Table of contents
Deprecated: This routine has been superseded by the Mice routines
cspice_ilumin, cspice_illumg and cspice_illumf. This routine is
supported for purposes of backward compatibility only.
CSPICE_ILLUM_PL02 returns the illumination angles---phase, solar
incidence, and emission---at a specified point on a target body at a
particular epoch, optionally corrected for light time and stellar
aberration. The target body's surface is represented by a triangular
plate model contained in a type 2 DSK segment.
Given:
handle the DAS file handle of a DSK file open for read
access.
[1,1] = size(handle); int32 = class(handle)
This kernel must contain a type 2 segment that provides a
plate model representing the entire surface of the target
body.
dladsc the DLA descriptor of a DSK segment representing
the surface of a target body.
[SPICE_DLA_DSCSIZ,1] = size(dladsc); int32 = class(dladsc)
target the name of the target body.
[1,c1] = size(target); char = class(target)
or
[1,1] = size(target); cell = class(target)
`target' is case-insensitive, and leading and trailing blanks
in `target' are not significant. Optionally, you may supply
a string containing the integer ID code for the object.
For example both 'MOON' and '301' are legitimate strings
that indicate the moon is the target body.
This routine assumes that the target body's surface is
represented using a plate model, and that a DSK file
containing the plate model has been loaded via cspice_dasopr.
et the epoch, represented as seconds past J2000 TDB, at
which the illumination angles are to be computed.
[1,1] = size(et); double = class(et)
When aberration corrections are used, `et' refers to the
epoch at which radiation is received at the observer.
abcorr indicates the aberration corrections to be applied to
the position and orientation of the target body and the
position of the Sun to account for one-way light time
and stellar aberration.
[1,c2] = size(abcorr); char = class(abcorr)
or
[1,1] = size(abcorr); cell = class(abcorr)
See the discussion in the -Particulars section for
recommendations on how to choose aberration corrections.
`abcorr' may be any of the following:
'NONE' Apply no correction. Use the geometric
positions of the Sun and target body
relative to the observer; evaluate the
target body's orientation at `et'.
The following values of `abcorr' apply to the
"reception" case in which photons depart from the
target's location at the light-time corrected epoch
et-lt and *arrive* at the observer's location at
`et':
'LT' Correct for one-way light time (also
called "planetary aberration") using a
Newtonian formulation. This correction
uses the position and orientation of the
target at the moment it emitted photons
arriving at the observer at `et'. The
position of the Sun relative to the
target is corrected for the one-way light
time from the Sun to the target.
The light time correction uses an
iterative solution of the light time
equation (see -Particulars for details).
The solution invoked by the 'LT' option
uses one iteration.
'LT+S' Correct for one-way light time and stellar
aberration using a Newtonian formulation.
This option modifies the positions
obtained with the 'LT' option to account
for the observer's velocity relative to
the solar system barycenter (note the
target plays the role of "observer" in the
computation of the aberration-corrected
target-Sun vector). The result is that the
illumination angles are computed using
apparent position and orientation of the
target as seen by the observer and the
apparent position of the Sun as seen by
the target.
'CN' Converged Newtonian light time correction.
In solving the light time equation, the
'CN' correction iterates until the
solution converges (three iterations on
all supported platforms).
'CN+S' Converged Newtonian light time
and stellar aberration corrections.
obsrvr the name of the observing body.
[1,c3] = size(obsrvr); char = class(obsrvr)
or
[1,1] = size(obsrvr); cell = class(obsrvr)
This is typically a spacecraft, the earth, or a surface point
on the earth. `obsrvr' is case-insensitive, and leading and
trailing blanks in `obsrvr' are not significant. Optionally,
you may supply a string containing the integer ID code for
the object. For example both 'EARTH' and '399' are
legitimate strings that indicate the earth is the
observer.
spoint a surface point on the target body, expressed in
rectangular body-fixed (body equator and prime meridian)
coordinates.
[3,1] = size(spoint); double = class(spoint)
`spoint' need not be visible from the observer's location at
time `et'.
the call:
[phase, solar, emissn] = cspice_illum_pl02( handle, dladsc, target, ...
et, abcorr, obsrvr, ...
spoint )
returns:
For all of the angles below, if `spoint' does not lie on
one of the *exterior* plates comprising the DSK type 2
surface representation, the "intercept" style
"sub-observer point" corresponding to `spoint' is used
in the illumination angle computations in place of
`spoint'. The selected point will always be on the
*outermost* plate intersected by a ray emanating from
the target body's center and passing through `spoint'.
See the header of CSPICE_SUBPT_PL02 for details
concerning the definition of the sub-observer point.
In all cases, the normal vector is taken from the plate
on which the sub-point corresponding to `spoint' lies.
If this sub-point lies on an edge or vertex, a normal
vector for one of the bordering plates is selected.
phase the phase angle at `spoint', as seen from `obsrvr' at
time `et'.
[1,1] = size(phase); double = class(phase)
This is the angle between the spoint-obsrvr vector and the
spoint-sun vector. Units are radians. The range of `phase'
is [0, pi].
See -Particulars below for a detailed discussion of the
definitions of this angle.
solar the solar incidence angle at `spoint', as seen from
`obsrvr' at time `et'.
[1,1] = size(solar); double = class(solar)
This is the angle between the surface normal vector at
`spoint' and the spoint-sun vector. Units are radians.
The range of `solar' is [0, pi].
See -Particulars below for a detailed discussion of the
definitions of this angle.
emissn the emission angle at `spoint', as seen from `obsrvr'
at time `et'.
[1,1] = size(emissn); double = class(emissn)
This is the angle between the surface normal vector at
`spoint' and the spoint-obsrvr vector. Units are radians.
The range of `emissn' is is [0, pi].
See -Particulars below for a detailed discussion of the
definitions of this angle.
None.
Any numerical results shown for this example may differ between
platforms as the results depend on the SPICE kernels used as input
and the machine specific arithmetic implementation.
1) Find the illumination angles at both the sub-observer point and
sub-solar point on Phobos as seen from Mars for a specified
sequence of times. Perform each computation twice, using both the
"intercept" and "ellipsoid near point" options for the sub-observer
point and sub-solar point computations. Compute the corresponding
illumination angles using an ellipsoidal surface for comparison.
(Note that the surface points on the plate model generally will
not lie on the ellipsoid's surface, so the emission and solar
incidence angles won't generally be zero at the sub-observer
and sub-solar points, respectively.)
Use the meta-kernel shown below to load the required SPICE
kernels.
KPL/MK
File: illum_pl02_ex1.tm
This meta-kernel is intended to support operation of SPICE
example programs. The kernels shown here should not be
assumed to contain adequate or correct versions of data
required by SPICE-based user applications.
In order for an application to use this meta-kernel, the
kernels referenced here must be present in the user's
current working directory.
The names and contents of the kernels referenced
by this meta-kernel are as follows:
File name Contents
--------- --------
mar097.bsp Mars satellite ephemeris
pck00010.tpc Planet orientation and
radii
naif0010.tls Leapseconds
\begindata
KERNELS_TO_LOAD = ( 'mar097.bsp',
'pck00010.tpc',
'naif0010.tls' )
\begintext
End of meta-kernel
Use the DSK kernel below to provide the plate model representation
of the surface of Phobos.
phobos_3_3.bds
Example code begins here.
function illum_pl02_ex1( meta, dsknam )
%
% Constants
%
NCORR = 2;
NSAMP = 3;
NMETHOD = 2;
FIXREF = 'IAU_PHOBOS';
ILUM_METHOD = 'ELLIPSOID';
TOL = 1.d-12;
%
% Initial values
%
abcorrs = { 'NONE', 'CN+S' };
methods = { 'Intercept', 'Ellipsoid near point' };
obsrvr = 'Mars';
target = 'Phobos';
%
% Load the meta kernel.
%
cspice_furnsh( meta )
%
% Open the DSK file for read access.
% We use the DAS-level interface for
% this function.
%
handle = cspice_dasopr( dsknam );
%
% Begin a forward search through the
% kernel, treating the file as a DLA.
% In this example, it's a very short
% search.
%
[dladsc, found] = cspice_dlabfs( handle );
if ~found
%
% We arrive here only if the kernel
% contains no segments. This is
% unexpected, but we're prepared for it.
%
fprintf( 'No segments found in DSK file %s\n', dsknam )
return
end
%
% If we made it this far, `dladsc' is the
% DLA descriptor of the first segment.
%
% Now compute sub-points using both computation
% methods. We'll vary the aberration corrections
% and the epochs.
%
et0 = 0.0;
stepsize = 1.d6;
for i = 0:(NSAMP-1)
%
% Set the computation time for the ith sample.
%
et = et0 + i * stepsize;
timstr = cspice_timout( et, ...
'YYYY-MON-DD HR:MN:SC.### ::TDB(TDB)' );
fprintf( '\n\nObservation epoch: %s\n', timstr )
for coridx = 1:NCORR
abcorr = abcorrs( coridx );
fprintf( ' abcorr = %s\n', char(abcorr) );
for midx = 1:NMETHOD
%
% Select the computation method.
%
method = methods( midx );
fprintf( '\n Method =%s\n ', char(method) )
%
% Compute the sub-observer point using a plate
% model representation of the target's surface.
%
[xpt, alt, plid] = cspice_subpt_pl02( handle, dladsc, ...
method, target, ...
et, abcorr, ...
obsrvr );
%
% Compute the illumination angles at the
% sub-observer point.
%
[phase, solar, emissn] = cspice_illum_pl02( handle, ...
dladsc, target, et, ...
abcorr, obsrvr, xpt );
%
% Represent the surface point in latitudinal
% coordinates.
%
[ xr, xlon, xlat] = cspice_reclat( xpt );
fprintf( ...
'\n Sub-observer point on plate model surface:\n' )
fprintf( ...
' Planetocentric Longitude (deg): %f\n', ...
xlon * cspice_dpr() )
fprintf( ...
' Planetocentric Latitude (deg): %f\n', ...
xlat * cspice_dpr() )
fprintf( ...
'\n Illumination angles derived using a\n' )
fprintf( ' plate model surface:\n' )
fprintf( ...
' Phase angle (deg): %f\n', ...
phase * cspice_dpr() )
fprintf( ...
' Solar incidence angle (deg): %f\n', ...
solar * cspice_dpr() )
fprintf( ...
' Emission angle (deg): %f\n\n', ...
emissn * cspice_dpr() )
%
% Compute the illumination angles using an ellipsoidal
% representation of the target's surface. The role of
% this representation is to provide an outward surface
% normal.
%
[trgepc, srfvec, phase, solar, emissn] = ...
cspice_ilumin( ILUM_METHOD, ...
target, et, FIXREF, ...
abcorr, obsrvr, xpt);
fprintf( ...
' Illumination angles derived using an\n' )
fprintf( ' ellipsoidal reference surface:\n' )
fprintf( ...
' Phase angle (deg): %f\n', ...
phase * cspice_dpr() )
fprintf( ...
' Solar incidence angle (deg): %f\n', ...
solar * cspice_dpr() )
fprintf( ...
' Emission angle (deg): %f\n\n', ...
emissn * cspice_dpr() )
%
% Now repeat our computations using the
% sub-solar point.
%
% Compute the sub-solar point using a plate model
% representation of the target's surface.
%
[xpt, dist, plid] = cspice_subsol_pl02( handle, dladsc, ...
method, target, ...
et, abcorr, ...
obsrvr );
%
% Compute the illumination angles at the
% sub-solar point.
%
[phase, solar, emissn] = cspice_illum_pl02( handle, ...
dladsc, target, et, ...
abcorr, obsrvr, xpt );
%
% Represent the surface point in latitudinal
% coordinates.
%
[ xr, xlon, xlat] = cspice_reclat( xpt );
fprintf( ' Sub-solar point on plate model surface:\n' )
fprintf( ...
' Planetocentric Longitude (deg): %f\n', ...
xlon * cspice_dpr() )
fprintf( ...
' Planetocentric Latitude (deg): %f\n', ...
xlat * cspice_dpr() )
fprintf( ...
'\n Illumination angles derived using a\n' )
fprintf( ' plate model surface:\n' )
fprintf( ...
' Phase angle (deg): %f\n', ...
phase * cspice_dpr() )
fprintf( ...
' Solar incidence angle (deg): %f\n', ...
solar * cspice_dpr() )
fprintf( ...
' Emission angle (deg): %f\n\n', ...
emissn * cspice_dpr() )
%
% Compute the illumination angles using an ellipsoidal
% representation of the target's surface. The role of
% this representation is to provide an outward surface
% normal.
%
[trgepc, srfvec, phase, solar, emissn] = ...
cspice_ilumin( ILUM_METHOD, ...
target, et, FIXREF, ...
abcorr, obsrvr, xpt);
fprintf( ...
' Illumination angles derived using an\n' )
fprintf( ' ellipsoidal surface:\n' )
fprintf( ...
' Phase angle (deg): %f\n', ...
phase * cspice_dpr() )
fprintf( ...
' Solar incidence angle (deg): %f\n', ...
solar * cspice_dpr() )
fprintf( ...
' Emission angle (deg): %f\n\n', ...
emissn * cspice_dpr() )
end
end
end
%
% Close the DSK file. Unload all other kernels as well.
%
cspice_dascls( handle )
%
% It's always good form to unload kernels after use,
% particularly in Matlab due to data persistence.
%
cspice_kclear
When this program was executed on a Mac/Intel/Octave6.x/64-bit
platform, with the following variables as inputs
meta = 'illum_pl02_ex1.tm';
dsknam = 'phobos_3_3.bds';
the output was:
Observation epoch: 2000-JAN-01 12:00:00.000 (TDB)
abcorr = NONE
Method =Intercept
Sub-observer point on plate model surface:
Planetocentric Longitude (deg): -0.348118
Planetocentric Latitude (deg): 0.008861
Illumination angles derived using a
plate model surface:
Phase angle (deg): 101.596824
Solar incidence angle (deg): 98.376877
Emission angle (deg): 9.812914
Illumination angles derived using an
ellipsoidal reference surface:
Phase angle (deg): 101.596824
Solar incidence angle (deg): 101.695444
Emission angle (deg): 0.104977
Sub-solar point on plate model surface:
Planetocentric Longitude (deg): 102.413905
Planetocentric Latitude (deg): -24.533127
Illumination angles derived using a
plate model surface:
Phase angle (deg): 101.665306
Solar incidence angle (deg): 13.068798
Emission angle (deg): 98.408735
Illumination angles derived using an
ellipsoidal surface:
Phase angle (deg): 101.665306
Solar incidence angle (deg): 11.594741
Emission angle (deg): 98.125499
Method =Ellipsoid near point
Sub-observer point on plate model surface:
Planetocentric Longitude (deg): -0.264850
Planetocentric Latitude (deg): 0.004180
Illumination angles derived using a
plate model surface:
Phase angle (deg): 101.596926
Solar incidence angle (deg): 98.376877
Emission angle (deg): 9.812985
Illumination angles derived using an
ellipsoidal reference surface:
Phase angle (deg): 101.596926
Solar incidence angle (deg): 101.593324
Emission angle (deg): 0.003834
Sub-solar point on plate model surface:
Planetocentric Longitude (deg): 105.857346
Planetocentric Latitude (deg): -16.270558
Illumination angles derived using a
plate model surface:
Phase angle (deg): 101.663675
Solar incidence angle (deg): 16.476730
Emission angle (deg): 118.124981
Illumination angles derived using an
ellipsoidal surface:
Phase angle (deg): 101.663675
Solar incidence angle (deg): 0.422781
Emission angle (deg): 101.541470
abcorr = CN+S
Method =Intercept
Sub-observer point on plate model surface:
Planetocentric Longitude (deg): -0.348101
Planetocentric Latitude (deg): 0.008861
Illumination angles derived using a
plate model surface:
Phase angle (deg): 101.592246
Solar incidence angle (deg): 98.372348
Emission angle (deg): 9.812902
Illumination angles derived using an
ellipsoidal reference surface:
Phase angle (deg): 101.592246
Solar incidence angle (deg): 101.690861
Emission angle (deg): 0.104971
Sub-solar point on plate model surface:
Planetocentric Longitude (deg): 102.408894
Planetocentric Latitude (deg): -24.533381
Illumination angles derived using a
plate model surface:
[...]
Warning: incomplete output. Only 100 out of 435 lines have been
provided.
The term "illumination angles" refers to following set of
angles:
solar incidence angle Angle between the surface normal at the
specified surface point and the vector
from the surface point to the Sun.
emission angle Angle between the surface normal at the
specified surface point and the vector
from the surface point to the observer.
phase angle Angle between the vectors from the
surface point to the observing body and
from the surface point to the Sun.
The diagram below illustrates the geometric relationships defining
these angles. The labels for the solar incidence, emission, and
phase angles are "s.i.", "e.", and "phase".
*
Sun
surface normal vector
._ _.
|\ /| Sun vector
\ phase /
\ . . /
. .
\ ___ /
. \/ \/
_\ s.i./
. / \ /
. | e. \ /
* <--------------- * surface point on
viewing vector target body
location to viewing
(observer) location
Note that if the target-observer vector, the target normal vector
at the surface point, and the target-sun vector are coplanar, then
phase is the sum of incidence and emission. This is rarely true;
usually
phase angle < solar incidence angle + emission angle
All of the above angles can be computed using light time
corrections, light time and stellar aberration corrections, or
no aberration corrections. The way aberration corrections
are used is described below.
Care must be used in computing light time corrections. The
guiding principle used here is "describe what appears in
an image." We ignore differential light time; the light times
from all points on the target to the observer are presumed to be
equal.
Observer-target body vector
---------------------------
Let `et' be the epoch at which an observation or remote
sensing measurement is made, and let et - lt ("lt" stands
for "light time") be the epoch at which the photons received
at `et' were emitted from the body (we use the term "emitted"
loosely here).
The correct observer-target vector points from the observer's
location at `et' to the target body's location at et - lt.
The target-observer vector points in the opposite direction.
Since light time corrections are not symmetric, the correct
target-observer vector CANNOT be found by computing the light
time corrected position of the observer as seen from the
target body.
Target body's orientation
-------------------------
Using the definitions of `et' and `lt' above, the target
body's orientation at et - lt is used. The surface
normal is dependent on the target body's orientation, so
the body's orientation model must be evaluated for the correct
epoch.
Target body -- Sun vector
-------------------------
All surface features on the target body will appear in a
measurement made at `et' as they were at the target at epoch
et-lt. In particular, lighting on the target body is dependent
on the apparent location of the Sun as seen from the target body
at et-lt. So, a second light time correction is used in finding
the apparent location of the Sun.
Stellar aberration corrections, when used, are applied as follows:
Observer-target body vector
---------------------------
In addition to light time correction, stellar aberration is
used in computing the apparent target body position as seen
from the observer's location at time `et'. This apparent
position defines the observer-target body vector.
Target body-Sun vector
----------------------
The target body-Sun vector is the apparent position of the Sun,
corrected for light time and stellar aberration, as seen from
the target body at time et-lt. Note that the target body's
position is not affected by the stellar aberration correction
applied in finding its apparent position as seen by the
observer.
Once all of the vectors, as well as the target body's
orientation, have been computed with the proper aberration
corrections, the element of time is eliminated from the
computation. The problem becomes a purely geometric one,
and is described by the diagram above.
If any of the listed errors occur, the output arguments are
left unchanged.
1) If `spoint' is the zero vector, the error SPICE(ZEROVECTOR) is
signaled by a routine in the call tree of this routine.
2) If either of the input body names `target' or `obsrvr' cannot be
mapped to NAIF integer codes, the error SPICE(IDCODENOTFOUND)
is signaled by a routine in the call tree of this routine.
3) If `obsrvr' and `target' map to the same NAIF integer ID codes, the
error SPICE(BODIESNOTDISTINCT) is signaled by a routine in the call
tree of this routine.
4) If frame definition data enabling the evaluation of the state
of the target relative to the observer in target body-fixed
coordinates have not been loaded prior to calling cspice_illum_pl02,
an error is signaled by a routine in the call tree of this
routine.
5) If the specified aberration correction is not recognized, an
error is signaled by a routine in the call tree of this
routine.
6) If insufficient ephemeris data have been loaded prior to
calling cspice_illum_pl02, an error is signaled by a
routine in the call tree of this routine.
7) If a DSK providing a DSK type 2 plate model has not been
loaded prior to calling cspice_illum_pl02, an error is signaled by a
routine in the call tree of this routine.
8) If the computation method is "near point" and radii of the
target body have not been loaded into the kernel pool, an
error is signaled by a routine in the call tree of this
routine.
9) If PCK data supplying a rotation model for the target body
have not been loaded prior to calling cspice_illum_pl02, an error is
signaled by a routine in the call tree of this routine.
10) If the segment associated with the input DLA descriptor does not
contain data for the designated target, the error
SPICE(TARGETMISMATCH) is signaled by a routine in the call tree
of this routine.
11) If the segment associated with the input DLA descriptor is not
of data type 2, the error SPICE(WRONGDATATYPE) is signaled by a
routine in the call tree of this routine.
12) If the sub-point associated with `spoint' cannot be computed
because the line segment from a suitably scaled-up `spoint' to
the target body's center fails to intersect the target surface
as defined by the plate model, the error SPICE(NOINTERCEPT) is
signaled by a routine in the call tree of this routine. See the
routine subpt_pl02 for details.
13) Use of transmission-style aberration corrections is not
permitted. If `abcorr' specified such a correction, the
error SPICE(NOTSUPPORTED) is signaled by a routine in the call
tree of this routine.
14) The observer is presumed to be outside the target body; no
checks are made to verify this.
15) If any of the input arguments, `handle', `dladsc', `target', `et',
`abcorr', `obsrvr' or `spoint', is undefined, an error is signaled
by the Matlab error handling system.
16) If any of the input arguments, `handle', `dladsc', `target', `et',
`abcorr', `obsrvr' or `spoint', is not of the expected type, or it
does not have the expected dimensions and size, an error is
signaled by the Mice interface.
Appropriate DSK, SPK, PCK, and frame data must be available to
the calling program before this routine is called. Typically
the data are made available by loading kernels; however the
data may be supplied via subroutine interfaces if applicable.
The following data are required:
- DSK data: a DSK file containing a plate model representing the
target body's surface must be loaded. This kernel must contain
a type 2 segment that contains data for the entire surface of
the target body.
- SPK data: ephemeris data for target, observer, and Sun must be
loaded. If aberration corrections are used, the states of
target and observer relative to the solar system barycenter
must be calculable from the available ephemeris data. Typically
ephemeris data are made available by loading one or more SPK
files via cspice_furnsh.
- PCK data: triaxial radii for the target body must be loaded
into the kernel pool if the "Near Point" method is selected.
Typically these data are made available by loading a text PCK
file via cspice_furnsh.
- Further PCK data: rotation data for the target body must
be loaded. These may be provided in a text or binary PCK file.
Either type of file may be loaded via cspice_furnsh.
- Frame data: if a frame definition is required to convert
the observer and target states to the body-fixed frame of
the target, that definition must be available in the kernel
pool. Typically the definition is supplied by loading a
frame kernel via cspice_furnsh.
In all cases, kernel data are normally loaded once per program
run, NOT every time this routine is called.
1) This routine assumes that the origin of the body-fixed reference
frame associated with the target body is located in the interior
of that body.
2) This routine does not compute illumination angles for surface
points on interior plates, for example plates representing
the interior of a cave or tunnel. See the -I/O section.
3) Illumination angles on an irregular target body surface may
differ greatly from those on a reference ellipsoid for the same
surface, as illustrated by the example program shown above.
Users may want to consider using the ellipsoid formulation of
this algorithm, which is implemented in the Mice routine
cspice_illum.
MICE.REQ
ABCORR.REQ
DSK.REQ
PCK.REQ
SPK.REQ
TIME.REQ
None.
N.J. Bachman (JPL)
J. Diaz del Rio (ODC Space)
E.D. Wright (JPL)
-Mice Version 1.1.0, 26-OCT-2021 (EDW) (JDR)
Edited the header to comply with NAIF standard.
Added -Parameters, -Exceptions, -Files, -Restrictions,
-Literature_References and -Author_and_Institution sections.
Eliminated use of "lasterror" in rethrow.
Removed reference to the function's corresponding CSPICE header from
-Required_Reading section.
Index lines now state that this routine is deprecated.
-Mice Version 1.0.0, 25-JUL-2016 (NJB) (EDW)
DEPRECATED illumination angles using DSK plate_model
DEPRECATED lighting angles using DSK triangular plate_model
DEPRECATED illumination angles using DSK type_2 plate_model
DEPRECATED lighting angles using DSK type_2 plate_model
DEPRECATED phase angle using DSK triangular plate_model
DEPRECATED emission angle using DSK triangular plate_model
DEPRECATED solar incidence angle using DSK plate_model
DEPRECATED phase angle using DSK type_2 plate_model
DEPRECATED emission angle using DSK type_2 model
DEPRECATED solar incidence angle using DSK type_2 model
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