# Computing the Correct Signal/Noise Ratios

## Abstract

In short, if your flux is L in the line and C in the continuum,
where you have multiplied the Continuum flux density in Jy by:
(delta lambda) x 2.99279 10^(-12) / lambda^2

to get C in the same flux units as L (W/m^2),
then the total Signal to Noise (Line and Continuum), SN', is related
to the desired continuum in the line, SN, by:

SN' = ( L + C ) / L x SN

It is this SN' value that you must type into the PGA-MDB and the time
estimator, lws-te.

## Preface

If you are running the time estimator (lws-te) and you
want to get an estimate of your required integration time
accurate to 5%, then you can simplify your life and
enter the input value for the
continuum flux in the
detector as zero, enter your anticipated line flux
and enter your desired Signal/Noise value for the line measurement
(which we call SN) and get a pretty good answer.
But, for running the lws-te accurately, and for
running the PGA-MDB in all cases to prepare the Mission Data Base for
you observation, you MUST include the continuum flux and the
"Line + Continuum" Signal/Noise, which we will call SN'.

## SN'

A good way to remember the following formula is to understand it:
From the instrument's point of view, there is no distinguishing
line and continuum photons in a detector. For the medium resolution
AOTs LWS01 and LWS02, the line is typically much narrower than the
resolution element observing it and the continuum photons typically
make up a significant fraction of the total number of incoming photons.

The observer tells the PGA-MDB (and thus the instrument software) what
the Line (in 10^-17 W/m^2) and Continuum flux (in Jy) are expected to be.
The software computes a total ( L + C ) expected signal and adjusts internal
parameters, such as integration ramp times to prepare for this flux.
The software knows the total signal and wants to know what the total
signal to noise is expected to be, so as to integrate long enough to reach
the correct noise level. While the observer has a "Line" signal to noise
in mind:

SN = L / N

the software wants to know the SN':

SN' = ( L + C ) / N, where L = Line Flux and C = Continuum Flux.

The relation between SN' and SN is then:

SN' = ( L + C ) / L x SN

Note this may also be written:

SN' = SN ( 1 + C / L ), as in equation 6.13 in the LWS Observers Manual.

It is this SN' value that you should input into the lws-te and the PGA-MDB.
Putting in a Line SN will result in a bad measurement. Note that for a
a desired SN of 10 a SN' of 100 might be necessary.

Two caveats are in order: If C / L > 100 you may be in trouble - see
section 6.1 of the LWS Observers Manual.

If C > 50 Jy, and/or C / L > 50, fast scanning may be warranted. See
this FAQ page.

Finally, the conversion between Flux Density in Jy and Flux in
W / m^2 for a resolution element in um (remember for LWS01 & LWS02
the resolution element is: (0.29 um for lambda < 90.5, 0.60
um for lambda > 90.5) is given at the start of this page, where
a resolution element = delta lambda.