vda Velocity Detrended 4-Minute Averaged Image 
 vai Velocity 4-Minute Averaged Image 
 mai Modulation 4-Minute Averaged Image 
 iai Intensity 4-Minute Averaged Image

AVER detrends the calibrated velocity images; applies a 17-minute, moving-average, anti-alias filter to these detrended velocity images and to the calibrated velocity, modulation, and intensity images; and resamples these images to a four-minute cadence.

Before filtering and resampling, the input images are registered (using bi-cubic polynomial interpolation) to a common orientation, position, and size in sky (or camera) coordinates. The calibrated velocity images are detrended then registered. These detrended velocity images along with the velocity, modulation, and intensity images are then registered, filtered, and resampled to produce the vda, vai, mai, and iai images.

Registration corrects the pixel aspect ratio from rectangular to square which changes the apparent shape of the Sun from elliptical to circular. In the unregistered images, the position of the center of the Solar disk and the size of the disk can change temporally at one observing site and among the sites. If the camera rotator is off, the unregistered images will rotate temporally about the line of sight from the observer to the Sun. If the camera rotator is on, there can be a small, non-zero, pole-position angle from the alignment of the camera and the camera rotator. This pole-position angle can vary from site to site and at one site whenever the camera assembly is adjusted. Until mid-1996, the correction applied by AVER was determined from the reduction of the instrument drift scan data. After the mid-1996, the correction angles determined by ] COPIPE are used. The registered images are oriented so that the north pole of the Sun is aligned with the y-pixel axis (the axis associated with the second dimension of the pixel array). Pixel mirror operations in the x-direction are applied so that the east limb of the Sun is at low x-pixel indices and the sense of rotation of a time series of images is from low to high x-pixel indices. If these registered images are displayed on a monitor that displays the origin in the lower-left corner of the screen, the Solar images will appear to be circular disks with the north pole of the Sun on top and the sense of Solar rotation from left to right.

The calibrated velocity images are detrended using observer motion calculated from ephemeris data and a simple model for Solar rotation and limb shift. To compensate for effects that are not removed by the calibration, six parameters are fit to a subset (e.g., every fifth image) of the calibrated velocity images for each site day. Subsequently, the entire time series of images is detrended; the fit parameters are applied, the observer motion is removed, and the Solar rotation and limb shift are removed. This detrended velocity time series is then registered, filtered, and resampled.

The observer motion is calculated using a polynomial expansion for the J2000 ephemeris data. The location and velocity of the observer are determined from the observer's geographic position (latitude, longitude and elevation) and the time of the observation. The component of the Solar Doppler velocity that results from the observer's motion is the projection of the observer's velocity onto the line of sight from the observer to the Sun. The observer's motion includes the rotation of the Earth, the orbital motion of the Earth relative to the Sun, perturbations of the Earth's motion caused by the Earth's moon, and the light travel time delay between the Sun and the Earth. The observer motion correction consists of a constant, a velocity gradient in the x-direction (Solar equator) and a velocity gradient in the y-direction (Solar meridian). The scalar term is on the order of 500 m/s, the x-direction gradient is on the order of 100 m/s at the limb, the y-direction gradient is on the order of 25 m/s at the limb. Higher order terms in the observer motion contribute less than 1 m/s at the limb and are discarded.

The Solar rotation model includes solid body and differential rotation

$v(lat)\; =\; v$₀ + v₁ * sin(lat)² + v₂ * sin(lat)⁴

where:

lat = < the Solar latitude >
v₀, v₁, v₂ = < the Solar rotation model parameters in m/s >
v₀ = 1984
v₁ = -238
v₂ = -330

The limb shift model is

$v\; *\; (1.0\; -$cos(rho))^p,

where:

rho = < the angle measured from the center of the Sun between
the sub-earth point and a point on the surface of the Sun >
v = 631 m/s
p = 3

The anti-alias filter is a tapered Gaussian with a length of 17 minutes:

for $t\; =\; -T\; +\; 1,\; T\; -\; 1$
$wa(t\; +\; T)\; =\; e(-t2\; /\; (2.0\; *\; tau2))-\; (\; (1.0+(T2-\; t2)\; /\; (2.0\; *\; tau2))\; *\; e(-T2\; /\; (2.0\; *\; tau2)))$

where:

tau = < an adjustable parameter controlling the taper >
T = < length of the filter in pixels > / 2 + 1
wa = < filter coefficients >

Click here for a graph which displays the filter coefficients normalized such that the sum of the coefficients is one.

Follow this link for a graph which displays the amplitude spectrum of the anti-alias filter. The filter provides greater than 40 db (10^-2) of attenuation for all frequencies above 2.08333 mHz, the Nyquist frequency for four-minute sampling.

To compensate for the effects of lead-in, tail-out, and bad input images, each image output from the filter is normalized by the sum of the filter weights that are associated with the input images that contribute to the filtered output image.

If the sum of the filter weights associated with the good input images is less than half of the sum of the filter weights (1.0); then the filtered output image is considered to be bad. Bad output images are identified by adding the parameter, FILLED, to the FITS header.

Processing parameters are written into the FITS image headers. For the detrended, resampled velocity images, these include the bias and scale factors that adjust the average velocity, the three parameters from the observer's motion, the three parameters from the Solar rotation model, and the parameters from the Solar limb shift model. For all resampled images, these header parameters include the filter weights and the status of the images input to the filter operator.

Bad input images are those that contain the header parameter, FILLED, or that fail a threshold test that is applied to the VMICAL QA statistics. The test begins by fitting the spatially averaged modulation and velocity statistics from VMICAL to fourth-order polynomials. Then the RMS is computed from the residuals. If an averaged modulation or velocity value is greater than 4.0 times the RMS, the sample is rejected. The RMS is recomputed using the time series without the rejected samples and the threshold test is reapplied. The rejected samples from the velocity and modulation times series are combined into one list of bad images.

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