The AstroImaging Channel June 17 th 2018 Dr Gaston Baudat Innovations Foresight LLC 1 c Innovations Foresight 2016 Dr Gaston Baudat Why autoguiding 2 c Innovations Foresight 201 Dr Gaston Baudat ID: 708577
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Slide1
Introduction to optimal auto-guidingThe Astro-Imaging ChannelJune 17th, 2018Dr. Gaston BaudatInnovations Foresight, LLC
1
(c) Innovations Foresight 2016 - Dr. Gaston BaudatSlide2
Why auto-guiding?2(c) Innovations Foresight 201 - Dr. Gaston BaudatStay on target within a fraction of an arc-second all the time. - Could become challenging for long focal lengths (>1m). Correct for mount and/or model errors, such:- Drifts, noises, artefacts, flexures, accidents, unforeseen...
guide
star open
loop errorSlide3
Deterministic setup tracking errors(open loop)3(c) Innovations Foresight 2016 - Dr. Gaston BaudatPeriodic error PE (unless direct drive):Can be learned and partially corrected (PEC), high resolution encoders (on RA & DEC shafts)Polar alignment errors
and drift
:
Minimized by good alignment (
)
Limited by atmospheric refraction to about
Can be learned/predicated and partially corrected, sky model
Flexure (OTA, mount, focuser/accessories, pier, …)
Minimized with a rigid setup. Can be learned & partially corrected, sky model
Slide4
Random setup tracking errors (open loop)4(c) Innovations Foresight 2016 - Dr. Gaston BaudatMount gear mechanical noise after PEC:Random errors ~0.1” to 1” rms (bandwidth ~ 0.001Hz)Minimized with a good mount (almost gone with direct drive and/or high resolution encoders)
Wind burst, accidents (bumping mount, cables, mirror flop, …):
Minimized by dropping frames
Unforeseen (Mr. Murphy is very creative and works in team)
Minimized by dropping frames
All of those errors are
fully correlated across the all FOV
!Slide5
The different types of noise5(c) Innovations Foresight 2016 - Dr. Gaston BaudatA noise is defined by its distribution (Gaussian, Poisson, …), its bandwidth [Hz] and its rms value
(noise mean = 0)
A “white” noise has a larger bandwidth
relative to the sampling rate
, hence
. There is no correlation, nor predictability, between any sample
A “pink” noise has a narrower bandwidth
relative to the sampling rate
, hence
. There is some level of correlation/predictability between samples
Seeing, electronic, thermal & "shot" noise are often "white" noises, they are either weakly or not at all correlated across the FOV
Mount mechanical noise is usually a “pink” noise fully correlated across the all FOV
Slide6
Mount mechanical noise (after PEC, no drift)6(c) Innovations Foresight 2016 - Dr. Gaston BaudatLow frequency (“pink”) noise (RA in the plot below)(almost gone with high resolution encoders and/or direct drive)
Time constant
t
=
122 secondsSlide7
Seeing limited conditions7(c) Innovations Foresight 2016 - Dr. Gaston BaudatAstronomical seeing is the blurring of astronomical objects caused by Earth's atmosphere turbulenceIt impacts the intensity
(scintillation) and the
shape (phase) of the
incoming wave front
Scintillation is usually
not a major problem, at
least for exposures above one second. Phase is the main concern (wandering stars) since the wavefront tilt/tip contribution >85% of the total seeing phase variance
Star under seeing limited condition (short exposure << 1s)
Credit: John HayesSlide8
Wavefront and phase distortion(c) Innovations Foresight 2016 - Dr. Gaston Baudat8An incoming plane wave (star) is perturbed by the Earth atmosphere turbulent structure leading to phase errors.
l
l
l
l
l
lSlide9
The Fried’s parameter9(c) Innovations Foresight 2016 - Dr. Gaston BaudatThe Fried’s parameter is the average turbulence cell size
z
zenith angle,
the wavelength
and
is the atmospheric turbulence
strength at the altitude
h
.
Diffraction limited images can only be achieved with aperture sizes no more then few inches!
Diffraction limited
FWHM [“]
1
1.5
2
2.5
3
[mm/inch]
110 / 4.3
74 / 2.9
56 / 2.2
44 / 1.7
37 / 1.5
FWHM [“]
1
1.5
2
2.5
3
110 / 4.3
74 / 2.9
56 / 2.2
44 / 1.7
37 / 1.5Slide10
Seeing versus diffraction limit10(c) Innovations Foresight 2016 - Dr. Gaston Baudat is the equivalent diameter of a seeing limited scope of aperture D>. Therefore diffraction limited images can only be achieved with aperture sizes no more then few inches!
FWHM [“]
1
1.5
2
2.5
3
[mm/inch]
110 / 4.3
74 / 2.9
56 / 2.2
44 / 1.7
37 / 1.5FWHM [“]11.522.53110 / 4.374 / 2.956 / 2.244 / 1.737 / 1.5Diffraction limited
Seeing limited
Slide11
Aberrations and seeing11(c) Innovations Foresight 2016 - Dr. Gaston BaudatWave-front Zernike’s decompositionZernike’s polynomials: F. Zernike (1934)
The strongest seeing induced optical aberrations are on the lower-order Zernike’s modes, mainly the tilt/tip (wandering stars).
(Noll, 1976)
decreases as
Type
of a
berration
Phase variance contribution
Tilt/tip (wandering star)
~87%
Defocus
~2%
Astigmatism
~2%
Coma (3
rd
order)
~2%
Spherical (4
th
order)
<1%
Trefoild
(3
rd
order)
<1%Slide12
Isoplanatic patch12(c) Innovations Foresight 2016 - Dr. Gaston BaudatThe angle for which the total wavefront error remains almost the same (~l/6) is known as the isoplanatic angle: ~ 5km,
is usually few
arc-second across (@550nm):
= 50mm
~ 0.6”
= 200mm
~ 2.6”
increases as
Slide13
Effect of the isoplanatic angle on AO13(c) Innovations Foresight 2016 - Dr. Gaston BaudatAO operation is usually only effective in a very narrow FOV.
Credit R. Dekany, Caltec
Palomar AO system
IR bands: 1200nm, 1600nm, and 2200nm
Guide
star offset
[“]
FWHM
[“]
0
0.2
5.5
0.3
130.45-0.59230.51-0.68Slide14
Isokinetic patch(wavefront tilt/tip component)14(c) Innovations Foresight 2016 - Dr. Gaston BaudatThe angle for which the wavefront tilt/tip component error remains almost the same (~l/6) is known as the isokinetic angle:
~ 5km,
few arc-second across:
= 200mm (~8 inches)
~ 3”
= 1m (~40 inches)
~ 13”
Conclusions:
For most setups the seeing
is not correlated across the FOV
! (unless you have a very narrow FOV, arc-second wide)
uide
star behavior is not correlated with the target
Slide15
Seeing wavefront tilt/tip (wandering star) power spectrum15(c) Innovations Foresight 2016 - Dr. Gaston BaudatThe wavefront tilt/tip seeing component is the dominant effectThe tilt/tip component is a large (“white”) bandwidth noiseSlide16
The mechanical noise bandwidth is typical ~0.001Hz, or less, while the seeing (tilt/tip) noise bandwidth is ~10Hz, or more, a ratio ~10,000xBoth noises have different consequences for auto-guiding For guider exposures (sampling periods) ~
:
->
Sampled seeing noise remains an unpredictable “white” noise under all seeing conditions (good or poor):
->
Sampled mechanical noise remains a partially predictable “pink” noise, samples are similar from one to the next:
Mechanical and seeing noise bandwidths
16
(c) Innovations Foresight 2016 - Dr. Gaston Baudat Can not be corrected
Can be corrected
Slide17
Total open loop noise(PEC, accidents & drift removed)17(c) Innovations Foresight 2016 - Dr. Gaston BaudatThe mount mechanical noise and seeing noise are uncorrelated to each other, their variances and
add in quadrature.
Therefore the total tracking noise variance
(open loop) is:
The total tracking noise
rms
is then:
+
Slide18
Auto-guiding error (close loop) on a target18(c) Innovations Foresight 2016 - Dr. Gaston BaudatThe classical auto-guiding strategy calls for a mount (or AO-tilt/tip) correction c[n] proportional to the close loop error e[n
].
At the
n
th
guider frame the close loop correction is:
K
is known as the “aggressiveness”, usually
The guiding error (close loop) impacts the target image quality
The guiding error is function of mount/setup error & seeing
There are two basic parameters (“knobs”) to control it:
Guider exposure time
= correction period, usuallyAggressiveness K (one for RA and one for DEC)
Slide19
Understanding the auto-guiding(proportional control)19(c) Innovations Foresight 2016 - Dr. Gaston BaudatOne can use the Z-transform to derive the transfer function of a digital control system, which is similar to the
MTF in an optical system. It describes how a system responds to a disturbance
at different sampling times.
relates any disturbance/perturbation
applied to the mount/setup to the close loop error
,
after correction.
Slide20
Auto-guiding system stability(step response diverged)20(c) Innovations Foresight 2016 - Dr. Gaston Baudat stable without overshoot for
stable
with
overshoot for
unstable for
K
K
KSlide21
Auto-guiding analysis3 basic situations21(c) Innovations Foresight 2016 - Dr. Gaston BaudatTo understand how a basic auto-guiding algorithms acts on error let’s analyze its close loop response on 3 classical perturbations (this is done with its Z transform ).Step response:
A one time perturbation, a “bump”
Drift response:
A constant drift perturbation
Noise response
:
A random perturbation, “white”, or “pink” noise (average = 0)
P.S:
Under the linearly assumption the superposition theorem holds.
The total response is the sum of the individual responses.
Slide22
Auto-guidingThe step response22(c) Innovations Foresight 2016 - Dr. Gaston BaudatThe plots below show the typical step response (no noise):
decays exponentially from guider frame to frame (
n
).
The error decayed by
~63% after one time constant
t
:
= auto-guiding period, ex.
= 2s,
= 0.2,
9s
Close loop error
K
= 0.5
K
= 0.8Slide23
Auto-guiding
The drift response
23
(c) Innovations Foresight 2016 - Dr. Gaston Baudat
The plots below show the typical drift response (no noise):
increases with
n
, then settles
. Same
t
than for a step
The final close loop error
is (a constant bias): Close loop errorK = 0.5
K
= 0.8
= drift during
,
ex.
,
= 0.5,
2 pixels
Even if a constant drift may appear in the tracking rms error, it may not be a real issue, just an image offset
Slide24
ex.
,
= 0.8,
1.29 pixels
Auto-guiding
The “white” noise response
24
(c) Innovations Foresight 2016 - Dr. Gaston Baudat
The plots below show the response to a “white” (broadband) noise of variance
(
=
rms value):The error is a noise too with
, its
rms
value is:
Close loop error
K
=0.8
K
= 0.8Slide25
Auto-guidingThe “pink” noise response25(c) Innovations Foresight 2016 - Dr. Gaston BaudatThe plots below show the response to a typical “pink” noise of variance (
= rms
value ):
The stronger the noise correlation the smaller the close loop error for the same
.
The mathematics are more complex than for a “white” noise but trackable.
K
=0.8
K
=0.8
Slide26
Optimal auto-guiding26(c) Innovations Foresight 2016 - Dr. Gaston BaudatStepwise perturbations are eventually fully corrected within few time constant
, usually few guider frames.
Drift perturbations eventually settle to a quasi constant close loop error
within few
, usually few guider frames.
The above conditions would lead to a large
(
1), but:
Noises are the main concern,
and
must be chosen wisely:
Optimal auto-guiding aims at
minimizing the total close loop noise rms value on a target, in other words: Given a mount performance (
) & local seeing (
)
what should be the best
and
values for minimizing
?
Slide27
Auto-Guiding System Proportional Corrector
Auto-guiding loop: The big picture
27
(c) Innovations Foresight 2016 - Dr. Gaston Baudat
Actual Mount & Setup
+
+
Perfect
Mount & Setup
-
+
+
+
Guider
+
Centroid
Slide28
Target error on imagerFinal FWHM28(c) Innovations Foresight 2016 - Dr. Gaston BaudatImager
Assumptions/Validity:
Imager exposure time >> mount time constant >> 1 minute typically
Guider exposure time ~
Seeing limited condition
Under average seeing 2.5”
Target outside the guide star isokinetic patch
Under average seeing 2.5”
[“],
D
in meter
Mount close loop and target seeing errors add in quadrature
+
+
Slide29
Effect of guider exposure for
29
(c) Innovations Foresight 2016 - Dr. Gaston Baudat
The guider sensor integrates (averages) the noise during exposure. Acting as a low pass filter with cut-off frequency
:
Mechanical noise bandwidth is typical around 0.001Hz, hence essentially left untouched (unfiltered) by the guider,
Seeing (tilt/tip) noise bandwidth is typically around 10Hz, or more, hence low pass filtered by the guider,
Those two very different bandwidths provide a way to filter the seeing, which cannot be corrected, while correcting, at least partially, the mechanical noise leading to optimal guiding.
Slide30
Effect of guider exposure on seeing power spectrum 30(c) Innovations Foresight 2016 - Dr. Gaston BaudatLonger guider exposures
lead to lower seeing
rms
contribution values on auto-guiding
(
)
Seeing limit,
= 0
= 0.1s
= 1s
= 10s
Seeing limit,
= 0
Slide31
Guiding rms error contributions v.s. Mid-range mount under average seeing 31(c) Innovations Foresight 2016 - Dr. Gaston BaudatMid-range mount with 4” peak-peak, after PEC, K
=1
Longer guider exposures
lead to lower seeing rms contribution while increasing the mount rms contribution.
(
)
Slide32
Guiding rms error contributions v.s. High-end mount under average seeing 32(c) Innovations Foresight 2016 - Dr. Gaston BaudatHigh-end mount with 1” peak-peak, after PEC, K
=1
Under identical conditions a mount with a lower tracking error performs better at longer guider exposure time values
.
(
)
Asymptotic mount errorSlide33
Aggressiveness & close loop target FWHM
mid-range mount (4” peak-peak, after PEC)
33
(c) Innovations Foresight 2016 - Dr. Gaston Baudat
For a given mount, seeing &
, the close loop target FWHM exbibits a minimum value for some
Slide34
Aggressiveness & close loop target FWHMhigh-end mount (1” peak-peak, after PEC)34(c) Innovations Foresight 2016 - Dr. Gaston BaudatA lower mount error leads to smaller close loop target FWHM under same seeing. Most guide exposures give the same result.
Improvement between 1s and 10s = ~0.02”Slide35
Target FWHM versus guider exposure time and aggressiveness K 35(c) Innovations Foresight 2016 - Dr. Gaston Baudat
Seeing = 2.5”, mid-range mount = 4” peak-peak (after PEC)
is not recommended (prone to scintillation/aberration)
Not recommended
+ 1 mag.
+ 2 mag.
+ 2.5 mag.Slide36
Open loop seeing error scatter plot36(c) Innovations Foresight 2016 - Dr. Gaston BaudatPerfect mount open loop error scatter plot (100 samples).SNR=6 dB (2x), 4 stars (same mag.), seeing 2 pixel rms.Red diamond: One star centroid.Green dot: Full frame guiding ADIC (uses the all frame). Blue dot: Multi-star centroids (uses 4 star centroids).Slide37
Close loop target FWHM v.s.information in guider FOVmid-range mount (4” peak-peak, after PEC)37(c) Innovations Foresight 2016 - Dr. Gaston BaudatMore information (like many stars) reduces guider seeing rms error
improving target FWHM
Target FWHM for various number of guide star in the guider FOV (same mag.)
One guide star
Four guide stars
Many stars,
0
Slide38
Lucky imaging and seeing38(c) Innovations Foresight 2016 - Dr. Gaston BaudatSome time a short exposure image can be close to the diffraction limit. The probability P for a rms phase error at, or below, ~l/6 (one radian) is (Fried 1977):
>3.5, t<<
,
inside isoplanatic patch
Example:
D=279mm (11”)
=50mm @550nm
0.04 (~1/25 frame)
=84mm @850nm
>0.7 (~18/25 frame)
Slide39
M83 with full spectrum guiding (OAG)NIR guiding helps reducing open loop seeing error contribution
39
(c) Innovations Foresight 2015 - Dr. Gaston Baudat
39
(c) Innovations Foresight 2016 - Dr. Gaston Baudat
Mario Motta’s relay telescope 32” (0.8m) @ f/6
M86 images with STL11000 + AO-L (OAG or ONAG)
Challenge:
FWHM>3”,
,
windy, near horizon (in purpose)
M83 with NIR guiding >750nm (ONAG)
FWHM improvement ~40% before processingSlide40
Optimal-guiding calculator(Excel spreadsheet)40(c) Innovations Foresight 2016 - Dr. Gaston BaudatAn optimal guiding calculator can be downloaded from here: https://www.innovationsforesight.com/support/download/Slide41
Clear skies!Thank you!41(c) Innovations Foresight 2016 - Dr. Gaston Baudat
Innovations
Foresight, LLC
A s t r o n o m y