11.3 Handling Errors
165
11.3.2 Random Errors
Random errors (also known as noise) are an inherent and unavoidable aspect of any
system. However, you should endeavour where possible to minimise random errors
by, for example, cooling the CCD, and quantify those you cannot.
In order to reduce errors, you should perform many observations if possible. As
you can see from (11.1), increasing the sample reduces the standard deviation by
1
N
. A simple way of determining whether your errors are large is to look at the
spread of your sample, i.e., the maximum value minus the minimum value minus the
mean value, and multiply this by 0.66. This will give you an approximate indication
of your 1σ error, although you should beware of large outliers and small sample
sizes, which tend not to be Gaussian in distribution.
A common trick used by marketers of beauty products is to say, for example,
that 85% of women agree that the product has a positive effect. This result seems to
suggest that the product works, when in fact, the sample size is likely too small for
the results to be meaningful.
Many of your errors will take the form of shot noise, which is noise that is a
result of the quantised nature of both electrons and photons. Shot noise has a Poisson
distribution rather than a Gaussian. The Poisson distribution is characterised by the
following equation:
P μ (υ) = e
μ μ
υ
υ!
.
(11.2)
In terms of astrometry, your random errors are largely caused by the pixel size
and the seeing, which is the Gaussian distribution of the starlight caused by random
changes in the atmosphere. There will also be errors from the distortion of the image
due to the optics and problems inherent in the processes used.
You will often hear the term signal-to-noise ratio, or SNR, in astronomy. The SNR
is just the ratio of the part of your signal (or count in the case of optical astronomy)
that is coming from your image to that part that is coming from random fluctuation
in the system. If you have low signal to noise, your uncertainties will be high, and
vice versa. Generally speaking, an SNR of 100 is good, and 200 is excellent, but it
depends on what you are trying to achieve.
Many of the astronomical applications used for photometry, and some for imaging,
will report the signal-to-noise ratio (or just the noise of the background). You should
use these figures with the understanding that they are underestimating the uncertainty.
The measurement report is based on the variation in the sky signal. However, many
things contribute to errors in observational astronomy, and although the sky signal
is the most dominant source of error, it is not the only one. Flat field, dark, and bias
noise, interpolation errors, and charge transfer errors as well as scintillation noise all
contribute to the overall noise level.
When you are dealing with spectrography rather than imaging, the problem of
noise becomes more important. Although we vertically collapse the spectra so that
we have a single line of pixels with each pixel representing a small wavelength range,
165
11.3.2 Random Errors
Random errors (also known as noise) are an inherent and unavoidable aspect of any
system. However, you should endeavour where possible to minimise random errors
by, for example, cooling the CCD, and quantify those you cannot.
In order to reduce errors, you should perform many observations if possible. As
you can see from (11.1), increasing the sample reduces the standard deviation by
1
N
. A simple way of determining whether your errors are large is to look at the
spread of your sample, i.e., the maximum value minus the minimum value minus the
mean value, and multiply this by 0.66. This will give you an approximate indication
of your 1σ error, although you should beware of large outliers and small sample
sizes, which tend not to be Gaussian in distribution.
A common trick used by marketers of beauty products is to say, for example,
that 85% of women agree that the product has a positive effect. This result seems to
suggest that the product works, when in fact, the sample size is likely too small for
the results to be meaningful.
Many of your errors will take the form of shot noise, which is noise that is a
result of the quantised nature of both electrons and photons. Shot noise has a Poisson
distribution rather than a Gaussian. The Poisson distribution is characterised by the
following equation:
P μ (υ) = e
μ μ
υ
υ!
.
(11.2)
In terms of astrometry, your random errors are largely caused by the pixel size
and the seeing, which is the Gaussian distribution of the starlight caused by random
changes in the atmosphere. There will also be errors from the distortion of the image
due to the optics and problems inherent in the processes used.
You will often hear the term signal-to-noise ratio, or SNR, in astronomy. The SNR
is just the ratio of the part of your signal (or count in the case of optical astronomy)
that is coming from your image to that part that is coming from random fluctuation
in the system. If you have low signal to noise, your uncertainties will be high, and
vice versa. Generally speaking, an SNR of 100 is good, and 200 is excellent, but it
depends on what you are trying to achieve.
Many of the astronomical applications used for photometry, and some for imaging,
will report the signal-to-noise ratio (or just the noise of the background). You should
use these figures with the understanding that they are underestimating the uncertainty.
The measurement report is based on the variation in the sky signal. However, many
things contribute to errors in observational astronomy, and although the sky signal
is the most dominant source of error, it is not the only one. Flat field, dark, and bias
noise, interpolation errors, and charge transfer errors as well as scintillation noise all
contribute to the overall noise level.
When you are dealing with spectrography rather than imaging, the problem of
noise becomes more important. Although we vertically collapse the spectra so that
we have a single line of pixels with each pixel representing a small wavelength range,