Accurate calibration and debiasing

Accurate calibration and debiasing of DIMM results becomes
critical for site comparison and for prediction of telescope
seeing.
Early comparisons of telescope seeing with “DIMM seeing”
have shown their good agreement. Nowadays, the quality of
large telescopes has increased significantly, and they are, generally, nearly free from both optical defects and locally generated seeing. Image quality in telescopes such as the VLT is
sometimes found to be better than the “seeing” measured by
a local DIMM, calling for a better interpretation of DIMM
data. This article addresses the problem by taking into account
both DIMM biases and the effects of turbulence outer scale.
The article is structured as follows. In § 2, I review the
relation between differential image motion in DIMMs and
the Fried parameter within the framework of the standard r
0
(Kolmogorov) turbulence model and provide updated formulae
for DIMM data reductions. In § 3, I show how to correct DIMM
data for the two most important biases, finite exposure time
and centroiding noise. This section contains the results of numerical simulations of DIMM operation. Then in § 4, the image
size in large telescopes is addressed for the von Ka´rma´n turbulence model and an engineering formula to compute seeing
is suggested to replace the standard (infinite-scale) relationship.
The conclusions and practical recommendations are given in
§ 5. The Appendix gives additional insights into DIMM interactions with the atmospheric turbulence spectrum.
2. RELATIONSHIP BETWEEN DIFFERENTIAL
IMAGE MOTION AND r
0
2.1. Centering Algorithms
The theory of the DIMM assumes that the centers of images
formed by subapertures are defined as centers of gravity, also
called centroids, G-tilts (Tyler 1994), or angle-of-arrival fluctuations. They are related to wave-front gradients averaged over
subapertures. If is the light intensity distribution in a I(x, y)
stellar image at the telescope focal plane (with background
subtracted), its centroid in the x-direction, , is defined to be x
G

1
x
G
p I xI(x, y) dx dy, (1) tot

where Itot is the total flux (the integral of image intensity). For
a discrete (pixelized) CCD image, the integral is replaced by
the sum over all pixels.
The centroid is the optimum position estimator for a Gaussian image profile that gives the lowest possible error for a
perfect detector with only Poisson noise and zero readout noise
(Irwan & Lane 1999). However, even for a perfect diffraction
image, this is no longer the case because the intensity of image
wings does not decrease fast enough. The readout noise contributes additional centroid error. Hence, it is not possible to
compute true image centroids in either a DIMM or an S-H
wave-front sensor.
Two centroiding methods, thresholding and windowing, are
commonly used in practice. For thresholding, the integration
(or summation) in equation (1) is extended only over pixels
where , where is the peak intensity in the I(x, y) 1 gI I max max
image and g is the relative threshold. There is an alternative
way of thresholding: first, the threshold is subtracted from the
image, then the centroid is computed for nonnegative pixels:


x p   xI I Z
, (2) T ij ij ij


I 0 I 0
where . This kind of thresholding reduces the

I p I
gI ij ij max
weight of pixels in the image wings and is closer to an optimal
fit.
In thewindowingmethod, only the pixels within some radius
from image center are taken into account. The center itself
must be known, at least approximately. Usually, the coordinates
of the brightest pixel are taken as window center. The ∗ ∗ x , y
centroid is then x
W
x
W ij ij ij p   xI I Z
. (3)
window window
The centering window can be of any shape, but here I consider
only a circular window of radius r. It is reasonable to selectr
equal to the radius of the first dark ring in the Airy image,1158 TOKOVININ
2002 PASP, 114:1156–1166
Fig. 2.—Displacement of the image center computed by different methods
(without noise) as a function of the coma aberration. Solid line: true centroid;
dashed line: thresholding with ; g p 0.2 dash-dotted line: windowing with
r p 1.22l/D; dotted line: Gaussian fit.
1.22l/D rad, in order to minimize the effects of truncati