Physics 114: Lecture 8 Measuring Noise in Real

Data

Dale E. Gary

NJIT Physics Department

February 12, 2010

Mean and Standard Deviation

Sample Mean

Parent population mean

Standard Deviation from sample mean

Standard Deviation from parent population mean

1ix x

N

1lim iN

xN

21.

1 is x xN

21lim .iN

xN

Homework 1 Data The HAT-P-6 b transit data are

shown at the right. If in MatLAB you type

mean(a(:,6)) and std(a(:,6)), you will find that the data have a mean of 10.50, and standard deviation of 0.015.

The plot at lower right shows the histogram of the measurements with an overlay of a Gaussian (normal distribution) bell curve using the parameters above.

February 12, 2010

10.44 10.45 10.46 10.47 10.48 10.49 10.5 10.51 10.52 10.530

5

10

15

20

V Magnitude

Num

ber

of p

oint

s

3 3.5 4 4.5

10.44

10.45

10.46

10.47

10.48

10.49

10.5

10.51

10.52

10.53

Time (UT hours)

V M

agni

tude

Hat-P-6 b TransitNote, “data” is plural

February 12, 2010

Homework 1 Data As an example of evaluating data in a real application, consider the

HAT-P-6 data from homework 1. This is data taken during an eclipse of a star by a planet (that is, the

planet is crossing in front of the star, causing a very small decrease in light level). Unfortunately, I could not get everything set up in time, and I only got the time at the end of the eclipse (egress).

The data came from images of the star field, and there are several steps to obtaining the light curve.

Two examples of eclipses by others,

with more completelightcurves.

February 12, 2010

Here is a fit to the measurements that you read in. The curve is the expected eclipse lightcurve obtained from “forward fitting” using a model for the eclipse.

Note the “trend removed” curve, which is an example of a systematic error.

Homework 1 Data

February 12, 2010

Homework 1 Data The magnitude measurements are themselves made with images from a CCD

camera, which have their own systematic and random errors. The systematic errors can be removed through calibration, and as mentioned

before, they include both additiveand multiplicative errors.

To remove such systematic errors,we want to make the random errorsin the calibration data as small aspossible.

Let’s go through the process andintroduce CCD cameras.

2010 Feb 13

How CCDs Work

• Photons to Analog/Digital Units (Counts)

These 2 parametersgive conversion ofphotons to counts

m m

One photon has 73%chance to cause releaseof an electron (e-). It takes1.6 e- to give 1 count. So 100 photons will result in 100*0.73/1.6 = 45 counts. Each well can hold 120,000 e- = 55000 counts

2010 Feb 13

How CCDs Work

• Bias (additive)

These 2 parametersgive noise output

m m

Even with 0 s exposure,just reading out the imagegives (on average) 17 e-, or about 10 counts. This iscalled bias, and is neithertemperature nor time dep.

2010 Feb 13

How CCDs Work

• Dark Current (additive)

These 2 parametersgive noise output

m m

With a time exposure,say a 1 min exposure at -30 C, will have 19 more counts. This is BOTH temperature and time dep.

2010 Feb 13

Imaging First Principles• The last step is to take calibration frames: Bias, Dark, and Flat frames.• I take 20 Bias and 20 Dark (set camera cooler to temperature first,

and take dark frames for same duration as imaging frames). I take 10-20 flat frames (need even illumination—set duration for mid-range exposure).

• Bias frames are instantaneous, for subtraction of read noise.• Dark frames are same duration as imaging frames, for subtraction of

dark current and correction of hot pixels.• Flat frames are for removal of non-uniform illumination (vignetting and

dust). Images are divided by flat frames.

2010 Feb 13

Imaging First Principles• Noise is the enemy, so average calibration frames.

2010 Feb 13

Imaging First Principles

Image without calibration• Flat field light box

Image with Calibration