Brian Schmidt
Principles in Data Reduction
Producing “Good Data”
• Essential to have a good working instrument
• Essential to take all of the relevant
calibration data
• Correct data reduction procedures for the
data
– Check constantly for errors
– All data is not alike. Do what is appropriate for
your data.
CCDs
•
•
•
•
•
•
•
Linear
Reasonable dynamic range (7 magnitudes)
Superb Efficiency (90%)
Low noise ( 4 counts per pixel)
Sensitivity X-ray to 1100nm
No colour discrimination (except in X-ray)
Can be run as fast as 100Hz (but typically has 20120 second down time between exposures)
• Largest Astro CCDs are currently 16 Million pixels
CCDs are the detector of choice in
most situations
• Exceptions are where fast readout times are
needed (photocounting preferred)
• And where backgrounds are extremely low
(sometime photocounting devices preferred)
• Very large Fields (CCDs very expensive)
where Photography has been used.
Noise in Astronomical Detectors
Poisson Noise in most
situations
Counts = photons detected within aperture or resolution
element
Sky Noise  sky counts
Shot Noise  object counts
Detector (Read Noise) Noise  counts
Dark Noise  dark counts
Signal/Noi se 
object counts
Sky Noise 2  Shot Noise 2  Det Noise 2  Dark Noise 2
SNR of two detectors


QE1t (obj count rate)


2
SNR1  QE1t (sky count rate)  QE1t (obj count rate)  Det Noise 1 

SNR2 

QE 2t (obj count rate)


2
 QE 2t (sky count rate)  QE 2t (obj count rate)  Det Noise 2 
UV Observation of a faint object from space
count rate obj
count rate sky
CCD QE
MAMA QE
CCD noise
MAMA noise
t seconds
0.5 count per sec
0.0001 count per sec
0.5
0.25
25.29822
0
1
8 per pixel
10
100
10 pixels
1000
10000
signal CCD
noise CCD
SNR CCD
0.25
2.5
25
250
2500
25.30316 25.34759 25.78769 29.83371 56.04016
0.00988 0.098629 0.969455 8.379784 44.61086
signal MAMA
noise MAMA
SNR MAMA
0.125
1.25
12.5
125
1250
0.353589 1.118146 3.535887 11.18146 35.35887
0.353518 1.117922 3.53518 11.17922 35.3518
Glossary of CCD terms (from http://www.pha.jhu.edu/~cat/seminar.html)
gate
A thin layer of metal or heavily doped polycrystalline attached to an electrode forms the gate. A bias
voltage may be applied to the gate in order to change the shape of the underlying potential.
oxide layer
The ~0.1 micron thick oxide layer (usually SiO2) beneath the gate functions as the dielectric of the
capacitor. The oxide is thickened to ~0.5 - 1.5 microns above the channel stops to insulate them from
changes in the gate voltage.
channel stop
The function of the channel stop regions is to confine charge. They are made of heavily doped p-type
materials with an extra thickness of oxide over top. This makes them relatively insensitive to voltages
applied to the gate and thus an effective potential barrier.
n-type buried channel
Most modern CCDs have buried channels. A buried channel is created by the addition of a n-type layer
(~1 micron thick) between the gate and the oxide. A n-type (negative) semi-conductor is one which
has been doped with impurities of higher atomic number yielding an excess of free electrons in the
conduction band. The effect of the n-type layer is to move the potential minimum back from the SiSiO2 interface eliminating "fast surface states" which cause problems with charge transfer. The region
where the signal charge collects, termed a channel, is within the n-type region.
p-type substrate
A p-type (positive) semi-conductor is one which is doped with with impurities of lower atomic
number, resulting in "holes" in the valence states. The substrate is usually at least 15 microns thick.
depletion region
In the depletion region, electrons from the n-type region have combined with holes from the p-type
region. The result is the establishment of a potential difference because the n-type region becomes
positively charged, while the p-type region becomes negatively charged. When photons are absorbed
in the depletion region they form electron-hole pairs. The electrons are attracted to the the n-type
region. The holes diffuse away into the p-type region.
Traditionally, CCDs were illuminated on the front side, meaning the side with the
gates. Many of the blue photons were absorbed by the relatively thick (0.5
micron) gates. Today it is possible to make back-side illuminated devices. In
these devices the silicon substrate is thinned to ~15 microns and the gate side is
mounted against a rigid surface. An enhancement layer is also added which
creates an electric field that forces electrons toward the potential wells. Back-side
illuminated devices have higher QE especially in the blue and UV portion of the
spectrum.
Optical depth of 1 for 1 micron photons is roughly 100 microns
in Silicon (CCDs typically 15 microns thick) So red light travels
straight through CCDs. Plus, whole CCD acts as a Fabry-Perot
cavity and one gets fringing!
High-resistivity Thick CCDs
The solution is to make the CCDs thicker than the absorption depth of the silicon,
incident photons will then be absorbed on their first pass and reflection from the
rear surface will be greatly reduced. Lincoln Labs and E2V have made 40 microns
thick CCDs, LBL is experimenting with 100 micron thick CCDs. Standard silicon
cannot be used for this process since it cannot sustain the high electric field
throughout the full depth of the device that is so important for good QE. Instead a
special grade of high-purity high-resistivity silicon must be used.
CCDs
Charge Transfer Efficiency (CTE)
• Each transfer has the possibility of leaving
some of the electrons behind.
• In a 2048x4096 pixel CCD, pixel 1,1
undergoes 4096 parallel transfers and 2048
serial transfer. If the parallel transfers are
99.99% efficient then pixels in last row lose
• 1-(.9999)4096=.334 of their charge relative to
CTE
loss of charge
first row! Bad!
0.9999
0.336
0.99999
0.040
0.999999
0.004
0.9999999
0.0004
QE of Modern CCDs
http://www.ing.iac.es/PR/newsletter/news5/simon2.gif
Gain, Bias, Readnoise, Dark Current
At the end of the Serial Register, the transferred electrons need
to amplified to create a voltage which can be measured by
our electronics (and which are converted typically by a 16 bit
[0-65536] Analog to Digital converter)
Gain: How many counts does each electron get (typically 1-8
e/ADU). Match so 65536->is where the CCD saturates (is
not longer linear)
Bias: When there is no charge, how many counts do we read.
Readnoise: How many electrons of noise does the CCD
amplifier add. (good CCDs now 4e-s, best are 1e-))
Dark Current: If CCDs are cold (<180K), they usually produce
only a few electrons per hour per pixel. Warmer, they
produce more.
Measuring the Gain
•
•
Fe55 source: Fe-55 produces a 5.9
KeV soft X-ray. When this interacts
with the CCD, 1620 electrons are
generated in a volume much smaller
than a pixel. Measure how many
counts there are in each interaction,
that tells us the # of ADUs per 1620
electrons. Very accurate at one input
level, but not easy for the
astronomer
Take two images of the dome with
lots of counts in them (30000 or
so), subtract these, use the poisson
noise to tell you how many
electrons per ADU.
 1 2 

2
1

  22  2 1
gain (e - /ADU) counts(ADU )
gain (e - /ADU)
2 counts (ADU)
gain (e - /ADU) 
2
 1 ( ADU ) 
 12 
Counts=ADU - BIAS
Measuring the Read Noise
• Take two BIAS images (0 second
exposures) – only need one if
the bias has no structure in it
like any decent modern CCD
system should provide!
• Readnoise is just
measured from the std
deviation of the BIAS
image (or their
difference) adjusted by
the gain.
 12 

2
1

  22  2 1
 1 ( ADU )  ReadNoise (e-)/gain (e - /ADU)
ReadNoise( e-)   1 ( ADU )gain (e - /ADU)
1
ReadNoise( e-) 
 1 2 ( ADU )gain (e - /ADU)
2
Measuring the Linearity
• In the lab, use a diode, and take different length exposures
to see brightness of diode.
or
• Under take gain test with several pairs of images at different
light levels. Does the gain remain the same.
or
• Take an image of a field of stars as twilight
ascends/descends. Measure the aperture magnitude of the
stars in increasing background light. Do the stars magnitude
change as background increases.
Measuring the Charge Transfer
Efficiency.
• Fe55 source: Fe-55 produces a 5.9 KeV soft X-ray. When this interacts
with the CCD, 1620 electrons are generated in a volume much smaller
than a pixel. Measure how many counts there are in each interaction
as a function of position. that tells us the # of ADUs per 1620
electrons. Comparison in the Y direction tells you charge loss in
parallel registers, Comparison in X direction tells you charge loss in
serial registers. Again, for the technition.
• Look at Cosmic Rays or defects in your images, do they bleed in
either the X, or Y direction.
• In a dense field on a photometric night, image a set of stars in the 4
corners of the chip and compare their photometry.
Atmospheric Refraction
• As stellar light passes through
the atmosphere, it is refracted
just as through a lens. Blue light
is refracted more than red light.
Like extinction, the amount of
atmospheric refraction depends
on the amount of air mass that
light has to traverse.
r  k tan z
r is increased altitude due to
refraction
z is the zenith angle
k is a constant which depends on
wavelength, pressure (altitude),
temperature, and humidity.
k

3000Å 63.4
5000Å 60.6
1
40
59.6
59.3
Differential Refraction
At 1.5 airmasses, an image at 4000Å is displaced towards the
zenith by 1.1” relative to the image at 6000Å. If you are
trying to observe over this wavelength range using a 2 arcsec
slit, you will suffer large amount of light-loss unless the slit
happens to be aligned at the parallactic angle, i.e., the
position angle on the sky that results in the slit being
perpendicular to the horizon.
• Try to always observe at low airmasses.
• Rotate the spectrograph so that the slit is near the parallactic
angle.
• Broad Blue filters are to be avoided when imaging (e.g.
Gunn-g) at high airmass.
• Use Atmospheric Dispersion Compensator
Atmospheric Extinction
1
X
cos z  0.025e 11cos z
Extinction per airmass depends on
wavelength, on altitude, and on the
night.
Mauna kea, for example as an
average pressure which is 60% that at
Sea Level. So to go to Asiago’s curve,
should multiple Mauna Kea’s by
1.4.
Atmospheric Dispersion
Compensator (ADC)
• Many large telescopes are with a
ADC – typically a pair of
``Risley prisms", which rotate
relative to each other and
provide excellent atmospheric
dispersion compensation. They
can cause ghosting and scattered
light, and chew up a lot of the
light below 4000Å, but they do
get rid of differential refraction.
Sky Brightness
Days
from
New
moon
U
B
V
R
I
0
22
22.7
21.8
20.9
19.9
3
21.5
22.4
21.7
20.8
19.9
7
19.9
21.6
21.4
20.6
19.7
10
18.5
20.7
20.7
20.3
19.5
14
17
19.5
20
19.9
19.2
Sky Brightness
Sky transmission in Red
• Telluric Lines caused by molecules in atmosphere
(O2, H20…)
• Do not scale linearly as airmass
• Observe smooth spectrum standard to remove
Telescopes
• Alt-Az
– Orientation of image changes as telescope tracks,
corrected by an image rotator
– Choose rotator angle to orient image (N-E) or
slit (parallactic or desired galaxy position) to
desired angle
• Equatorial
– Orientation of image typically fixed, but should
choose angle so slit is at parallactic angle or to
minimise galaxy gradient.
Telescope Optics
• Each mirror surface looses about 8% of the
light
• Each coated transmissive surface looses about
4% of the light
• Large glass correctors tend to remove UV
(Fused Silica is the best)
Slit Spectrographs
• Classic long slit
spectrograph is very
simple
• (But orders overlap, so
tough to get more
than a factor of two in
wavelength coverage)
And hard to optimise
system from 320nm1micron
Double Beam Spectrograph
• Split light into two
channels using a
dichroic.
Orders now do not
overlap
Can use Blue and Red
optimised CCDs on
either arm
Echelle Spectrograph
• Echelle Spectrographs use a
highly dispersive grating
which output many order.
• Orders are separated out by
an additional grating
(Cross-disperser) which
moves the light as a
function of wavelength and
position.
Integral Field Spectrographs
• Image slicer (or fiber
pod) creates a series of
slit spectra covering a
2-d piece of sky…
Taking a Spectrum
• Object Bright – center directly
into slit
• Object Faint – offset from bright
star
• Always good to take more than
one spectrum of each object to
remove Cosmic Rays Even betterdefine two places on the slit, take
two spectra of each object
(standards as well). Subtract the
two images to remove most of the
sky lines for each object. (not so
useful if not photometric)
Taking a spectrum
• Choosing slit size. As you increase your slit size,
– you loose resolution,
– increase the fraction of light of object landing on
detector
– Increase the background hitting the detector
• Typically we choose a slitsize about 1.5 FWHM of
image seeing, unless we are trying to get absolute
spectrophotometry (then need a larger slit)
Nod and Shuffle
– a. An observation is taken at a first position on the sky (position
A), while guiding.
– b. The shutter is closed, and charge is shuffled by +Y pixels.
– c. The telescope is moved to a second position (position B).
– d. The shutter is opened at position B and the observation is
continued.
– e. The shutter is closed and charge shuffled by –Y pixels.
– f.
The telescope is moved back to position A.
– g. The procedure is iterated until the exposure is complete. The
final exposure time is the product of the sub-integration time and
the number of sub-integrations.
Data taken at position A has sky spectrum taken at position B
subtracted, removing very effectively, the sky lines.
Calibration Data-Spectra
•
•
•
•
•
Arc at the position of each object
A smooth spectrum standard
A spectral flux standard star
Biases – 10 or so are usually sufficient
Internal flats (quartz) – Make sure that you have
lots of counts in the blue – You might need to take
two exposure times – one where you saturate the
red part of the spectrum
• Skyflat – Not that important, but useful if you are
planning to use more than one position on the slit.
Smooth Spectral Standards
• Smooth spectrum stars
have basically no
features (EG 131 is a
flux standard and has
no features) – a unique
star in the sky
Flux Standards
Spectral Reductions
• Cadillac treatment requires
– Bias frames
– Dark Frames
– Quartz Flats (removes high frequency CCD variations) at
each object if fringing
– Sky Flat (removes variations across slit)
– Flux standard stars at several airmasses
– Smooth spectrum standard stars at several airmasses
– ARCs at each position of each object and standard star
Minimum observing Set to get good
Spectral data
•
•
•
•
Bias (Dark if required)
ARC at each position
Observe object a parallactic angle
Smooth flux spectrum standard star at close
to same airmass as your program object.
(parallactic angle)
– Or smooth spectrum+flux standard
• Quartz at position if fringing and flexture
Imagers
• Imagers typically give a focal plane a the desired
scale (arcseconds per pixel) and with some field size.
• The key to a good imager is no scattered light
(telescope baffled), high throughput (little glass),
well characterised filters, and good CCDs. Many
imagers are parts of spectrographs (FORS), and
while convenient, it ultimately hampers their
performance compared to a dedicated instrument.
Taking an Image
• Point to object
• Do you need to guide (short exposure maybe not)
• How many counts do you need? SNR > 200 is a
waste as photometry of SN is always dominated by
other errors. But make sure you have several
comparison stars with this type of SNR as well.
• Multiple exposure on different places on the chip
are preferred over single long exposure – exception
is U band if you are read noise limited.
Basic Calibration Data imaging
• Bias images (10)
• Standard stars (Landolt are easiest at present)
– Covering colour range of your objects
– Airmass range if photometric
• Twilight flats (>=3 in each filter > 10000 counts in
each)
• (Dome flats are sometime useful)
• Dark images (>=3 of 1/2 hour each)
• Random data taken with instrument of pieces of
uninteresting sky.
BASIC CCD reduction
Most people find IRAF a good package to undertake
their Basic CCD reduction. However, do not be
scared to do it yourself!
Iraf: imred.ccdred.ccdproc
A good tutorial is available at
iraf.noao.edu/iraf/web/tutorials/tutorials.html
But do not necessarily believe this is the bible of data
reduction
Data Reduction -BIAS
•
•
•
Data
Data
overscan
•
•
Always take ~10 BIAS frames every
night to assess the state of your
instrument
Determine trim area (area which has
data you want)
Determine position of Overscan
Median 10 Bias frames together,
subtract overscan from image
(either a constant value for the
whole image, or a fit value (as a
function of y) as necessary.
Look at the ZERO frame. Is just
noise with Zero level? If so, you DO
NOT need to apply it (instead
simply subtract off the overscan
from each image). If it has structure,
then you will need to subtract this
from every frame along with the
overscan.
DATA Reduction - Flatfield
•
Each pixel has a slightly difference response from each other intrinsic to the CCD
(and due to the telescope + instrument), and this depends on colour, and
somewhat on time.
• Typically one takes images of the twilight sky (pictures off the dome are also) –
need 5000-40000 counts per image.
U, B, Z, V, I, R is the best order to take your flats in
in the evening
(reverse order in morning). Depending
on readout time of CCD, you may
only be able to do a subset.
• Remove overscan/BIAS of all images. For each colour, measure the median
value in a region near the centre of each image, and then scale each of the set so
this middle region has a value of 1. Then Take median of each the set of sky flat
images to create a flatfield for this colour.
• All data then first is trimmed, overscan subtracted, bias subtracted as necessary,
and then is divided through by the flatfield. This is generally your reduced data…
For spectra, need to use a lamp inside the spectrograph to get flatfield. But must
remove the wavelength shape of the lamp before dividing.
Imaging Illumination
Correction/Fringes
• After flatfielding your data, it is often found that their remains a
residual in the data. This is because the twilight and night sky are a bit
different.
• Create superflat: Take all of your data that has been reduced to flatfield
stage. Measure median value within a patch in the centre, and scale each
image to a common value (e.g. divide the average value for all images
by each images value). For each pixel, take median value over all images
(removing some fraction of the top values to avoid print through). This
requires >50 images to really work.
Alternatively mask all objects out of each image, and only median the
left over pixels in the stack of image.
A well constructed Superflat image will have no stellar print through, and
should reveal any imperfections in the twilight flat (illumination
correction) and fringes.
Illumination Correction
• Hopefully created superflat is essentially featureless. If it is not (and does
not have fringes), then you can either divide all the data by this frame (if
sufficient signal), or smooth the superflat by, for example, a 10 pixel
gaussian, and divide through by this.
• NOTE: If you are observing in dense stellar fields, this isn’t going to
work, it is impossible to get a superflat. If you are observing large
galaxies, DO NOT put them in the same part of CCD each time, or you
will not be able to make superflat.
For spectra. Take an spectrum of the twilight sky. This should be uniform
across the slit (but note the twilight sky is the spectrum of the sun, so
need to look at each spectra line and fit the slit profile as a function of
position on the CCD).
Fringe Correction- Imaging
• If you are observing in Z, I, and possibly R (or other filters beyond
700nm) the superflat may show fringing. Fringing is additive (imaging
data) because it is caused by the narrow night sky lines + the
transparency of the CCD at >700nm.
• Best is to fit a low order surface to the superflat and assume this is the
illumination correction. You will divide this through each image. What
remains is the fringe frame.
• Fringe frame needs to be scaled and subtracted from each image. Scaling
to first order is exposure time, but improvement can be done by
tweaking the scaling to minimize residuals in the image.
Fringing depends on the night sky, and this is often not stable even across a
night.
Thick deep depletion devices have much less fringing than thin devices, and
fringing gets worse as one moves to the red.
Fringing - Spectra
• Fringing is caused by the monochromatic light dispersed by
the spectrograph to undergo interference within the CCD.
• Since everything is affected, works more as a multiplication
effect.
For spectra - divide through by a smooth star, and ensure that you
place the standard at the same place as your program
objects. Challenging if spectrograph has lots of flexture.
- if lots of flexture, take a internal flat after each program
object and use this to flatfield each image – or ask for new
spectrograph!
Shutter Correction - Imaging
• Shutter correction: Short exposures often have a
different illumination than long exposures due to
the shutter.
• median 20x1 second exposure of dome/sky (scale
to same mean)
• Median 3x20s dome/sky (scale to same mean).
• Scale short and long image to same mean
• Shutter corr image = long/short
• Image correction to be multiplied through is
– Shutterimage /(exptime), [assuming long image is
perfect]
Absolute Photometry
Astronomers have
developed a peculiar
system of magnitudes
as their reference
system.
 S x ( ) F , ( )d 
mx  K x  2.5 log 

S
(

)
F
(

)
d

 x

Vega systems have !approximately! Kx defined such that Vega
has a magnitude of 0.03 in each band.
AB magnitudes use F in units of ergs/cm2/s/Hz with Kx
!approximately! Defined as -48.6.
In reality, all systems are defined as the observed magnitudes of
a set of standards. Landolt, Cousins, Gunn, etc, which
themselves are attempted to be tied to a fundamental standard
such as Vega, serious, or BD13….
Absolute Photometry at the
Telescope
• Typically need a photometric night (unless you have
standard stars on your CCD frame)
• Observe Standards over the relevant range of Airmasses,
and range of Colours of your program stars.
Find constants of the transformation of the form…
 counts 
1
2
3
mx  2.5 log 

K

K
(
X
)

K
x
x
x ( mx  m y ) 

 second 
K x4 (mx  m y ) X  K x5 (mx  m y ) 2  ...


Absolute Photometry at the
Telescope (2)
For example, for V, the form of the transformation is,
neglecting 2nd order terms.
 counts 
1
2
3
V  2.5 log 

K

K
(
X
)

K
V
V
V (B  V )

 second 
Which we solve for constants K1-K3, by minimizing



 counts 
1
2
3

Min  Vi   2.5 log 
 KV  KV ( X i )  KV ( B  V )i  

 second  i
all stars 


2
Where I have assumed that the observational uncertainty in each
observation is the same
Common Errors Photometry (1)
• Most people observe at at too large a range
of colour, or too large a range of Airmass,
and then apply linear corrections on their
program stars at neutral colour and low
airmass. Bracket your program stars
Airmasses and colours. If you must observe
at 2 airmasses, use 2nd order terms, and fit
your equations for objects at high airmass
only.
Common Errors Photometry (2)
• Landolt observed his stars in 14” radii. People
often use smaller apertures of each star.
Since, especially in redder bands, there are
often fainter stars, mags are systematically
off because of these non-included
interlopers. Either use apertures matched to
the standard star (thereby incurring
skynoise), or cull all stars with interlopers as
standards.
Aperture Photometry
• Aperture to maximise signal (r=0.75xFWHM
theoretical best), but then subject to variations of
PSF shape!
• Background fit in annulus around each star
– Often this is largest source of error, especially if on
complex galactic background
• Extend aperture to ∞ by determining mag
difference between small and large apertures for
bright stars (only for absolute photometry)
PSF photometry
• Model shape of a star, and fit this to each object. For each
star fit (X,Y, height of PSF, background). Collectively across
several stars fit shape.
– DAOPHOT (gaussian or other + sub-pixel residual)
– DOPHOT (truncated power-series of a gaussian)
For SN photometry applications, not so important to get PSF
exactly right, so no disadvantage of purely analytic PSF.
Extend PSF to ∞ aperture by determining mag difference
between PSF mag and large apertures for bright stars (only
for absolute photometry)
Beware of variable PSFs
Image subtraction
• For SN photometry, subtract off a template, and
perform photometry (aperture or PSF) with a
clean background.
• Match PSF with Drew Phillips Fourier-based
convolution (implemented in IRAF)
• Alard-Lupton least squares (sum of 3 gaussians)
• Gal Yams – convolve each image with the other
PSF.
Getting subtractions to work perfectly in all situations
is a bit of an art form.
Intrapixel Sensitivity
• CCD pixels vary in sensitivity across the pixel by ~20%. If > 2
pixel fwhm stars, then pixel is fully illuminated, and average
holds.
• If undersampled image (fwhm < 1.5 pixels), then brightness
of stars depends on where they hit in pixel.
• Pixel to pixel, the nature of the sensitivity is very uniform.
Do dithering with many exposures per field to mostly
eliminate. To do better, model the shape of the pixel, and
include it in the analysis.
see Lauer, T 1999PASP..111.1434L
Imaging Gotcha’s
• Is your CCD linear? – test by observing stars in
twilight
• Does your filter transform nicely to the standard
system – one photometric night, takes lots of
standards stars and really see how the equations
work
• Does your Imager deliver uniform photometry
across the focal plane
– CTE problems (check CRs if really bad)
– Scattered light mascquerading as Flatfield
Ultimate Check for Photometry
• Determining the uniformity of your photometry as a
function of position is hard. The best way to do it is to do a
stellar flat – that is, check photometry of stars as a function
of position.
Take a series of exposures of a relatively dense field (e.g. R149
field is nice) – doesn’t need to be a standard field, doesn’t
need to be photometric. Offset each exposure so as to place
your favourite set of stars into 16 different parts of the chip.
After fully flatfield, biased, fringed, etc., photometer all
bright stars on each image. Find average magnitude offset
for each image to bring it into consistency with the first
image, and calculate average relative mag for each star. Make
a table of X,Y,m-mave for each star on each image. Plot X/Y
versus m-mave and look for residuals.