the formula above.

Probability of False Detection
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Another quality measure for a fringe is given by the probability of false detection:

.. math::

   \mathrm{PFD} = 1 - \left(1 - \exp\left(-\frac{\mathrm{SNR}^2}{2}\right)\right)^{N_{\mathrm{search}}}

When this evaluates to less than 0.01, the small-PFD approximation is used instead:

.. math::

   \mathrm{PFD} \approx N_{\mathrm{search}} \cdot \exp\left(-\frac{\mathrm{SNR}^2}{2}\right)

where :math:`N_{\mathrm{search}}` is the total number of independent points
searched in the coarse grid.