Chromospheric Cloud-Model Inversion Techniques
The Physics of Chromospheric Plasmas
ASP Conference Series, Vol. 368, 2007
Petr Heinzel, Ivan Dorotovič and Robert J. Rutten, eds.
Chromospheric Cloud-Model Inversion Techniques
Kostas Tziotziou
National Observatory of Athens, Institute for Space Applications and
Remote Sensing, Greece
Abstract. Spectral inversion techniques based on the cloud model are ex-
tremely useful for the study of properties and dynamics of various chromospheric
cloud-like structures. Several inversion techniques are reviewed based on simple
(constant source function) and more elaborated cloud models, as well as on grids
of synthetic line profiles produced for a wide range of physical parameters by
different NLTE codes. Several examples are shown of how such techniques can
be used in different chromospheric lines, for the study of structures of the quiet
chromosphere, such as mottles/spicules, as well as for active region structures
such as fibrils, arch filament systems (AFS), filaments and flares.
1. Introduction
Observed intensity line profiles are a function of several parameters describing
the three-dimensional solar atmosphere, such as chemical abundance, density,
temperature, velocity, magnetic field, microturbulence etc (which one would like
to determine), as well as of wavelength, space (solar coordinates) and time. How-
ever, due to the large number of parameters that an observed profile depends
on, as well as data noise, model atmospheres have to be assumed in order to
restrict the number of these unknown parameters. The term “inversion tech-
niques” refers to the procedures used for inferring these model parameters from
observed profiles. We refer the reader to Mein (2000) for an extended overview of
inversion techniques. In this paper, we will review only a class of such inversion
techniques known in the solar community as “cloud models”.
Cloud models refer to models describing the transfer of radiation through
structures located higher up from the solar photosphere, which represents the
solar surface, resembling clouds on earth’s sky (see Fig. 1). Such cloud-like
structures, when observed from above, would seem to mostly absorb the radi-
ation coming from below, an absorption which mostly depends on the optical
thickness of the cloud, that is the “transparency” of the cloud to the incident
radiation and also on the physical parameters that describe it. The possibility of
observed emission from such structures cannot, of course, be excluded when the
radiation produced by the cloud-like structure is higher than the absorbed one.
The aforementioned processes are described by the radiative transfer equation
I(∆λ) = I0(∆λ) e
−τ(∆λ) +
∫ τ(∆λ)
−t(∆λ) dt , (1)
where I(∆λ) is the observed intensity, I0(∆λ) is the reference profile emitted by
the background (the incident radiation to the cloud from below), τ(∆λ) is the
http://arxiv.org/abs/0704.1558v1
218 Tziotziou
Figure 1. Geometry of the cloud model. D is the geometrical thickness
of the cloud at height H above the solar surface and V its velocity. From
Heinzel et al. (1999).
optical thickness and S the source function which is a function of optical depth
along the cloud. The first term of the right hand part of the equation represents
the absorption of the incident radiation by the cloud, while the second term
represents emission by the cloud itself.
The simple cloud model method introduced by Beckers (1964) arose from
the need to solve fast the radiation transfer equation and deduce the physical
parameters that describe the observed structure. Beckers assumed that a) the
structure is fully separated from the underlying chromosphere, b) the source
function, radial velocity, Doppler width and the absorption coefficient are con-
stant along the line-of-sight (hereafter LOS) and c) the background intensity is
the same below the structure and the surrounding atmosphere; hence it can be
extrapolated from a neighboring to the structure under study region. Under the
above assumptions the radiative transfer equation is simplified to
I(∆λ) = I0(∆λ) e
−τ(∆λ) + S(1− e−τ(∆λ)) (2)
and can be rewritten as
C(∆λ) =
I(∆λ)− I0(∆λ)
I0(∆λ)
I0(∆λ)
(1− e−τ(∆λ)) , (3)
where C(∆λ) defines the contrast profile. A Gaussian wavelength dependence
is usually assumed for the optical depth as follows
τ(∆λ) = τ0 e
∆λ−∆λI
, (4)
Cloud-model Inversion Techniques 219
where τ0 is the line center optical thickness, ∆λI = λ0v/c is the Doppler shift
with λ0 being the line center wavelength, c the speed of light and ∆λD is the
Doppler width. The latter depends on temperature T and microturbulent ve-
locity ξt through the relationship
∆λD =
ξ2t +
, (5)
where m is the atom rest mass. Other wavelength dependent profiles than the
Gaussian one can also be assumed for the optical depth, e.g., a Voigt profile
(Tsiropoula et al. 1999).
The four adjustable parameters of the model are the source function S, the
Doppler width ∆λD, the optical thickness τ0 and the LOS velocity v. All these
parameters are assumed to be constant through the structure. There are some
crucial assumptions concerning Beckers’ cloud model (hereafter BCM):
– the uniform background radiation assumption, which is not always true es-
pecially for cloud-like structures that do not reside above quiet Sun regions.
Moreover, the background radiation plays an important role in the correct
quantitative determination of the physical parameters.
– the neglect of incident radiation, the effects of which are of course not directly
considered in BCM, but does play an important role in non-Local Thermody-
namic Equilibrium (hereafter NLTE) modeling, since it determines the radia-
tion field within the structure, that is the excitation and ionization conditions
and hence the source function.
– the constant source function assumption which is not realistic especially in
the optically thick case or not valid in the presence of large velocity gradients.
However, the cloud model works quite well for a large number of optically thin
structures and can provide useful, reasonable estimates for the physical param-
eters that describe them. We refer the reader to Alissandrakis et al. (1990) for
a detailed discussion on the validity conditions of BCM for different types of
contrast profiles.
2. Cloud Model Variants
Since the introduction of the BCM method several improvements have been
suggested in the literature. When looking at the radiative transfer equation
(Eq. 1), it is obvious that all efforts concentrate on a better description of the
source function S which in BCM is considered to be constant. In the following
subsections some of these suggested improvements are described.
2.1. Variable source function
Mein et al. (1996a) considered a source function that is a function of optical
depth and is approximated by a second-order polynomial
St = S0 + S1
τ0,max
τ0,max
220 Tziotziou
Figure 2. Left: Geometry of the cloud model in the case of first-order differ-
ential cloud model. From Heinzel et al. (1992). Right: The “3-optical depths”
procedure for solving the differential cloud model case which is described in
Sect. 3.5 (from Mein & Mein (1988)).
with the optical depth at the center of the line τ0 taking values between 0 and
the total optical thickness at line center τ0,max, while S0, S1 and S2 are functions
of τ0,max. This formulation was further improved by Heinzel et al. (1999), who
included also the effect of cloud motion by assuming that S0, S1 and S2 are now
not only functions of τ0,max, but also depend on the velocity v of the structure.
Tsiropoula et al. (1999) assumed the parabolic formula
St = S0
1 + α
as an initial condition for the variation of the source function with optical depth,
where S0 is the source function at the middle of the structure, τ0 the optical
depth at line center, and α a constant expressing the variation of the source
function. However, their final results on the dependence of the source on optical
depth were in good agreement with the results of Mein et al. (1996a).
2.2. Differential cloud models
First and second order differential cloud models (hereafter DCM1 and DCM2)
were introduced by Mein & Mein (1988) to account for fast mass flows observed
on the disc, where BCM is not valid due to fluctuations of the background and
strong velocity gradients along the LOS. DCM1 assumes that the source function
S, temperature T and velocity v are constant within a small volume contained
between two close lines of sight P and R (see Fig. 2, left panel). If we assume
that the variation of the background profile is negligible (I0P ≃ I0R) for such
close points then the differential cloud contrast profile can be written as
C(P,R, λ) =
IP (λ)− IR(λ)
IR(λ)
IR(λ)
(1− e−δτ(λ)) (8)
Cloud-model Inversion Techniques 221
δτ(λ) = δτ0 e
λ− λ0 − vλ0/c
, (9)
where ∆λD is the Doppler width. The zero velocity reference wavelength λ0
is obtained by averaging over the whole field of view. DCM1 is a method for
suppressing the use of the background radiation. If velocity shears are present
between neighboring LOS then DCM2 can be used instead which requires the
use of three neighboring LOS. We refer the reader to Mein & Mein (1988) for the
precise formulation of DCM2 and to Table 1 of the same paper which summarizes
the validity conditions, constrains and results of the two models in comparison
to the classical BCM.
2.3. Multi-cloud models
The multi-cloud model (Gu et al. 1992, 1996) – hereafter MCM – was intro-
duced for the study of asymmetric, non-Gaussian profiles, such as line profiles
of post-flare loops, prominences and surges and was based on the BCM and
DCMs models. These asymmetric line profiles are assumed to be the result of
overlapping of several symmetric Gaussian profiles along the LOS, formed in
small radiative elements (clouds) which have a) different or identical physical
properties and b) a source function and velocity independent of depth. The
profile asymmetry mostly results from the relative Doppler shifts of the different
clouds. The total intensity Iλ emitted by m clouds is then given by the relation
Iλ = I0,λ e
−τλ +
Sj(1− e
−τλ,j ) exp
 , (10)
where τλ,0 = 0, I0,λ is the background intensity, τλ =
j=1 τλ,j is the total
optical depth of the m clouds and
τλ,j = τ0,j e
λ− λ0 −∆λ0,j
∆λD,j
∆λ0,j = λ0vj/c, Sj, τ0,j, vj , ∆λD,j are respectively the optical depth, Doppler
shift, source function, line-center thickness, velocity and Doppler width of the
jth cloud.
A somewhat similar in philosophy, two-cloud model method was used by
Heinzel & Schmieder (1994) in their study of black and white mottles. It was
assumed that the LOS intersects two mottles treated as two different clouds c1
and c2 with optical depths τ1 and τ2 respectively. Hence the emerging intensity
from the lower mottle I1 is assumed to be the background incident intensity for
the second upper mottle. Then, the equations describing the radiation transfer
through the two mottles are
I2(∆λ) = I1 e
−τ2(∆λ) + Ic2(∆λ)
I1(∆λ) = I0 e
−τ1(∆λ) + Ic1(∆λ) , (12)
222 Tziotziou
where I0 is the background chromospheric intensity and Ic1, Ic2 the intensity
emitted by the two clouds respectively. The novelty of the method is that for
the emitted by the clouds intensity, a grid of 140 NLTE models was used which
was computed for prominence-like structures by Gouttebroze et al. (1993). So
this method is a combination of MCM with NLTE source function calculations
which will be further discussed in Section 2.6.
2.4. The Doppler signal method
The Doppler signal method (Georgakilas et al. 1990; Tsiropoula 2000) can be
used when filtergrams at two wavelengths −∆λ and +∆λ (blue and red side of
the line) are available and a fast determination of mass motions is needed. Then
the Doppler signal DS can be defined from the BCM equations as
I − 2I0
− e−τ
2− e−τ
− e−τ
, (13)
where ∆I = I(−∆λ) − I(+∆λ),
I = I(−∆λ) + I(+∆λ) and τ± = τ(±∆λ).
The Doppler signal DS has the same sign as velocity and can be used for a
qualitative description of the velocity field. The left hand side of the above
equation can be determined by the observations while the right hand clearly
does not depend on the source function. Quantitative values for the velocity can
be obtained when τ0 < 1; then the Doppler signal equation reduces to
τ− − τ+
τ− + τ+
and the velocity v – once DS is calculated from the observations and a value of
the Doppler width ∆λD is obtained from the literature or assumed – is given by
the equation
1 +DS
. (15)
2.5. Avoiding the background profile
Liu & Ding (2001) in order to avoid the use of the background profile needed
in BCM assumed that it is symmetric, that is I0(∆λ) = I0(−∆λ). Then it can
easily be shown that we can obtain the relationship
∆I(∆λ) = I(∆λ)− I(−∆λ) = [I(∆λ)− S][1− eτ(∆λ)−τ(−∆λ)] , (16)
which does not require the use of the background for the derivation of the phys-
ical parameters.
2.6. NLTE methods
As Eq. 1 shows, in the general case, the source function S within a cloud-like
structure is not constant, but usually depends on optical depth. In order to cal-
culate this dependence, the NLTE radiative transfer problem within the struc-
ture has to be solved, taking into account all excitation and ionization conditions
within the structure. Several efforts have been undertaken in the past for such
Cloud-model Inversion Techniques 223
Figure 3. Geometry of a two-dimensional cloud model slab. The incident
radiation comes not only from below, but also from the sides of the structure.
From Vial (1982).
NLTE calculations, usually for the case of filaments or prominences. Such NLTE
calculations started from the one-dimensional regime, where the cloud-like struc-
ture is approximated by an infinite one-dimensional slab (see Fig. 1) or a cylin-
der. We refer the reader to the works of Heasley et al. (1974), Heasley & Mihalas
(1976), Heasley & Milkey (1976), Mozozhenko (1978), Fontenla & Rovira (1985),
Heinzel et al. (1987), Gouttebroze et al. (1993), Heinzel (1995), Gouttebroze
(2004) for an overview of such one-dimensional NLTE models. The philosophy of
two-dimensional NLTE models is similar to the one-dimensional models, but now
the cloud-like structure is replaced by a two-dimensional slab or cylinder which
is infinite in the third dimension, allowing both vertical, as well as horizontal
radiation transport (see Fig. 3). Furthermore, the incident radiation is treated
as anisotropic and comes now not only from below, but also from the sides of
the structure. We refer the reader to the works of Mihalas et al. (1978), Vial
(1982), Paletou et al. (1993), Auer & Paletou (1994), Heinzel & Anzer (2001),
Gouttebroze (2005) for an overview of such two-dimensional NLTE models.
A general recipe for such NLTE models, which is modified according to the
specific needs, i.e. the line profile used and the structure observed, has as follows:
– The cloud-like structure is assumed to be a 1-D or 2-D slab or cylinder at
a height H above the photosphere. This slab/cylinder can be considered
to be either isothermal (e.g., Heinzel 1995) or isothermal and isobaric (e.g.,
Paletou et al. 1993).
– The incident radiation comes in the case of 1-D models only from below and
in the case of 2-D models also from the sides and determines the radiation
field within the structure, that is all excitation and ionization conditions.
– A multi-level atom plus continuum is assumed. The larger the number of
atomic levels used, the more computationally demanding the method is.
Complete or partial redistribution effects (CRD or PRD) are also assumed
224 Tziotziou
depending on the formation properties of the line. Methods with CRD are
computationally much faster so sometimes CRD is used but with simulated
PRD effects taken into account (e.g., Heinzel 1995).
– Some physical parameters are assigned to the slab/cylinder, like temperature
T , bulk velocity v, geometrical thickness Z, electronic density Ne or pressure
p. Calculations with electronic density are usually faster than calculations
with pressure.
– The radiative transfer statistical equilibrium equations are numerically solved
and the population levels are found and hence the source function as a func-
tion of optical depth for a set of selected physical parameters.
Once the source function S is obtained as a function of optical depth, Eq. 1 can
be solved in order to calculate the emerging observed profile from the structure
which is going to be compared to the observed one.
3. Solving the Cloud Model Equations
In the following subsections some of the methods used to solve the cloud model
equations are reviewed. We remind the reader that whenever the background
profile is needed, either the average profile of a quiet Sun region is taken or the
average profile of a region close to the structure under study.
3.1. Solving the constant-S case with the “5-point” method
Mein et al. (1996a) introduced the “5-point method” for solving the BCM equa-
tion with constant S. According to this method five intensities of the observed
and the background profile at wavelengths λ1, λ2 (blue wing of the observed
profile), λ3, λ4 (red wing of the the observed profile) and the line-center wave-
length λ0 are used for solving Eqs. 3 and 4. It is an iterative method that works
as follows:
– The line-center wavelength λ0 profile and background intensities are used for
calculating S, where τ0, ∆λD and v are determined in a previous iteration.
At the first step of the iteration some values can be assumed and S can be
taken as equal to zero.
– Profile and background intensities at wavelengths λ1 and λ3 are used for
calculating a new τ0.
– Afterwards a new ∆λD is calculated using the other two remaining wave-
lengths λ2 and λ4.
– Finally a new velocity is calculated from wavelengths λ1, λ2, λ3, and λ4 and
then a reconstructed profile obtained using the derived parameters which is
compared to the observed one. If any of the departures between the recon-
structed and the observed profile is higher than an assumed small threshold
value (i.e. 10−4) then the aforementioned procedure is repeated until con-
vergence is achieved. If no convergence is gained after a certain number of
iterations then it is assumed that no solution exists.
We refer the reader to Mein et al. (1996a) for a detailed description of the ana-
lytical equations described above.
Cloud-model Inversion Techniques 225
3.2. Solving the constant-S case with an iterative least-square fit
This method which was used by Alissandrakis et al. (1990) and further described
in Tsiropoula et al. (1999) and Tziotziou et al. (2003) fits the observed contrast
profile with a curve that results from an iterative least-square procedure for
non-linear functions which is repeated until the departures between computed
and observed profiles are minimized. The coefficients of the fitted curve are
functions of the free parameters of the cloud model. At the beginning of the
iteration procedure initial values have to be assumed for the free parameters
and especially for the source function S which is usually estimated from some
empirical approximate expressions that relate it to the line-center contrast. This
method is very accurate and usually converges within a few iterations. The more
observed wavelengths used within the profile, the better the determination of
the ohysical parameters is. However, as Tziotziou et al. (2003) have reported,
the velocity calculation can overshoot producing very high values, if the wings of
the profile are not sufficiently covered by observed wavelengths. The suggested
way to overcome the problem is to artificially add two extra contrast points near
the continuum of the observed profile where the contrast should be in theory
equal to zero.
This iterative method can also be successfully used not only in the case
of a constant source function S, but also for cases with a prescribed expres-
sion for the source function, such as the parabolic expression of Eq. 7 used by
Tsiropoula et al. (1999).
3.3. Solving the constant-S case with a constrained nonlinear least-
square fitting technique
The constrained nonlinear least-square fitting technique, used by Chae et al.
(2006) for the inversion of a filament with BCM, was introduced by Chae et al.
(1998). According to the method a) expectation values pei of the ith free param-
eters, b) their uncertainties εi, as well as c) the data to fit are provided (M wave-
lengths along the profile) and then a set ofN free parameters p = (p0, p1, ...pN−1)
are sought, i.e. p = (S, τ0, λ0,∆λD), that minimize the function
H(p) =
Cobsj − C
j (p)
pi − p
, (17)
where Cobsj and C
j are respectively the observed and calculated with the
expectation values contrasts and σj the noise in the data. The first term of the
sum H represents the data χ2, while the second term the expectation χ2 which
regularizes the solution by constraining the probable range of free parameters.
For very small values of εi the solution will not be much constrained by the
data and will be close to the chosen set of expectation values pei , while for large
values of εi it will be mostly constrained by the data and not by the expectation
values. We refer the reader to Chae et al. (2006) for a detailed discussion of the
effects of constrained fitting.
3.4. Solving the variable-S case
Apart from the iterative least-square procedure described above which can be
used when the source function varies in a prescribed way, Mein et al. (1996a)
226 Tziotziou
have introduced also the “4-point method” for solving the case of a source func-
tion that is described by the second order polynomial of Eq. 6. According to
the method an intensity I ′(∆λ) can be defined as follows
I ′(∆λ) = I(∆λ)−
1− [τ(∆λ) + 1] e−τ(∆λ)
τ(∆λ)
2− [τ2(∆λ) + 2τ(∆λ) + 2] e−τ(∆λ)
τ2(∆λ)
S2 (18)
and then the radiative transfer equation reduces to
I ′(∆λ) = S0 + (I0 − S0) e
−τ(∆λ) . (19)
This equation can be solved now using the iteration procedure described in
Sect. 3.1, with the modification that I(∆λ) is now replaced by I ′(∆λ) and
that the source function calculation in the first step is replaced by the assumed
theoretical relation for S given by Eq. 6.
3.5. Solving the DCM cases
A method for solving the differential cloud model cases is the “3-optical depths”
procedure introduced by Mein & Mein (1988). According to this procedure:
– the zero velocity reference is obtained from the average profile over the whole
field of view;
– a value S is assumed between zero and the line-center intensity (in principle
it could even work also for emitting clouds) and a function δτ(λ) is derived
from Eq. 8. The latter is characterized by the maximum value δτ0 and δτ1,
and the values δτ2 (see right panel of Fig. 2) which correspond to the half
widths ∆λ1 and ∆λ2 respectively and are given by the following relations
δτ1 = δτ0 e
−(∆λ1/∆λD)
δτ2 = δτ0 e
−(∆λ2/∆λD)
; (20)
– the code fits S and ∆λD by the conditions of Eq. 20 coupled with Eq. 8 and
the solutions are assumed to be acceptable when the radial velocities v1 and
v2, which correspond to widths ∆λ1 and ∆λ2 respectively and are defined
as the displacement of the middle of these chords compared with the zero
reference position, are not that different. When convergence is achieved the
δτ(λ) curve is well represented by a Gaussian and the Doppler width ∆λD is
independent of the chord ∆λ.
3.6. Solving the MCM case
We refer the reader to the papers by Li & Ding (1992) and Li et al. (1993, 1994)
for a detailed description of the methods and mathematical manipulations used
for fitting observed profiles with the multi-cloud method, which unfortunately
are not easy to concisely describe within a few lines.
Cloud-model Inversion Techniques 227
3.7. Using NLTE Methods
The most straightforward method for deriving the parameters of an observed
structure with NLTE calculations would be the calculation of a grid of models
for a wide range of the physical parameters used to describe the structure. How-
ever, the calculation of such a grid is computationally demanding, especially in
the case when a) a large number of atomic levels is assumed and/or b) partial
redistribution effects (PRD) are taken into account and/or c) a two-dimensional
geometry is considered. For such cases, either a very small grid of models is con-
structed and thus only approximate values for the observed structure are derived
or “test and try” methods are used where the user makes a “good guess” for the
physical parameter values, proceeds to the respective NLTE calculations, com-
pares the derived profile(s) with the observed one(s) and applies the necessary
adjustments to the model parameters according to the derived results.
However, nowadays the construction of a large grid of models, although
time-demanding, becomes more of a common practice with the extended ca-
pabilities of modern computers. We refer the reader to Molowny-Horas et al.
(2001) and Tziotziou et al. (2001) for two such examples, both considering a
one-dimensional isothermal slab for a cloud-like structure, which is the same
filament observed and studied in the Hα in Ca II 8542 Å lines respectively. The
general methodology used in the case of grids of models is the following:
– a grid of synthetic line profiles for a wide range of model parameters is
computed using NLTE calculations for the source function, as described in
Sect. 2.6;
– these synthetic profiles are convolved with the characteristics of the instru-
ment used for the observations in order to simulate its effects on the observed
profiles;
– each observed profile is compared with the whole library of convolved syn-
thetic profiles and the best fit is derived, that is the synthetic profile with the
smallest departure, and hence the physical parameters that describe it;
– an interpolation (linear or parabolic) between neighboring points in the pa-
rameter space can also be used, for a more accurate quantitative determina-
tion of the physical parameters that best describe the observed profile.
Grid models based on NLTE calculations have many advantages since pre-
ferred geometries, temperature structures, etc can be used, no iterations are
required, errors can be easily defined from the parameter space and inversions
are nowadays becoming faster with modern computers.
4. Validity of the Cloud Models
The validity of the cloud model used for an inversion obviously strongly de-
pends on a) the method used, b) the assumptions that were made for the model
atmosphere describing the structure and c) the specific characteristics of the
structure under study. Most of the reviewed papers in Sect. 5, concerning ap-
plications of different cloud models, have extended discussions on the validity
of the cloud model method and the results obtained, as well as the limitations
228 Tziotziou
Figure 4. Two left panels: The calculated optical depth τ0,max and velocity
v with BCM (constant source function) versus the assumed optical thickness.
The dashed curve is the model, the solid curve the inversion. Two right panels:
Same plots but with added Gaussian noise. From Mein et al. (1996a).
Figure 5. Two left panels: The calculated optical depth τ0,max and velocity
v using a cloud model with variable source function (see Eq. 6) depending
only on line-center optical thickness versus the assumed optical thickness.
Gaussian noise has also been taken into account. Two right panels: Same plots
but for an over-estimated chromospheric background profile. FromMein et al.
(1996a).
of the method for the specific structure. However, below, some studies found in
literature about the validity of cloud models are presented.
Mein et al. (1996a) presented a rather detailed study about the validity of
BCM (constant source function), as well as of cloud models with a variable source
function as described in Eq. 6 (depending only on line-center optical thickness)
by inverting theoretical profiles produced with a NLTE code and comparing
the resulting model parameters from the inversion with the assumed ones. Fig-
ure 4 (two left panels) shows the results of the inversion versus the assumed
model optical thickness for the BCM inversion (constant source function). The
calculated optical thickness is smaller, with the difference increasing with the
thickness of the cloud, while the difference in velocity is no more than 20% and
only for high values of the thickness. The figure shows that for optically thin
structures there is practically no difference in the obtained results. When noise
is included (Fig. 4, two right panels) the error increases for increasing thickness
but the mean values stay almost the same. Again for optically thin structures
the difference in the results is very small.
Figure 5 (two left panels) shows the results of the inversion versus the
assumed model optical thickness for a cloud model with variable source function
Cloud-model Inversion Techniques 229
Figure 6. Comparison of the results obtained with method (a) represented
by dots and with method (b) represented by asterisks (see text for de-
tails of the methods) with the assumed model values (solid curve). From
Heinzel et al. (1999).
according to Eq. 6 depending only on line-center optical thickness with an added
Gaussian noise; without noise the results are perfectly reproduced. We see
that the differences are now almost negligible for a large range of the assumed
optical thickness and the parameters are better determined. However, when
taking a slightly brighter background (Fig. 5, two right panels) we see that the
calculated values of the optical thickness are larger than the assumed ones, while
the estimation of velocity is still rather good. This shows the importance of a
correct background profile choice in cloud model calculations.
Heinzel et al. (1999) has repeated the same exercise (inversion of NLTE
synthetic profiles) for a cloud model with a variable source function according
to Eq. 6 depending a) only on line-center optical thickness (method a) and b) on
line-center optical thickness and velocity (method b). Some of their results are
shown in Fig. 6. We see that although with method (a) there are some differences
in the calculation of optical thickness, similarly to Mein et al. (1996a), method
(b) gives exact solutions. Heinzel et al. (1999) have also applied the two methods
in observed profiles of a dark arch filament. Figure 7 shows the comparison of
the results obtained with the two methods.
We refer also the reader selectively to a) Molowny-Horas et al. (2001) (Fig. 12
of their paper) for a comparison of inversion results for a filament with a NLTE
method and a cloud model with a parabolic S, b) Schmieder et al. (2003) (Fig. 16
of their paper) for a comparison of inversion results for a filament with a NLTE
230 Tziotziou
Figure 7. Comparison of the results obtained with the two methods (a)
and (b) (see text for details) from the inversion of observed profiles of a dark
arch filament. Scatter plots are shown for (1) optical thickness, (2) velocity
(in km s−1), and (3) Doppler width (in Å). From Heinzel et al. (1999).
method and a constant source function cloud model, c) Tsiropoula et al. (1999)
(Fig. 5 of their paper) for a comparison of inversion results for mottles for
cloud models with a constant and parabolic S, and d) Alissandrakis et al. (1990)
(Fig. 8 to 11 of their paper) for a comparison of inversion results for an arch
filament system with Beckers’ cloud model, the Doppler signal method and the
differential cloud model.
5. Examples of Cloud Model Inversions
Cloud models have been so far successfully applied for the derivation of the
parameters of several cloud-like solar structures of the quiet Sun, such as mot-
tles/spicules, as well of active region structures, such as arch filament systems
(AFS), filaments, fibrils, flaring regions, surges etc. Below, some examples of
such cloud model inversions are presented.
5.1. Application to filaments
Filaments are commonly observed features that appear on the solar disc as dark
long structures, lying along longitudinal magnetic field inversion lines. When
observed on the limb they are bright and are called prominences. Filaments
were some of the first solar structures to be studied with cloud models (see
for example Maltby 1976, and references therein). Since then several authors
used different cloud models to infer the dynamics and physical parameters of
filaments. Mein, Mein & Wiik (1994), for example, studied the dynamical fine
structure (threads) of a quiescent filament assuming a number of identical –
except for the velocity – threads seen over the chromosphere and using a variant
of BCM, while Schmieder et al. (1991) performed a similar study for threads by
using the DCM. Morimoto & Kurokawa (2003) developed an interesting method
applying BCM to determine the three-dimensional velocity fields of disappearing
filaments.
Molowny-Horas et al. (2001) and Tziotziou et al. (2001) studied the same
filament observed in Hα and Ca II 8542 Å respectively with the Multichannel
Subtractive Double Pass (MSDP) spectrograph (Mein 1991, 2002) mounted on
the German solar telescope VTT in Tenerife. The filament was studied by
using two very large grids of models in Hα and Ca II 8542 Å respectively which
were constructed with the NLTE one-dimensional code MALI (Heinzel 1995), as
Cloud-model Inversion Techniques 231
Figure 8. Top row: A filament observed in Hα and the two-dimensional
parameter distributions derived with a Hα NLTE inversion using a grid of
models. From Molowny-Horas et al. (2001). Bottom row: Same filament
observed in Ca II 8542 Å and the two-dimensional parameter distributions
derived with a Ca II 8542 Å NLTE grid model inversion. From Tziotziou et al.
(2001).
described in Sect. 2.6 Two-dimensional distributions of the physical parameters
were obtained (see Fig. 8) which are not that similar due to the different physical
formation properties and formation heights of the two lines. Schmieder et al.
(2003) performed a similar NLTE grid inversion of a filament combined with a
classical BCM inversion in a multi-wavelength study of filament channels. More
recently, Chae et al. (2006) used Hα images obtained with a tunable filter and a
BCM inversion to obtain detailed two-dimensional distributions of the physical
parameters describing a quiescent filament.
5.2. Application to arch filaments (AFS)
Arch filaments systems (AFSs) are low-lying dark loop-like structures formed
during the emergence of solar magnetic flux in active regions. Georgakilas et al.
(1990) have used the Doppler signal method described in Sect. 2.4 to study
mass motions in AFSs observed in Hα, while Alissandrakis et al. (1990) and
Tsiropoula et al. (1992) used the standard BCM to obtain the physical param-
eters describing arch filament regions observed in the same line (see Fig. 9). An
example of the use of the differential cloud model described in Sect. 2.2 for the
232 Tziotziou
Figure 9. Contours maps of source function (top right panel) and the ve-
locity (bottom right panel) derived with the cloud model for the AFS shown
in Hα in the left panel of the figure. From Alissandrakis et al. (1990).
study of the dynamics of AFSs can be found in Mein et al. (1996b) who applied
the method to Hα observations from a two-telescope coordinated campaign. Fi-
nally Mein et al. (2000) present a study of AFSs in Ca II 8542 Å using a fitting
done with NLTE synthetic profile calculations – as described in Sect. 2.6 – with
the one-dimensional MALI code (Heinzel 1995).
5.3. Application to fibrils
Fibrils are small dark structures, belonging to the family of “chromospheric fine
structures”, found in active regions surrounding plages or sunspots (penum-
bral fibrils). One of the first studies of fibrils was conducted by Bray (1974)
who compared observed profiles of fibrils with profiles calculated with BCM.
Alissandrakis et al. (1990) used the standard BCM to obtain two-dimensional
maps of several physical parameters distributions describing fibrils using Hα ob-
servations obtained at Pic du Midi Observatory. Georgakilas et al. (2003) used
filtergrams obtained at nine wavelengths along the Hα to study the Evershed
flow in sunspots and reconstruct the three-dimensional velocity vector using
the Doppler signal method (see Fig. 10), while Tsiropoula (2000) used also the
Doppler signal method to determine LOS velocities of dark penumbral fibrils.
5.4. Application to mottles
Mottles are small-scale structures (appearing both bright and dark) belonging
also to the family of “chromospheric fine structures” and occurring at quiet
Sun regions at the boundaries of supergranular cells. Mottles are believed to
be the counterparts of limb spicules. They form groups called chains (when
they are almost parallel to each other) or rosettes (when they are more or less
Cloud-model Inversion Techniques 233
Figure 10. Image of a sunspot observed in Hα (a) and Doppler veloc-
ity maps computed with the Doppler signal method from filtergrams in
Hα±0.35Å (b), in Hα±0.5Å (c) and in Hα±0.75Å (d). The intensity gray
scale bar corresponds to normalized intensities while the Doppler velocity gray
scale bars to velocities in km s−1. From Georgakilas et al. (2003).
circularly aligned, pointing radially outwards from a central core) depending on
their location at the chromospheric network.
First cloud studies of mottles started with a controversy about the ability of
BCM to explain their contrast profiles. Bray (1973) and Loughhead (1973) who
studied bright and dark mottles found that their contrast profiles are in good
agreement with BCM. However, Loughhead (1973) used also BCM to deduce
that it could not explain the contrast of individual bright and dark mottles
observed in Hα near the limb, while Cram (1975) claimed that the parameters
inferred from an application of BCM to contrast profiles of chromospheric fine
structures are unreliable.
Since then cloud models have been established as a reliable method for the
study of physical parameters of mottles. Tsiropoula et al. (1999) studied several
bright and dark mottles to derive physical parameters assuming a constant as
well as a varying source function according to Eq. 7. Tsiropoula & Schmieder
(1997) applied Beckers’ cloud model to determine physical parameters in Hα
dark mottles of a rosette region, while Tsiropoula et al. (1993, 1994) studied
the time evolution and fine structure of a rosette with BCM and first showed
an alternating behaviour with time for velocity along mottles (see Fig. 11, left
panel). A similar behaviour has also been found by Tziotziou et al. (2003) using
BCM for a chain of mottles (see Fig. 11, right panel), while the dynamics of an
enhanced network region were also explored in high resolution Hα images by
Al et al. (2004).
234 Tziotziou
Figure 11. Left panel: Cloud velocity as a function of position and time
along the axis of a dark mottle belonging to a rosette. From Tsiropoula et al.
(1994). Right panel: Cloud velocity as a function of position and time along
the axis of a dark mottle belonging to a chain of mottles. White contours
denote downward velocities, black upward velocities, while the thick gray
curve is the zero velocity contour. From Tziotziou et al. (2003).
5.5. Application to post-flare loops
Post-flare loops are loops generally observed between two-ribbon flares. We
refer the reader to Bray & Loughhead (1983) for one of the first post-flare loop
studies, who constructed theoretical curves based on the cloud model to fit
observed contrast profiles of active region loops. Later Schmieder et al. (1988)
and Heinzel et al. (1992) used a differential cloud model to study the structure
and dynamics of post-flare loops. Heinzel et al. (1992) also constructed several
isobaric and isothermal NLTE models of post-flare loops. Their results were
compared by Gu et al. (1997) with two-dimensional maps of Hα post-flare loop
cloud parameters obtained using a two-cloud model. Multi-cloud models like the
ones described in Sect. 2.3 were used by Liang et al. (2004) to study Hα post-
flare loops at the limb (see Fig. 12), by Gu & Ding (2002) for the study of Hα
and Ca II 8542 Å post-flare loops and by Dun et al. (2000) for the study of Hβ
post-flare loops. Liu & Ding (2001) obtained parameters of Hα post-flare loops
using the modified cloud model method presented in Sect. 2.5 that eliminates
the use of the background profile while Gu et al. (1992) presented an extensive
study using BCM, the differential cloud model and a two-cloud model to study
the time evolution of post-flare loops in two-ribbon flares. Finally, we refer the
reader to Berlicki et al. (2005) who studied Hα ribbons during the gradual phase
of a flare by comparing observed Hα profiles with a grid of synthetic Hα profiles
calculated with the NLTE code MALI (Heinzel 1995) which was modified to
account for flare conditions.
5.6. Application to surges
Surges are large jet-like structures observed in opposite polarity flux emer-
gence areas in active regions believed to be supported by magnetic reconnection.
Gu et al. (1994) studied a surge on the limb observed in Hα, using a two-cloud
model inversion as described in Sect. 2.3 (see Fig 13). The inversion result was
Cloud-model Inversion Techniques 235
Figure 12. The distributions of Doppler velocity (in km s−1) derived with
a multi-cloud method for Hα limb post-flare loops. Coordinates are in units
of arcsec, dashed curves show red-shifted mass motions, while solid curves
indicate blue-shifted ones. From Liang et al. (2004).
Figure 13. An Hα filtergram of a surge (left panle) and the two-dimensional
isocontours of Doppler velocity derived with a two-cloud model. Dashed
curves refer to blue-shifted velocities (middle panel), solid curves red-shifted
ones (right panel), while the unit of velocity is in km s−1. From Gu et al.
(1994).
detailed two-dimensional maps of the blue-shifted and red-shifted LOS velocity
distributions.
6. Conclusions
Several inversion techniques for chromospheric structures based on the cloud
model have been reviewed. Cloud models are fast, quite reliable tools for in-
ferring the physical parameters describing cloud-like chromospheric structures
located above the solar photosphere and being illuminated by a background radi-
ation. Cloud model techniques usually provide unique solutions and the results
do not differ – in principle – qualitatively, especially for velocity, when using
different cloud model techniques. However there can be quantitative differences
arising from a) the selection of the background intensity, b) the physical condi-
tions and especially the behaviour of the source function within the structure
under study, and c) the particular model assumptions. Cloud models are mainly
used for absorbing structures, however most of the techniques do work also for
236 Tziotziou
line-center contrasts that are slightly higher than zero, indicating an important
emission by the structure itself.
Several different variants for cloud modeling have been proposed in litera-
ture so far that mainly deal with different assumptions or calculations for the
source functions and span from the simple BCM that assumes a constant source
function to more complicated NLTE calculations of the radiation transfer and
hence the source function within the structure. Accordingly, several different
techniques – most of them iterative – have been proposed for solving cloud
model equations. The latest and more accurate inversion techniques involve
the construction of large grids of synthetic profiles, for different geometries and
physical conditions, which are used for comparison with observed profiles.
Cloud models can be applied with success to several, different in geometry
and physical conditions, solar structures both of the quiet Sun, as well as of
active regions. The resulting parameter inversions has shed light to several
problems involving the physics and dynamics of chromospheric structures.
The future of cloud modeling looks even more brighter. New high resolution
data from telescopes combined with an always increasing computer power and
the continuous development of new, state of the art, NLTE one-dimensional and
two-dimensional cloud model codes will provide further detailed insights to the
physics and dynamics that govern chromospheric structures.
Acknowledgments. KT thanks G. Tsiropoula for constructive comments on the
manuscript and acknowledges support by the organizers of the meeting and by Marie
Curie European Reintegration Grant MERG-CT-2004-021626.
References
Al N., Bendlin C., Hirzberger J., Kneer F., Trujillo Bueno J., 2004, A&A 418, 1131
Alissandrakis C. E., Tsiropoula G., Mein P., 1990, A&A 230, 200
Auer L. H., Paletou F., 1994, A&A 284, 675
Beckers J. M., 1964, A study of the fine structures in the solar chromosphere, Ph.D.
Thesis, Utrecht
Berlicki A., Heinzel P., Schmieder B., Mein P., Mein N., 2005, A&A 430, 679
Bray R. J., 1973, Solar Phys. 29, 317
Bray R. J., 1974, Solar Phys. 38, 377
Bray R. J., Loughhead R. E., 1983, Solar Phys. 85, 131
Chae, J., Yun, H.S., Poland, A.I., 1998, ApJS 114, 151
Chae, J., Park, Y.-D., Park, H.-M., 2006, Solar Phys. 234, 115
Cram L. E., 1975, Solar Phys. 42, 53
Dun J.P., Gu X.M., Zhong S.H., 2000, Ap&SS 274, 473
Fontenla J. M., Rovira M., 1985, Solar Phys. 96, 53
Georgakilas A. A., Alissandrakis C. E., Zachariadis Th. G., 1990, Solar Phys. 129, 277
Georgakilas A. A., Christopoulou E. B., Skodras A., Koutchmy S., 2003, A&A 403,
Gouttebroze P., 2004, A&A 413, 733
Gouttebroze P., 2005, A&A 434, 1165
Gouttebroze P., Heinzel P., Vial J.-C., 1993, A&AS 99, 513
Grossmann-Doerth U., Uexküll M., 1971, Solar Phys. 20, 31
Gu X. M., Ding M. D., 2002, Chin. J. Astron. Astrophys. 2, 92
Gu X. M., Lin J., Luan T., Schmieder B., 1992, A&A 259, 649
Gu X. M., Lin J., Li K. J., Xuan J. Y., Luan T., Li Z. K., 1994, A&A 282, 240
Cloud-model Inversion Techniques 237
Gu X. M., Lin J., Li K. J., Dun J. P., 1996, Ap&SS 240, 263
Gu X. M., Ding Y. J., Luo Z., Schmieder B., 1997, A&A 324, 289
Heasley J. N., Mihalas D., 1976, ApJ 205, 273
Heasley J. N., Milkey R. W., 1976, ApJ 210, 827
Heasley J. N., Mihalas D., Poland A., 1974, ApJ 192, 181
Heinzel P., 1995, A&A 299, 563
Heinzel P., Anzer U., 2001, A&A 375, 1082
Heinzel P., Schmieder B., 1994, A&A 282, 939
Heinzel P., Gouttebroze P., Vial J.-C., 1987, A&A 183, 351
Heinzel P., Mein N., Mein P., 1999, A&A 346, 322
Heinzel P., Schmieder B., Mein P., 1992, Solar Phys. 139, 81
Liang H., Zhan L., Xiang F., Zhao H., 2004, New Astron. 10, 121
Li K. J., Ding Y. J., 1992, Acta Astron. Sin. 33, 171
Li K. J., Ding Y. J., Gu X. M., Li Q. S., Zhong S. H., Li Q. Y., 1993, A&A 269, 496
Li K. J., Ding Y. J., Bai J. M., Zhong S. H., Xuan J. Y., Li Q. Y., 1994, Solar Phys.
150, 87
Liu Y., Ding M. D., 2001, Solar Phys. 200, 127
Loughhead R. E., 1973, Solar Phys. 29, 327
Maltby P., 1976, Solar Phys. 46, 149
Mein P., 1991, A&A 248, 669
Mein P., 2000, in J.-P. Zahn, M. Stavinschi (eds.), NATO ASIC Proc. 558: Advances
in Solar Research at Eclipses from Ground and from Space, 221
Mein P., 2002, A&A 381, 271
Mein P., Mein N., 1988, A&A 203, 162
Mein N., Mein P., Wiik, J. E., 1994, Solar Phys. 151, 75
Mein N., Mein P., Heinzel, P., Vial, J.-C., Malherbe, J. M., Staiger, J., 1996a, A&A 309,
Mein P., Démoulin P., Mein N., Engvold O., Molowny-Horas R., Heinzel, P., Gontikakis,
C., 1996b, A&A 305, 343
Mein P., Briand C., Heinzel, P., Mein, N., 2000, A&A 355, 1146
Mihalas D., Auer L. H., Mihalas B. R., 1978, ApJ 220, 1001
Molowny-Horas R., Heinzel P., Mein P., Mein N., 1999, A&A 345, 618
Morimoto T., Kurokawa H., 2003, PASJ 55, 503
Mozozhenko N. N., 1978, Solar Phys. 58, 47
Paletou F., Vial, J.-C., Auer L. H., 1993, A&A 274, 571
Schmieder B., Mein P., Malherbe J.-M., Forbes T. G., 1988, Advances in Space Re-
search 8, 145
Schmieder B., Raadu M. A., Wiik, J. E., 1991, A&A 252, 353
Schmieder B., Tziotziou K., Heinzel, P., 2003, A&A 401, 361
Tsiropoula G., 2000, New Astron. 5, 1
Tsiropoula G., Schmieder B., 1997, A&A 324, 1183
Tsiropoula G., Georgakilas A. A., Alissandrakis C. E., Mein P., 1992, A&A 262, 587
Tsiropoula G., Alissandrakis C. E., Schmieder B., 1993, A&A 271, 574
Tsiropoula G., Alissandrakis C. E., Schmieder B., 1994, A&A 290, 285
Tsiropoula G., Madi C., Schmieder B., 1999, Solar Phys. 187, 11
Tziotziou K., Heinzel P., Mein P., Mein N., 2001, A&A 366, 686
Tziotziou K., Tsiropoula G., Mein P., 2003, A&A 402, 361
Vial J.-C., 1982, ApJ 254, 780
