**TIME EMBEDDING FOR THE FUTURE STEREO RECONSTRUCTIONS**

###### I.S.Veselovsky

####### Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia, 

####### E-mail: veselov.dec1.sinp.msu.ru

**[Abstract.]{.underline}** Real and spurious topological changes seen
in the coronal images can be separated when singular projection effects
are eliminated by stereoscopic observations with a sufficient time
continuity

[Key words]{.underline}: solar corona, images, time, continuity

##### Introduction

The monocular tomography of the solar atmosphere when observed from the
Earth is difficult because of large distances and small angles. Closer
approach to the Sun could help, but it is not planned. Multipoint
stereoscopic observations are powerful, but not always sufficient for
unique solutions of 3D reconstructions. For example, the elliptic
cylinder (ellipsoid) in symmetric positions against two observers can be
not distinguished from the circular cylinder (sphere). Two static
projections are often not sufficient for unambiguous binocular 3D
reconstructions without some additional information (Vedenov, 1994) This
information can be provided a priori when the objects are well known and
only "labels" are searched for the recognition purposes. Unfortunately,
it is not the case in the solar atmosphere. Nevertheless, the needed
additional information can be obtained in many instances from the time
evolution of the object under the condition of reasonable understanding
of its dynamical and kinematic properties. We do not consider here the
"wave sounding". The aim of this paper is to discuss some global
geometry aspects in the future 3D reconstructions from stereo-movies for
CME investigations.

**3D objects and 2D images (projections)**

2D Images (I) are usually constructed from 3D objects (O) by the action
of some projective operator $(\hat{P})$ defined on the manifold {O}. The
projective transformation (mapping) is unique by definition

$$\left\{ I \right\} = \hat{P}\left\{ O \right\}\text{.}$$

The inverse transformation from {I} space to {O} space is not unique as
a rule. The inverse operator $(\hat{P})^{- 1}$ acts in the space {I).
One or more objects from {O} space correspond to each element I

$$O = {\hat{P}}^{- 1}I\text{.}$$

The one-to-one correspondence between O and I is possible only if the
continuum parametrization in {I} space is available. It is the necessary
but not sufficient condition. The necessity of continuum parameters is
understandable because of a more powerful manifold of the 3D space in
comparison with the 2D space.

The STEREO mission purpose consists mainly in the reconstruction of {O}
space. For this purpose several problems should be solved.

1\) The {I} space should be reasonably defined. Practically, we need the
answers to the questions: What to measure and how to do this? Let us
suppose that this problem is solved in any way and we have obtained the
needed information by remote sensing methods (e.g., brightness
distributions, spectral characteristics, polarisation diagrams etc.). We
should understand the quality of the information, its completeness,
precision, error bars. Naturally, because some pre-assumptions were
adopted during the formation of {I}, we should keep the reservation to
check their applicability a posteriori when the intermediate or final
results will be obtained and interpreted. Otherwise, pre-assumptions and
hypotheses easily turn to be prejudices, which are still so common in
the solar physics. To avoid this situation we need a caution and a deep
understanding of the physics behind the direct operators $(\hat{P})$ and
their working properties. This task is not easy and could be solved only
step by step. A good library of direct transformations is useful in this
respect.

2\) The resolving inverse operator $(\hat{P})^{- 1}$ should be properly
introduced in the selected {I} space with the necessary and sufficient
parametrization to reduce the unavoidable ambiguity of the
decompactification. In this ideal case, we are tending to have unique
solutions, i.e. correctly posed problems and stable algorithms of the
deconvolution. It is a very difficult task in the case when the expected
properties of the objects {O} are a priori not known. All kinds of the
additional information should be incorporated in the definition of
$(\hat{P})^{- 1}$ to increase the number of equations to be solved in
the hope to reject spurious solutions in this way. A danger exits here
from both sides. The solutions are strongly not unique if the
incorporated information in $(\hat{P})^{- 1}$ is too scanty and the
inverse problem is not completely determined. This situation is very
common in coronal physics. In the opposite case, when the available
information is overabundant, but not precise, we can loss the needed
unique solution, because the problem can be easily made over-determined
and we are facing "no solution" situations typical for "over-precision"
data and models, not so uncommon in solar physics. Hence, the needed
best solutions are marginally stable in this respect being at the
boundary of their existence. An accurate extension of {I} space
parametric dimensions and the corresponding $(\hat{P})^{- 1}$operators
is very promising for a better understanding and representation
attainable by the embedding of lower dimensional physical space into the
higher one with new independent variables.

3\) Some recognizible reference points should exist in binocular
stereo-images . If such points are absent or not found, the brain (or
computer) can not adequately reconstruct the 3D pattern: the
correspondence between left and right elements is lost. If such points
are too numerous, the relief is also lost or weak in the better case.
For example, in the eroded case when left and right images are identical
in the stereo-couple (the object is placed at infinity), the
correspondence is perfect, but we have no information about the depth.

Hence, the binocular parallax $p = \frac{d}{l}$ where $d$ is the
stereo-basis, i.e. the distance between the the 'eyes', $l$ is the
distance to the object, should be large enough and greater than some
threshold value related to the resolution limits (practically, 5-10
arcsec for the naked eye observations).

The relative binocular parallax $\delta p = d\frac{\delta l}{l^{2}}$,
where $\delta l$ is the depth difference, defines the direction and the
visible distance to the object. This quantity should be small enough and
less than some upper limit value $$${\delta p}_{\text{max}}$determined
by the nature of the object, allowing the reliable identification of the
same reference point by both eyes against the background. This last
quantity, ${\delta p}_{\text{max}}$,is difficult to evaluate when the
properties of the object are poorly known. Both local and global
geometry characteristics are important in this respect for the correct
interpretation. Global information is needed for the fixation of the
local picture planes of both eyes. After this fixation process the
recognition work takes place for the evaluation of more local details in
vicinities of the selected reference points .The coherent picture is not
possible when this points are lost. Practically, the limitations on the
value of $d$ appear when the broad-angle views in the left and the right
eyes look very different or noisy. The recognition process fails in this
case because of the lost reference points. This difficulty is most
severe in the static stereoscopy ('in vitro'). The situation is more
promising 'in vivo' when the object is moderately changing its attitude
due to its internal dynamical processes. In that case, we have a set of
consecutive stereo-couples (movies) taken from one position. When the
time cadence is sufficiently high to warranty the appropriate
conservation of the reference points between two snapshots, i.e. the
dynamics is not too fast, we obtain the needed supplementary
narrow-angle viewing information for the recognition process.

The time embedding of this kind seems to be practically very important
for the future STEREO purposes. Nevertheless, even in this case, we are
facing three problems of the first acquaintance, recognition and
classification.

##### Time embedding

Stereoscopic movies open new possibilities to deconvolve 3D {O} space
from parametrized 2D{I} space in the manner, which can be illustrated by
several simple examples, when 3D objects are reconstructed from the
usual simple movie with no additional parameters or projections.

In this case, reasonably understood dynamics or kinematics can suggest a
correct solution of the 3D reconstruction problem.

1\) Polar plumes. From nice SOHO/EIT movies it is clear now that plumes
are linear rather than curtain-like structures (De Forest, 1998). Some
of them demonstrate helical shapes (Veselovsky et al., 1999a), which can
be used for the electric current estimations of \~1GA and the geometry
model constructions (Fig.1, 2).

2\) Coronal streamers. Their rotation with the Sun allows an approximate
reconstruction of visual coronal shapes for the period of a half
rotation, which could be better than an assumption of a quasi-stationary
rotating pattern (Veselovsky at al., 1999b).

3\) Coronal mass ejections. Dynamical motions of foot points (local
rotations) for twisted loops and flux ropes add the trajectory
information in depth projected on the sphere and facilitate 3D
reconstructions. Evolving 3D coronal loops are easier de-convolved from
2D projections when foot points on the sphere are seen in their
perspective.

4\) Dynamical shadowing effects. Moderate relative changes in the
visible appearance of evolving and rotating loops allow better imagine
3D shapes when optically thin and thick parts are seen simultaneously in
movies with foot points on the reference sphere. Shadowing by the solar
disk also helps in some instances when the streamers, plumes are getting
partially occulted.

5\) Projection catastrophes are easier recognisable from real
singularities in the plasma distribution when the time history is
followed. Topological changes, by their definition, can be not removed
by continuous transformations. Geometrical changes can be removed in
this way, for example, when viewing from different aspect angles.
Spurious topological changes are unstable in this respect and removable.
Time evolution is a good example of a needed continuous transformation.

##### Coronal loops with low expansion factors

We can suggest the solution of one puzzle (Klimchuk, 2002): many coronal
loops do not expand and have nearly uniform cross-sections because they
are immersed not in the dipolar magnetic field assumed by Klimchuk
(2002) in his model, but in the magnetic field of the horizontal linear
current. Field lines of a linear current are concentric circular rings
with constant magnetic field intensity. The model electrojets with
currents 0.1-1.0 TA explain this puzzle and other salient features of
active region loops and prominence cavities (Panasenco and Veselovsky,
2002) (Fig.3).

The prominence cavity structure in EUV lines is better resolved at the
limb under the appropriate viewing angles: coronal bushes when seen
perpendicular to the electrojet and rings when seen along the current
axis. At the disk, the prominence cavity looks as a dark corridor with
bushes at its walls. Bushes represent brighter bottom parts of magnetic
loops, summits being not visible because of low contrast and luminosity.

##### Spurios reconnections: kinks

There is no standard definition of the term "reconnection" in the
current literature (Priest, 1982). This term and its older counterpart
"magnetic field annihilation" is excessive. It was not used by Faraday,
Maxwell and other authors of classical works on the electrodynamics, but
it can be found as a specific "argot" during last decades mostly in the
papers of some authors on (space) plasma physics. Faraday introduced the
field line concept and discussed the "merging" of field lines with the
clear geometrical interpretation of this situation, not more. The whole
building of the modern electrodynamics was constructed without using the
notions of "reconnection" or "field annihilation". Fortunately, this
last term is now nearly forgotten. Usually, but not always, (see e.g.,
Gosling, 1999), the topological change in the field line connectivity is
assumed under the term "reconnection". This makes the use of this term
somewhat ambiguous and sometimes arbitrary. For example, several
"sketches of successive steps in 3-dimensional reconnection in the
corona beneath a departing CME" (Gosling, 1999) are in reality couples
of identical topological configurations with the same preserved
connectivity, according to the standard mathematical definitions
(Fig.4). The topology is completely defined and controlled by zero
points and singularities. The topological change per se can be not a
source of energy or lead to an enhanced dissipation of free energy.
Curiously enough, but some authors write about "three dimensional
reconnection points".

What is really needed for the enhancement of the Joule dissipation, it
is an increased current strength when the current geometry and material
conditions are preserved or, if the total current strength is preserved,
an enhanced resistivity - longer thinner current paths. In the fixed
volume this practically means more complicated geometry with more
developed fine structures. Similar considerations are applicable for the
viscous dissipation.

Hence, we distinguish two types of energy releases: extensive and
intensive. The first of them is characteristic of open physical systems
driven by external forces and free energy sources situated outside the
system. Solar flares and CMEs of this type are associated with larger
scale reservoirs of the magnetic energy in comparison with the "local"
brightest manifestations in impulsive flares. Long duration events are
well known examples of bigger reservoirs. They are often accompanied by
filament and prominence eruptions. The main physical cause of these
events is an increase of the electric currents supplied from
subphotospheric interiors.

The second type is characteristic of the closed physical system, which
is getting internally unstable and leads to rapid local processes The
external energy supply to the system is not essential during the
characteristic "explosion" time, which is just the transition time to
the new state with a less free energy. The excessive free energy is
emitted as radiation or accelerated particles in impulsive flares (heat
fluxes, mass fluxes, cosmic rays).

Naturally, in addition to these two extreme cases of purely "open" and
"closed" evolution, intermediate cases are possible, when both factors
are essential. It is especially true for the most interesting marginally
stable structures, which are common in the solar atmosphere. Indeed, we
often see the dynamical phenomena with preserved topological
configurations and also events with drastic topological transformations.
Dimensionless Trieste numbers are useful quantitative characteristics of
the openness degree of physical systems in the configuration and
parametric space. They are determined as dimensionless ratios of
internal and linking energy, momentum and mass fluxes for a given
object.

The non-linear helical kink instability of active region loops seems to
be the attractive hypothesis for the explanation of twisted flare loops.
According to one scenario (Linton et al., 1999), right-handed flux ropes
will emerge rotated clockwise away from the usual Hale orientation, and
then will rotate counter-clockwise as they evolve. The opposite will
hold for tubes with left-handed twist. The entire concentrated kink
would emerge into the corona and only un-kinked portions of the tube
will intersect the photosphere.

Nevertheless, what we observe in numerous TRACE movies of kinked and
rotating eruptive prominences and coronal loops differs from this
picture (see, e.g., in Fig.5 for the February 26, 2002 twisted loops in
TRACE and also in SOHO/EIT movies). Kinks (electric currents) cross the
photosphere. Twisted and sheared loops often look as self-crossing
structures when projected at the picture plane. Spurious cusps appear in
the projections of deformed planar loops (see, e.g., Fig.4, 5 in Moses
et al (1998). In reality, many wavy threads (planar, cylindrical,
toroidal, more complicated and less symmetric), even if they are
homogeneous along the main axis, could form caustics when mapped on the
picture plane. Attempts of a mathematical classification of
singularities in mappings can be found in the literature (Arnold et al.,
1982). The same caution can be expressed regarding "sigmoids". The
detailed analysis of them with a higher resolution often shows
individual loops and two differently inclined loop systems tracing
different field lines and not just a tube of S-shaped field lines as
often assumed in the literature. The middle of bright coronal "S" is
often interrupted by the dark space corresponding to the gaps between
these two bright coronal loop systems individually rooted with their
legs in the photosphere. We should distinct between these topologically
different situations for the correct interpretation of images with real
S-shaped coronal field lines or projection effects

Many dynamical structures seen in the solar atmosphere preserve their
overall (or local) topology (see e.g. Uchida, 2000), but appear as
changing their connectivity due to projection catastrophes and
singularities which have nothing to do with real topological changes and
difficult to deconvolve from static images, but easier recognisable in
movies. Some of ambiguities in 3D reconstructions can be removed in this
way. It is especially true, when we have an additional physical
information, which can be included in the analysis a priori for the test
of different possible (often not unique) solutions.

This can be demonstrated effectively by STEREO on the examples of
coronal electrojets with their characteristic arcades seen already in
EUV SOHO/EIT images.

##### Discussion

Here we comment on several popular terminological concepts, which could
be of some interest to the three-dimensional aspects and the physics
related to the main STEREO aims.

1\) "Solar source of interplanetary magnetic fields" (Wilcox and Ness,
1969). The magnetic field has no sources by definition because it has no
divergence.

2\) "As in many other flare models, we consider magnetic reconnection as
the source of solar flare energy" (Voitenko and Goossens, 1999). Here
"reconnection" substitutes "energy dissipation".

3\) "A genetic magnetic field has the important property of always
obeying an ideal Ohm's law locally" (Boozer, 1999). The pre-assumption
of the ideal conductivity approximation turned to be prejudice because
"always". The correct statement should be "not always".

4\) "The essential point is that the electric current flows in the
moving frame of reference of the fluid or plasma, in which there is no
electric field, and therefore no powerful inductive effects" (Parker,
2001). The case of no dissipation is assumed. It is only the crude
approximation (see previous point 3), but not the general rule of nature
as stated. Contrary to this statement, inductive electric fields in the
solar atmosphere are playing very important role. Plasma often easily
drifts in the crossed inductive E fields and time dependent B fields
(see movies of dynamical loops in the solar corona).

5\) "It is easy to show from Ampere law that the net current flowing
along an isolated fibril is zero and hence that the net longitudinal
current carried by a distribution of fibrils is zero" (Parker, 1996). If
you assume isolator, you obtain isolator with zero currents. If you
assume zero, you obtain zero. We read further: "Insofar it is correct,
it follows that there are no mean longitudinal currents anywhere in the
convective zone". It is not correct because the conductivity is not
zero. The general conclusion about "actual, vanishing mean current" is
physically wrong and based on the prejudice of solar interiors as an
isolator.

6\) "The X-ray corona of the Sun consists of tenuous wisps of hot gas
enclosed in strong (10^2^ G) bipolar magnetic fields" (Parker, 1988).
Not only. The quantitative measure is not known and need observations
with better resolutions.

STEREO can help to overcome some of existing prejudices, which are based
on pre-assumptions of the ideal plasma and simple geometry. We expect to
see manifestations of non-local magnetic couplings, electric currents
and charges, mass motions along and across the magnetic fields, energy
and momentum transports in static and dynamics structures.

##### Figure captions

(Figures from the original poster presentation are not shown and only
listed below)

Fig.1. Helical polar plumes

Fig.2. Geometry of polar plumes

Fig.3. Coronal electrojet

Fig.4. Geometry changes associated and not associated with topological
transitions

Fig.5. Spurious reconnections. Side views of non-planar loops in their
time evolution (Trace, Yohkoh)

.

##### Conclusions 

STEREO observations could resolve real topological changes, which
obviously happen in the solar atmosphere and in the heliosphere.
Stereoscopic data can be used to overcome some of projection effects
when caustics are formed in the coronal images. We demonstrate that
twisted loops in many instances can be misinterpreted as
\"reconnection\" events because of the projection \"catastrophes\" in
the images. The formation of spurious singularities like cusps, folds,
islands, detached or disconnected elements is often observed in coronal
images. Stereoscopic movies are promising for the reducing the remaining
binocular reconstruction ambiguities. The likelihood of the correct
interpretation of the transparent dynamical object is generally higher
than for the static object of the same complexity because of the using
of the \"forth co-ordinate\" and the \"time embedding\". Examples were
presented when partial and restricted reconstruction is attainable with
TRACE movies for the 3D loop systems when the condition of their
moderate dynamics in active regions is fulfilled. The reasonable time
cadence of the order of 1 min or even better and the continuity of
observations is necessary in this case for the 3D reconstruction of the
prominence eruptions and eruptive flares. Long duration coronal events
associated with CMEs probably can be reconstructed with a lower cadence.

Acknowledgements. This work is partially supported by the State Program
"Astronomy", the Federal Program "Universities of Russia", RFBR grants
01-02-16579, 00-15-96623 and the INTAS-ESA grant 99-00727.

##### References

Arnold, V.I., Varchenko, A.N., Husein-Zade, S.M., 1982, Singularities of
Differntiable Mappings. Classification of Critical Points, Caustics and
Wave Fronts. Nauka Publ. Co., Moscow, 304 pp. (in Russian).

Forest, C.E., 1998, Solar Jets and Coronal Plumes, ESA SP-421, 63

Gosling, J., 1999, Magnetic Helicity in Space and Laboratory Plasmas,
AGU Geophys. Monograph 111, 205.

Linton, M.G., 1999, Proc. 9th European Meeting on Solar Physics,
"Magnetic Fields and Solar Processes", ESA SP-448, 609

Mosses, D. et al., 1988, New Perspectives on Solar Prominences, IAU
Colloquium 167, AIP Conference Series, V.150, 45

Panasenco, O.A., and Veselovsky, I.S., 2002 Yohkoh 10-th Years
Anniversary Meeting,

Kona, Hawai, 51

Parker, E.N., 2001, Astrophys. and Space Sci., 277, 1

Parker, E.N., 1998, Astrophys.J., 330, 474

Parker, E.N., 1996, Astrophys.J., 471, 489

Priest, E.R., 1982, Solar Magnetohydrodynamics, Reidel, Dordrecht

Priest, E.R., Forbes, T.G., 2000. Magnetic Reconnection, Cambridge Univ.
Press, Cambridge

Uchida, Y., 2000, Advances in Solar Research at Eclipses from Ground and
from Space.

Eds J.-P. Zahn and M. Stavinschi, Kluwer Academic Publishers, 105

Vedenov A.A., 1994, Computer-generated three-dimensional representations
of objects. Uspekhi Fizicheskich Nauk ( Also translated in English as
"Physics --Uspekhi", Vol. 164, No9, 967-978

Veselovsky, I.S., et al., 1999a, Proc. 8th SOHO Workshop, ESA SP-446,
675

Veselovsky, I.S., et al. 1999b, Romanian Astronomical Journal 9,
Supplement, 29