AIPS HELP file for BDEPO in 31DEC24
As of Sat Apr 20 0:01:13 2024
BDEPO: Task to find beam depolarization from RM gradient
INPUTS
INNAME First RM grad image name
INCLASS First RM grad image class
INSEQ 0.0 9999.0 First RM grad image seq. no.
INDISK 0.0 9.0 First RM grad image disk no.
IN2NAME Second RM grad image name
IN2CLASS Second RM grad image class
IN2SEQ 0.0 9999.0 Second RM grad image seq. no.
IN2DISK 0.0 9.0 Second RM grad image disk no.
BLC BLC of input image(s)
TRC TRC of input image(s)
OUTNAME Output DPR image name
OUTCLASS Output DPR image class
OUTSEQ 0.0 9999.0 Output DPR image seq. #
OUTDISK 0.0 9.0 Output DPR image disk drive
CPARM User supplied parameters
HELP SECTION
BDEPO
Type: Task
Use: Depolarization can be produced if the Rotation Measure
gradient is sufficiently large across the synthesized
beam. BDEPO tries to predict the depolarization ratio
(DPR) between two frequencies owing to a linear
Rotation Measure (RM) gradient across the beam. The
gradient image(s) may be calculated from your RM image
(with NINER), or may be models of a foreground screen
that you have generated in some other way. The output
DPR image can be plotted (with IMVIM) against an observed
DPR image in order to asses the importance of beam
depolarization. The observed DPR should be m(F1)/m(F2)
where F1 < F2 are the frequencies, and m is the degree of
polarization.
It assumed that the intrinsic degree of polarization is
constant across the beam. A beam of arbitrary
orientation must be present in the header and you must
supply the two frequencies between which to calculate the
DPR. If the RM structure is still possibly unresolved in
your highest resolution RM image, you will run the risk
of underpredicting the beam depolarization.
This task runs in 2 modes. First, the more general case
is treated if you input 2 RM gradient images. The first
must be the x (RA) direction gradient, and the second the
y (DEC) direction gradient. The second mode is invoked
if you input only one RM gradient image. In this case it
is assumed that the x and y components of the RM gradient
are equal and given by the input image.
The units of the gradient images should be reduceable to
RAD/M/M/PIXEL by scale factors which you may supply via
the CPARM(3) and CPARM(4) parameters. For example,
consider BDEPO working in the second mode mentioned
above. A generalised gradient image can be produced by
NINER with opcode 'SOBL'. If DPARM(1) = 0, NINER makes a
gradient of the form :
G = 8 * (g_x**2 + g_y**2)**0.5
where g_x and g_y are x and y directional derivatives.
Now, BDEPO assumes g = g_x = g_y so that
g = G / (8 * sqrt(2)). Thus, a total scale factor of
CPARM(3) = 1 / (8 * sqrt(2)) would be required. This
factor is built in as the default when CPARM(3) = 0 and
only one input image is specified.
If two input images are specified, CPARM(3,4) = 0 =>
CPARM(3,4) = 1.0
Adverbs:
INNAME........First (generalised or x-axis) RM gradient image
name
INCLASS.......First RM gradient image class
INSEQ.........First RM gradient image seq. #
INDISK........First RM gradient image disk drive #
IN2NAME.......Second (y-axis) RM gradient image name
IN2CLASS......Second RM gradient image class
IN2SEQ........Second RM gradient image seq. #
IN2DISK.......Second RM gradient image disk drive #
BLC...........BLC of input image(s)
TRC...........TRC of input image(s)
OUTNAME.......Output predicted DPR image name
OUTCLASS......Output predicted DPR image class
OUTSEQ........Output predicted DPR image seq. #
OUTDISK.......Output predicted DPR image disk drive #
CPARM.........Frequencies (Hz) at which to calculate the DPR
are stored in CPARM(1) and (2). The order is
unimportant. CPARM(3) and CPARM(4) are scale
factors by which the gradient images are
multiplied. If there is no second image,
CPARM(3) = 0 => invoke scale factor appropriate to
a NINER 'SOBL' gradient image. If there is a
second image, then both CPARM(3) and CPARM(4) = 0
give scale factors of unity.
EXPLAIN SECTION
TASK : BDEPO
DOCUMENTOR : Neil Killeen, NRAO
RELATED PROGRAMS : RM, COMB, NINER, IMVIM
PURPOSE
-----------
Depolarization can be produced by differential rotation across
the beam owing to a Rotation Measure gradient. This rotation
can occur anywhere along the line of sight. Thus, "beam"
depolarization is distinguished from "internal" depolarization
in which the rotating material is mixed in with the emitting
material. BDEPO tries to predict the beam depolarization
ratio (DPR) between two frequencies owing to a linear Rotation
Measure (RM) gradient across the beam. The depolarization ratio
is defined as DPR = m(F1) / m(F2) where m is the degree of
polarization at the freqeuncy F. The convention used in this
program is that F1 < F2 so that, in the absence of noise, DPR <
1.0. The output DPR image can be plotted (with IMVIM) against
an observed DPR image in order to asses the importance of beam
depolarization. The gradient image(s) may be calculated from
your RM images (with NINER), or perhaps may be models of a
foreground screen that you have generated in some other way.
Note that because we generally make images with about 3 pixels
per beam, you should set the XINC and YINC adverbs in IMVIM to
2 or 3 so that the plotted points are more or less independent
of each other. Also. if you use NINER, remember that it
operates on your image with a 3 x 3 matrix, so that there is a
loss of independence between points here as well.
BDEPO assumes that the intrinsic degree of polarization is
constant across the beam. If there is unresolved and
significant structure in your RM gradient image, then BDEPO
will underpredict the beam depolarization.
MODE 1:
Let us call the x (RA) and y (DEC) components of the RM gradient
g_x and g_y. If we also assume that m is constant across the
beam, then the DPR between wavelengths L_1 > L_2 is :
-- -- ----
|-g_x**2 * (L_1**4 - L_2**4) | [A * g_y/g_x + C]**2| |
DPR=exp|--------------------------- * |1+--------------------| |
| 4 * ln2 * A | [A*B - C**2] | |
-- -- ----
where
a**2 B_min**2 + b**2 B_maj**2
A = -----------------------------
B_maj**2 B_min**2
a**2 B_maj**2 + b**2 B_min**2
B = -----------------------------
B_maj**2 B_min**2
ab (B_maj**2 - b**2 B_min**2)
C = -----------------------------
B_maj**2 B_min**2
B_maj = FWHM of the beam major axis
B_min = FWHM of the beam minor axis
a = cos(theta)
b = sin(theta)
theta = angle between beam major axis and positive X-axis
measured positive counter clockwise, and with
0-180 degree limits.
This is BDEPO's most general mode. The x and y gradient images
should be input as the first and second images, respectively.
Before the DPR is calculated, g_x and g_y (as read directly from
the input images) can be multipled by factors stored in CPARM(3)
and CPARM(4), respectively. That is,
g_x = CPARM(3) * First_RM_gradient_image_value
g_y = CPARM(4) * Second_RM_gradient_image_value
If these CPARMS are 0.0 on starting up BDEPO, they default to
1.0. They might be used to account for weighting in the input
images, or conversion to radians from degrees. After
application of the scale factors, the units of the images must
be RAD/M/M/PIXEL.
In the case that the beam is round, this expression simplifies
to :
-- --
| - B_maj**2 * (g_x**2 + g_y**2) * (L_1**4 - L_2**4) |
DPR = exp | -------------------------------------------------- |
| 4 * ln2 |
-- --
MODE 2:
In the case that g = g_x = g_y and the beam is not round, we
have :
-- -- -- --
| - g**2 * (L_1**4 - L_2**4) | [A + C]**2 | |
DPR= exp| ----------------------- * |1 + ------------ | |
| 4 * ln2 * A | [A*B - C**2] | |
-- -- -- --
To invoke this mode in BDEPO, supply only the first gradient
image. The scale factor CPARM(3) must then reduce the units to
RAD/M/M/PIXEL
g = CPARM(3) * First_RM_gradient_image_value
If the beam is round, then :
-- --
| - (g * B_maj)**2 * (L_1**4 - L_2**4) |
DPR = exp | ------------------------------------ |
| 2 * ln2 |
-- --
As an example, consider the output produced by NINER with opcode
'SOBL' and DPARM(1) = 0.0. This is a generalised gradient
formed from a 3 x 3 matrix as follows :
| A1 A2 A3 |
| A8 A0 A4 |
| A7 A6 A5 |
NINER computes G = sqrt (X**2 + Y**2)
where X = (A3 + 2A4 + A5) - (A1 + 2A8 + A7)
Y = (A1 + 2A2 + A3) - (A7 + 2A6 + A5)
Now, we can associate X with g_x and Y with g_y such that
g_x ~ X / (4*2) and g_y ~ Y / (4*2)
The factor of 4 for weighting, the factor of two because a
gradient = DELTA Y / DELTA X and DELTA X = 2 pixels so that
G ~ 8 sqrt (g_x**2 + g_y**2)
Now in MODE 2, BDEPO assumes g = g_x = g_y so that
G ~ 8 * sqrt (2) * g
Thus, the scale factor in BDEPO input via CPARM(3) that is
necessary for a NINER 'SOBL' (DPARM(1)=0)) gradient image is
1 / (8 sqrt(2))
This is the MODE 2 default scale factor applied when CPARM(3)=0
Other parameters:
BLC, TRC : The usual windows for the input images. These
default to the entire image
CPARM(1 & 2) : Specify the frequencies (Hz) between which to
calculate the DPR. The order is unimportant and
there are no defaults.