Signal, Noise and Detection

O.Hainaut, 2005-Jun-01
(ohainaut@eso.org)
URL: http://www.sc.eso.org/~ohainaut/ccd/sn.html

This short document discusses the concept of Signal, Noise, Signal/Noise ratio, and their practical application in astronomy.

1.- Signal

The Signal (S) is the number of photons detected from a source. In practice, we don't detect all the photons that arrive on the detector, only a fraction of them are converted to electrons and detected (the ratio electron/photon is the quantum efficiency; it is typically 50-85% for a CCD). So, in what follows, we will work only in electrons.

The total signal S is related to the number of electrons we get per seconds, s [e-/s] (the [] marks the units and/or dimensions) and the total exposure time, t [s]:

S = s . t [e].

In many cases, the total exposure time t is split in various exposures. To keep the "infrared" notation,

t = NDIT . DIT

where NDIT is the number of exposures and DIT is the individual exposure times. So,

S = s . NDIT . DIT

2.- Noise

The Noise (N) is the total random contribution from various sources that affect the measurement of the signal. Also measured in [e]. We will see after that there are various sources of noises and that they behave differently. The noise is basically the error on the flux measured:

S +/- N,

like, 1532 +/- 231 electrons, or 123+/- 14 mJy.

3.- Signal-to-Noise Ratio

The Signal-to-Noise Ratio, SNR, S/N, measures how well an object is measured. Typical values:

S/N = 2-3: object barely detected
S/N = 5: object detected, one can really start to beleive what one sees
S/N = 10: we can stat to do measurements
S/N = 100: excellent measurment.

As N is the error on the measurement, 1 / (S/N ) is the relative error on the measurement:

S/N = 100: measurement at 1%
S/N = 10: at 10%
S/N = 5 : at 20% (+/- 20%)
S/N = 2: +/- 50% !

Important remark about the flux/magnitude conversion: by definition,

Mag = -2.5 log ( S ),

(to which one has to add a series of magic constants to take into account the conversion of units, correction for extinctions, etc... That is another story. As these are constants, they don't affect what follows). As we have an error N on the Signal, the error on the magnidude would be obtained from dMag = -2.5 log ( S ) + 2.5 log ( S + N ) = 2.5 log ( (S+N) / S ) = 2.5 log ( 1 + N/S). Here comes the "magic":

S/N = 100: measurement at 1%, error on magnitude = -2.5 log (1.01) = 0.0108 ~= 0.01 mag
S/N = 10: at 10%, error on mag = -2.5 log (1.1) = 0.103 ~ 0.1 mag
S/N = 5 : at 20% (+/- 20%); error on mag = -2.5 log (1.2 ) = 0.198 ~ 0.2 mag

Error on Mag ~ 1/ (S/N)

4.- Noises

Various noises are considered:

4.1. Shot Noise

Every time we deal with a source of photons arriving at random, the noise assiciated with that randomness is

N = sqrt( n )

Where n is the number of photons. As we work in electrons, same thing, with n in electrons. This applies to two sources of noises:

Object noise, associated to the number of electrons from the object itself:

N_object = sqrt (n_object)

By definition of S = s . t, the signal:

N_object = sqrt( S )
= sqrt ( s . t )
= sqrt ( s . NDIT . DIT )

Sky Noise: This also applies to the sky: when we measure the object, we have to measure the sky "under" the object at the same time, then take it into account by subtracting it. Bottom line: we have to take into account the shot noise of the sky:

N_sky = sqrt( Sky )

Where sky is the number of electrons coming from the sky in the region covered by the object. We define sky as the number of electron per second an per pixel, and n_pix the number of pixels covered by the object (this is the area un der the object, so related to seeing^2 ). With these, we have

N_sky = sqrt ( n_pix . s . t )
= sqrt ( n_pix . s . NDIT . DIT )

Dark Noise: Finally, the shot noise also applies to the Dark Current, i.e. the number of electrons that are coming from thermal radiation of the detector itself:

Dk = n_pix . dk . t

(where dk is the number of electrons per pixel per hour), so the dark shot noise is:

N_dk = sqrt( n_pix . dk . t )
= sqrt ( n_pix . dk . NDIT . DIT )

4.2. Read-out-noise

When reading the detector, the amplifier(s) involved add some noise to the signal. This is a characteristic of the chip and of the read-out-mode used. Let RON be the read-out-noise per read-out (one get the noise each time one reads) and per pixel (in electron). The total read out noise is then:

N_ron = sqrt( npix . RON^2)
= sqrt( n_pix) . RON

(all the individual read-out-noises add quadratically). If we read the chip NDIT times,

N_ron = sqrt ( n_pix . NDIT ) . RON

4.3. Adding noises and total signal to noise ratio

Random, uncorrelated noises (such those described) add quadratically, so, the generic formula for the noise is

N_total = sqrt( N^2 + N_sky^2 + N_dark^2 + N_ron^2 )

We can play with the formula, introducing the various definitions, to get the generic formula for the SNR:

S/N = S/N_tot = S / sqrt ( S + Sky + Dark + N_ron^2 )

S/N = s.t / sqrt ( s.t + n_pix . sky . t + n_pix . dark . t + n_pix . NDIT . RON^2 )

5. Special cases -- in practice

The generic S/N equation is used in details in the Exposure Time Calculators. However, in practice, it is useful to consider some special cases, to understand the behavior of the instrument, and why one can/should make many short or one long exposure.

5.1.- Bright source

Let's first consider the case of a bright star. In that case, the signal of the star is so bright that all the other sources of noises are negligible:
we can remove them from the equation:

S/N = s.t / sqrt ( s.t ) = sqrt( s . t)

More importantly:

S/N === sqrt ( t )

( === means "varies proprortionally to").

Application: bright standard stars in imaging and in spectro. Note that in this case, the problem is usually not to reach a good S/N, but not to saturate the detector.

5.2- Sky Noise Dominated case

After this warm up, let's consider a more useful case: a faint star with a bright sky background: sky >> s . The S/N equation becomes:

S/N = s.t / sqrt ( n_pix . sky . t)

S/N === sqrt ( t ) / sqrt (n_pix)

First aspect: if n_pix is smaller, S/N is larger. As n_pix === seeing^2, one sees that good seeing is critical

Second aspect: the signal-to-noise increases as the square root of the TOTAL time t = NDIT.DIT.
This means that the S/N will not be affected if we take 1 exposure of 3000s or 10 exposures of 300s (as long as each of the individual images is sky-noise dominated). In many cases it IS advantageous to split the exposure time in many exposures.

Applications:

broad band images in the visible (typical value: DIT = 300-600s, Sky = 5000adu = 10000 e, RON = 3 e). Taking many images improves the flat field of the final image
low resolution spectro in the visible (EFOSC, FORS, EMMI-RILD...). As soon as the sky reaches a few hundred e, one can split. Advantages: get rid of the cosmic rays.
Infra red imaging: the sky is so high that it would saturate the detector very quickly (few s to few 10s in JHK, few ms in thermal IR). There is no other choice than to split the total exp. with many short DIT. This is not a problem for the final S/N, since we are sky noise dominated

Counter applications: This is not the case in high resolutions spectroscopy, or in narrow band imaging, since the sky is then very low. Do not split in short DITs, cf next section.

5.3- Read-out-noise dominated case

In case the sky --and the dark-- are very small, they do not dominate in the general S/N equation. There are even cases where they are very small; in that case, the RON dominates:

S/N = s.t / sqrt ( n_pix . NDIT . RON^2 )

S/N === 1/sqrt(n_pix) . 1/sqrt(NDIT)

First aspect: 1/sqrt(n_pix): there is here a way to cheat. n_pix is the number of pixel read. We can decrease this number by binning the detector (i.e. reading only once for 2x2=4 pixels). There is a gain of sqrt(2x2) = 2 in S/N! Of course, there is a price: one loses some of the resolution: spacial resolution in case of imaging, beware of keeping at least 2 (binned) pix across the seeing) or spectral resolution in case of spectro, beware of keeping at least 2 (binned) pix across one spectral line.

Second: 1/sqrt(NDIT): one should keep the NDIT as small as possible, meaning keeping DIT as long as possible. In case of CCDs, the real limit becomes the number of cosmic rays. In practice, DIT_max is about 45min, but in some cases, it is worth making it even longer (e.g. if the observer does not care about cosmic rays).

Application:

narrow band imaging (sky = a couple of electrons), typical exposure times can be as long as 1h.
Echelle spectro: the sky light is so much dispersed that it does not count anymore. Keep the exp. time as long as possible.

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