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Yale Astrometry Workshop July ‘05

Yale Astrometry Workshop July ‘05. CCD Reductions Designed for Astrometry and Photometry. William van Altena Yale University Basic Astrometric Methods Yale University July 18-22, 2005. References. References S. Howell, “Handbook of CCD Astronomy”, Cambridge Univ. Press (2000)

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Yale Astrometry Workshop July ‘05

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  1. Yale Astrometry Workshop July ‘05

  2. CCD Reductions Designed forAstrometry and Photometry William van Altena Yale University Basic Astrometric Methods Yale University July 18-22, 2005

  3. References • References S. Howell, “Handbook of CCD Astronomy”, Cambridge Univ. Press (2000) A good up-to-date book on the CCD as a detector. G. Walker, “Observational Astronomy”, Cambridge Univ. Press (1987), Ch. 7. A good general introduction to all aspects of observational astronomy. M. Newberry 1991, PASP, 102, 122 A detailed discussion of signal-to-noise

  4. CCD Quantum Efficiency • Advantages of CCDs • High quantum efficiency. • Responsive QE • RQE = N(detected)/N(incident) 25% > RQE > 95% • Detective QE • DQE = (S/N)out/(S/N)in • Spatially stable silicon substrate. • Reasonable resolution Typically 10-20 microns • Modest format size (4k x 4k) • Disadvantage • Need to enhance blue response with down-converting phosphors.

  5. CCDs as Analogue Integrators • CCDs are analogue integrators • The charge is accumulated during the exposure and then read out. • Shot Noise is added by both the Detector and the Amplifier during the read out operation. • Take as “D” equivalent # of photoelectrons • DQE = q/{1 + [D/(Nqm2)]} • N = total # incident photons • q = responsive QE • (Nq) = # detected photons • M = modulation transfer efficiency ~ 1.0 for a single diode • Note that as D increases, the DQE is degraded, i.e. the quantum efficiency of the detector is effectively reduced.

  6. The Role of “Bias”, or “Pedestal” • CCD operation • Switch is closed and the diode is reverse-biased by 5 volts. • Depletion region about 2 microns thick is created at layer interface • Switch is disconnected & diode exposed to light • Photons generate charge-carrier pairs, which partially discarge the bias • Diffusion length of carriers ~25-50 microns >> depletion region thickness. • Close switch to end exposure & measure charge required to rebias > # photons absorbed.

  7. Astrometry and Photometry with CCDs • Front-side illumination • The electrical connections interfere with the access of the photons to the sensitive area • Potential for systematic “pixel-phase” position errors • Reduced sensitive area > lower QE • QE(max) ~ 25% • Back-side illumination • Thick semiconductor • Photons are absorbed and charge carriers created too far from the depletion layer • Thinning the backside to ~10 microns can yield QE ~ 95%!

  8. Read Noise • Noise sources • Bias noise • Johnson noise in the switch introduces an uncertainty in the bias voltage actually applied: • dV2 = kT/C volts • Or in the charge fluctuation • dQ = 2.5 x 109 (kTC) 1/2electrons • For a typical CCD, • C ~ 1 pf, or 10-12 farads • T ~ 150o K , which dQ ~ 280 electrons, or D = 78,000 • Recall, DQE = q/{1 + [D/(Nqm2)]} • Obviously, we need to minimize this noise.

  9. Dark Noise • Noise sources • Dark noise • Thermal production of charge carriers • j(T) = 1.94x1022 (wa/t)T1.5e-(7015/T) electrons/sec • w and a = width and area of junction > w*a = volume • t = lift-time of carriers • Result: dark current doubles for every ~8o K increase in temperature. • Room temperature > detector saturation in about one sec. • Dark current produces noise > [j(T)*exp. time]1/2 • Add dark current noise to DQE equation, and we get: • DQE = q/{1 + [(D + j(T).exposure)/(Nqm2)]}

  10. Reading a 3-Phase CCD • Read-out schematic for a 3-phase CCD • Columns are sequentially transferred to the read-out column which then reads each pixel in the column.

  11. Charge Transfer Efficiency • Reiss (STScI-ACS 2003-009) • If the transfer of charge is not 100% from pixel to pixel during the read out, then CTE loss occurs. • CTE loss occurs from radiation damage, temperature of CCD and scene characteristics (# counts, extent of image, local and global background) • CTE 1k chip 2k 4k • 0.999975 3% 5% 10% • 0.9999975 0.3% 0.5% 1% • Minimization of CTE loss: • “pre-flashing” chip • “charge injection” in front of image readout direction

  12. Charge is lost to defects in the Silicon • Charge-transfer efficiency effects on Astrometry and Photometry

  13. Saturation and the Full Well • CCD Full- Well/Saturation level • Full well is proportional to the volume of a pixel, i.e. the area times the thickness of the depletion region. • Typical: Manufacturer says full well = 250,000 e-1; using a 16-bit A/D converter (216 - 1 = 65535 bits) • Gain = 250,000/65535 = 3.81 e -1 / ADU (analog-to-digital units)

  14. CCD Linearity • Linearity = range over which the ADU to input photons is linear. • Evaluating linearity: Take a series of frames with exposures varying by factors of two, e.g. 1, 2, 4, 8, etc. seconds. • Plot ADU vs. Exposure time. Point where the relationship turns over and becomes nonlinear is saturation level.

  15. Binning the charge and Fringing effects • On-chip binning • Electronically add pixels before they are read • 2x2, 4x4, 3x1 (for spectroscopy) • Reduces read noise, since there is only one read for each “super pixel” • Much faster read speed, since fewer pixels • Lose resolution • Fringing • Newton’s rings in the silicon for the redder wavelengths (R-band and I-band) where the photons can penetrate the silicon • Interference from reflected light within the CCD • For narrow-band observations and nearly monochromatic light, e.g. emission line nebulae and night-sky lines. • Very difficult to correct through flat-fields. • Often a problem due to variations in the thickness of the silicon layer in the thinning process.

  16. Micro- and Macro-Noise • Micro-noise and Macro-noise - useful concepts • Macro: s2 = (1/N)SNi=1(<S> - Si)2 • Micro: s2 = [1/2*(N-1)]SN-1i=1(Si+1 - Si)2 • Macro-noise includes the large-scale non-uniformities (or errors) while Micro-noise includes only the point-to-point fluctuations • Astrometry is limited primarily by Micro-noise • Noise in defining the image center • Photometry is limited primarily by Macro-noise • Zero-point variations in the scale over the field

  17. Components of the CCD Signal • Components of the CCD signal • References • Howell p.58 • CCD Astronomy Magazine. 1994-96 • S = measured signal in a pixel • Ro = rate (photons/sec) of desired signal • t = exposure time of Ro • Q = responsive quantum efficiency • s = shadowing due to dust, etc. on filter, window, … • V = vignetting in optics • Rd = dark count rate (thermal) • td = dark exposure time (not target exp. time) • B = bias charge on each pixel

  18. Components of the Signal • The equation for each pixel: • S = [V*s*q]*(Ro*t) + Rd*td + B • We want Ro, the photon rate of the target! • Flat field = [V*s*q] • Usually dominated by the pixel-to-pixel variation in the responsive quantum efficiency - typically +/- several % • Dark current = Rd*td • Thermally induced. Exp. Time since the last read cycle, can have pixel-to-pixel variations of several % • Bias = B (normally has two components) • Bs = structure bias, which is usually a constant due to the cameral electronics. • Bo = offset bias, which tends to change with time • R = read-out noise

  19. Bias/Pedestal and Read-out Noise • How do we get the Bias? • Create Master Bias Frame -(MBF) Take several 0 sec. exps. and median average to eliminate cosmic rays and bad data points. • Divide into two components: Bs (constant) and Bo (frame to frame variation). • Bo from “overscan” on each line/frame. • Read-out noise = R • Bias is initial charge on pixel - Johnson noise in switch is R • R = FWHM/2.35, i.e. the Gaussian equivalent sigma of the noise in the Bias.

  20. Dark Noise • How do we get the Dark? • Dark is the thermal emission from each pixel = f(Temp), but not photon count. • Rd = Rdo*2(T-To)/dT • Usually dT ~ 8oK, so dark increases by 2x if temp. increases by 8o, and vice versa. • Some pixels will be “hot”, so this can be an important correction - several % variation pixel-to-pixel. • CCD must have a constant temperature!! • Create a Master-Dark-Frame (MDF) by median averaging several zero-sec dark exposures taken with the shutter closed.

  21. The Flat Field • How do we get the Flat-Field? • F-F = [V*s*q] • V = Vignetting is usually a slowly varying function of location on the frame • Obstructions in the light beam to the CCD • s = Shadowing is due to dust, etc. on the filter and window - therefore variable with time • Different for each filter and possibly filter change if mechanism is sloppy.

  22. Flat Field is a function of ….. • How do we get the Flat-Field? • F-F = [V*s*q] • q = RQE may be a function of CCD temperature with pixel-to-pixel variations of several percent. • Mirror and lens transmission are f(wavelength), Q(CCD) = f(wavelength), so we need a FF for each filter. • Spectral energy distribution of illumination should be same as target illum. • Twilight night sky is bluer that midnight sky • 13o east of zenith just after sunset ~ midnight sky (Chromey & Hasselbacker 1996 PASP 108, 944)

  23. Getting the Best Flat Field • Types of Flat fields • Median Flat-Field (MFF) from several exposures in each filter. • Median gets rid of cosmic ray hits and bad data. • Super-sky flat - median average all exposures during a night. • Random star locations in field-of-view average out in median. • Generally low S/N due to low sky background. • Dome-diffuser flat - plexiglass diffuser in front of objective or corrector in a Schmidt. • Good for wide fov. • High S/N since it is a dome flat and bright lights. • Zhou, Burstein, et al. 2004, AJ 127, 3642. S-S flat taken with wire objective grating (45o) in place. Diagonal streaks are due to partial overlapping of the stars in the deep and dense exposures. Fluctuations in the S-S flat are ~8%.

  24. Signal-to-Noise • References • Newberry 1991, PASP 103, 122. • Newberry 1994, CCD Astronomy • Howell, “Handbook of CCD Astronomy”, p. 53 • S/N sets the fundamental limit on our ability to measure the signal from the target. • Bias, Dark and Flat-field corrections all contribute to degrading the S/N in the measured signal.

  25. Components of the Signal and Noise • The Signal = S, and the Noise = N, both in total counts • S = So + Sd + Sb + Ss • N2 = No2 + Nd2 + Nb2 + Ns2 + Nr2 = So + Sd + Sb + Ss + Nr2 • S/N = {So + Sd + Sb + Ss}/{So + Sd + Sb + Ss + Nr2} 0.5

  26. S/N for an Area of the CCD • S/N of an image, rather than for a pixel. • Sum the counts, C, over a small area, say n pixels • Let C = ∑[So + Sd + Sb + Ss ] = Co + Cd + Cb + Cs • or, C = Co[1 + Cs/Co + (Cd + Cb)/Co] • The noise, N = ∑[So + Sd + Sb + Ss + Nr2]0.5 • Or, N = Co0.5[1 + Cs/Co + (Cd + Cb)/Co + nNr2] 0.5 • Assume bias and dark noise are small, i.e. you are dominated either by sky noise or read noise. • S/N = Co0.5[1 + Cs/Co + (Cd + Cb)/Co]/ [1 + Cs/Co + nNr2] 0.5

  27. Read-Noise vs Sky-Limited S/N • S/N of an image, rather than for a pixel. • Case 1: Read-noise limited (short exposures) • S/N = Co0.5[1 + Cs/Co + (Cd + Cb)/Co]/ [1 + nNr2] 0.5 • S/N is degraded by the CCD read noise - • get a better CCD • Bin pixels in 2x2 or 3x3 > only one read/group of pixels, but then under sampling may compromise astrometry • Case 2: Sky-noise limited (long exp. or bright sky) • S/N = Co0.5[1 + Cs/Co + (Cd + Cb)/Co]/ [1 + Cs/Co] 0.5 • S/N is degraded by the sky noise - • move to a darker site • Observe in better seeing, fewer sky pixels in image • Under sampling might be a problem with fewer pixels

  28. Noise in the Flat Field • In either case, for each pixel we have: • S = [V*s*q]*(Ro*t) + Rd*td + B, or • [V*s*q]*(Ro*t) = S - [Rd*td + B] • Assuming that the bias and dark are small and noise free • [V*s*q]*(Ro*t) ~ S • The flat field, [V*s*q], is an observed quantity with error, therefore • (Ro*t) = So ~ [V*s*q]true*S/[V*s*q]obs • By error propagation: • (∂So/So)2 ~ (∂FF/FF)2true + (∂S/S)2o + (∂FF/FF)2obs • where FF = V*s*q • Since the error of the “true” FF = 0.0, • (∂So/So)2 ~ (∂S/S)2o + (∂FF/FF)2obs • In other words, the final object S/N is only as good as the flat field!

  29. S/N for Multiple Exposures • S/N propagation • Assume n exposures: S1, S2, S3, …; N1, N2, N3, … • S/N = [S1 + S2 + ] / [N12 + N22 + ]0.5 • S/N = (S1/N1) / [1 + (N2/N1)2 + … ]0.5+ + (S2/N2) / [1 + (N1/N2)2 + … ]0.5 + … • If S1 ~ S2, etc … then N1 ~ N2, etc … and • ∑ [1 + (N2/N1)2 + … ]0.5 ~ n 0.5 • ∑ (Si/Ni) ~ n (Si/Ni) S/N ~ n 0.5 (Si/Ni)

  30. Fringing and Bad Pixels • Fringing • Newton’s rings from interference of nearly monochromatic light reflecting within the CCD, or passing through and reflecting back. • Generally only a problem for redder wavelengths (R, I & Z passbands) that can penetrate through the silicon layers. • Monochromatic light: Night-sky & emission lines in nebulae. • Very difficult to correct for with flat fields. • Night-sky lines are highly variable with time. • Fringing a function of angle of incidence of light on CCD, therefore can’t duplicate in a flat field. • Bad and “Hot” pixels • A mask of the pixel locations (x, y) is made to eliminate them from the analysis.

  31. Macro- and Micro-Noise • Micro and Macro noise • Latham & Furenlid 1976, AAS Photo-Bulletin 11, 11. • Useful concepts to characterize the noise on a frame • Macro noise > pixel-to-mean variations = sm sm2 = (1/N)*∑NI=1[Si - <S>] • Measures the large-scale variations in sensitivity and non-uniformity of the CCD. • Characterizes the photometric accuracy, i.e. the zero point of the photometry. • Not important for astrometry. • Micro noise > pixel-to-pixel variations = sµ sµ2 = {1/[2(N-1)]}∑N-1I=1[Si - Si+1] • Characterizes the photometric and astrometric precision of a measure.

  32. Statistics • Probability plots • Robust estimation of the dispersion of a sample. • Hamaker inversion • App.Stat. 27, 76, 1978. • Mode is sky background

  33. CCD Photometry References • References • “Astronomical CCD Observing and Reduction Techniques”, S. B. Howell, ed., 1992, ASP Conf. Series23. • Stetson, P. 1990, PASP 102, 932. • DaCosta, G., 1992, ASP Conf. Series23, 90.

  34. Surface Photometry • Surface Photometry • Generally trying to determine the surface brightness of a galaxy that is very faint in the presence of a “bright” sky. • Goal is to trace the galaxy out to, say, ≤ 1% of sky level. • I(x,y) = Galaxy(x,y) + Sky(x,y • Need Sky(x,y) to better than 1% accuracy, or ss(x,y) ≤ 0.01*S(x,y) • Macro noise is critical here, due to possible poor flat-field.

  35. Sky Background and its Error • <S> = corrected mean sky • sm = macro-noise in the sky • sm = Poisson + Read Noise + Large scale background fluctuations. • sm = [ S + Nr2 + ??]0.5 • s(<S>) = error of mean sky • X = % error desired in <S> • X = s(<S>) / <S>, e.g. 1% • s(<S>) = sm / N0.5 , or X = sm / [N0.5 * <S>] • N = {sm / [X * <S>]2 • <S> = 100 counts, Nr = 5 counts, X = 1%, then N=121 pixels • <S> = 100 counts, Nr = 5 counts, X = 0.1%, then N=12,100 pixels

  36. CCD Photometry r (arcsec) ~Gaussian image core mag. log I • King, PASP 83, 199, 1971 • Stellar profile observed through atmosphere. • Sky background near star is affected by presence of stellar wings r-2 wings r (arcmin)

  37. Aperture Photometry • DaCosta, G. ASP Conf. Series 23, 90, 1992. • Select an aperture of radius r that contains the image and sum all of the pixels that fall within the aperture • Isum = I* + <S> • What radius should be used to include all of the stellar flux? • Remember that the King stellar profile extends many arcsec. • What about the sky within the aperture?

  38. Aperture correction • Stetson, P. PASP 102, 932, 1990 • Large r to include all star light, but: • Includes other stars • Adds sky noise • All stars have the same psf, so all psfs scale with the # photons • psf is constant, except for • optical aberrations • seeing variations over field (short exposures) • CCD is linear, except for • saturation and CTE

  39. Aperture correction • Same psf, only volume scaled • The same X% of photons will fall within an aperture of radius, r. • dm = -2.5 log10I2/I1 • If I2 = X% of I1, • dm = -2.5 log X*I1/I1 • dm = -2.5 log X • Constant aperture correction, dm. • Star image must be accurately centered in aperture for this to work. • Typical ~4-5 FWHM for aperture radius • “Optimal” aperture = 1.35 FWHM Ap. corr. = 0.2 mag for a Gaussian

  40. Sky Background and Photometry Error • Sky background • Select many star-free spots and average the results • Probability plot analysis of background and scale to aperture • Error in photometry • So = total stellar signal, but noise and read-out noise are per pixel • s2 = So + n(Ss + Nr2) • mag = mo - 2.5 log10So • sm = smo - 2.5 (log10e)s(So)/So • sm = smo - 1.086 (1/So) s(So) • sm = smo - 1.086 (1/So) {So + n(Ss + Nr2)}0.5 • = Zero pt. Error + Poisson + Sky + Read-out noise • So is fixed, so the larger n, the greater the contribution of the sky and read-out noise is to the error.

  41. Sky Noise • Ideally the histogram of pixel values is unimodal and we determine the mode. • Mode is usually difficult to determine and poorly defined. • Kendal & Stuart, p.40, 1977 • mean-mode = 3(mean-median)

  42. PSF Photometry • King found the stellar profile to be approximately Gaussian in the core: • G(x) = (sx√2π)-1 exp-{0.5[(x - xc)/sx]2} • ∫xG(x) = 1.0 • G(x = xc) = 0.3989 • G(x = xc ± sx) = 0.2420: 0.607 height at x = xc • G(x = xc ± fwhm/2) = 0.1995: one-half the height at x = xc • Gaussian doesn’t fit in wings, so other functions are added • Modified Lorentzian: L(x) = C*{1 + (x2/s2)ß}-1 • Moffat function: M(x) = C*{1 + (x2/s2)}-ß • C = constant • Not even those are perfect so a table of corrections (H(x,y) is added to give the final model PSF: • PSF(x,y) = [a*G(x,y) + b*L(x,y) + c*M(x,y)] * [1 + H(x,y)] • DAOPHOT and IRAF: See Stetson, P. in PASP 102, 932, 1990.

  43. Passbands and Filters • UBVRI: Bessell PASP 102, 1181, 1990 • JHKLM: Bessell & Brett PASP 100, 1134, 1988 • IR: Astrophys. Quant. IV (AQ4), A. N. Cox, ed. Ch. 7.1-7.7 • Visual: AQ4 Ch 15.3 • Asiago Database on photometric systems (ADPS) • http://www.pd.astro.it/Astro/ADPS/

  44. Photometric Surveys

  45. Image Centering • References • van Altena and Auer: in “Image Processing Techniques in Astronomy”, p. 41, 1975 • Auer and van Altena: AJ 83, 531, 1978 • Lee and van Altena: AJ 88, 1683, 1983

  46. Image Centers vs. Centroids • Given an intensity distribution, S(x,y) • The Centroid , center of mass or 1st moment of the distribution. • See: van Altena and Auer: in “Image Processing Techniques in Astronomy”, p. 41, 1975 • <x> = ∑x,y{xi*[S(x,y) - B]} / ∑x,y[S(x,y) - B], • <y> = ∑x,y{yi*[S(x,y) - B]} / ∑x,y[S(x,y) - B] • where B is the assumed sky background around the image. • The centroid is very sensitive to the adopted sky background, but it is also works well for very faint images. • The Image Center • See: Auer and van Altena: AJ 83, 531, 1978 • and Stetson in DAOPHOT manuals. • The Marginal distributions are defined by: • rx(x) = Ny-1∑ySo(x,y) • ry(y) = Nx-1∑xSo(x,y)

  47. Gradient Search for Image Crowding • Image centering • Take the x-marginal, rx(x), on the upper right. • Image crowding noted where rx(x) increases at edges of the diagrams. • The derivative of the x-marginal, r’x(x), is on lower right. • Peaks at ± gaussian radius, Rx = FWHM / 2.36 • Zeros at image center and inflection points in rx(x) that indicate image crowding.

  48. Image Centering on the 1-D Marginalrx(x) = ax + bx(x-xc) + [1 + cx(1.5t - t3)]hxexp(-0.5t2) • Univariate Gaussian • t = (x-xc) / Rx • hx = Nx / [Rx√2π] • Rx = Gaussian radius • bx = sloping background • cx = skewness of image • Generally take bx = cx = 0.0, since there is usually a high degree of correlation between the odd terms and this degrades the image center precision, i.e. use a symmetric Gaussian for the fit.

  49. Image Centering on the 2-D Distribution F(x,y) = Doexp(-0.5r2 / R2) + B • Bivariate Gaussian • Lee and van Altena: AJ 88, 1683, 1983 • Do = image height at center • r2 = (x-xc)2 + (y-yc)2 • R = Gaussian radius • B = background • Precision • #1 Bivariate • #2 Univariate • #3 Centroid • Convergence • Inverse order of precision, I.e. the centroid is most stable, especially for faint images.

  50. Dealing with Saturated Photographic ImagesF(x) = a[2tan-1bc]-1{tan-1 [b(x-xo+c)] - tan-1 [b(x-xo-c)]} + do • Saturated photographic image usually has a flat top and Gaussian fits poorly • Two arctangent functions fit very nicely • Stock, J. ~1997 • a = image height above background • do = background • c is proportional to image width • b = scale factor for image slope and gradient • Winter (Ph.D. thesis 1995) - a generalized Lorentz profile also works well.

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