THE ASTROPHYSICAL JOURNAL, 535:475È488, 2000 May 20 ( The American Astronomical Society. All rights reserved. Printed in U.S.A.

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1 THE ASTROPHYSICAL JOURNAL, 535:475È488, 2000 May 20 ( The American Astronomical Society. All rights reserved. Printed in U.S.A. INVERSION OF STOKES PROFILES FROM SOLAR MAGNETIC ELEMENTS L. R. BELLOT RUBIO, B. RUIZ COBO, AND M. COLLADOS Instituto de Astrof sica de Canarias, E-38200, La Laguna (Tenerife), Spain Received 1998 July 28; accepted 1999 December 23 ABSTRACT We describe a new LTE inversion code for the analysis of Stokes proðles emerging from unresolved magnetic elements. It has been speciðcally designed to obtain the thermal, dynamic, and magnetic properties of these structures in a self-consistent manner by Ðtting the whole shape of the observed spectra. The inversion code is based on a previous scheme by Ruiz Cobo & del Toro Iniesta and implements the thin Ñux-tube model as a reasonable description of reality. All physical parameters considered relevant for the problem (including velocity Ðelds) are retrieved by means of a Marquardt nonlinear least-squares algorithm. We present the results of extensive tests aimed at characterizing the behavior of the code so as to understand its limitations for the analysis of real observations. The code is found to produce accurate results even with only two spectral lines and noisy Stokes I and V proðles. A detailed error treatment, in which the covariances between parameters are explicitly included, is also carried out in order to investigate the uniqueness and reliability of the inferred model atmospheres. Subject headings: line: formation È polarization È radiative transfer È Sun: magnetic Ðelds È Sun: photosphere 1. INTRODUCTION One of the most exciting Ðelds in solar physics is that of active regions outside sunspots. Thought to provide a signiðcant contribution to the global magnetism of the Sun, they have been the subject of many observational and theoretical e orts. It is now widely accepted that network and plage regions consist of bundles of small (\0A.5) elements whose magnetic Ðeld in the lower photosphere is of the order of 1500 G. Unfortunately, the inference of their properties is hampered by the fact that present-day telescopes are not able to resolve spatially the various atmospheres of which these structures are made. For this reason, techniques other than imaging have to be used. Among these, spectropolarimetry has become the preferred over the years. The particular character of the magnetic elements thus determines what we can measure and how we measure it. A second difficulty arises from the fact that the interpretation of polarization spectra in plages and the network requires some prior knowledge of the structures in which they originate. This is because the observed proðles are invariably compared to synthetic proðles emerging from prescribed model atmospheres. Indeed, the extent to which the adopted model approaches the real solar atmosphere determines the degree of reliability of the inversion itself. For this reason, a careful selection of a realistic physical model is mandatory. The analysis of polarization spectra provides us with a rich variety of information that is normally extracted either by synthesis or inversion methods. The former have been extensively used in the past because of the highly nonlinear dependence of the Stokes proðles on the various atmospheric parameters. Synthesis methods are based upon the perturbation of an initial guess atmosphere by trial and error until the synthetic proðles resemble the observed ones as closely as possible. In this process, each of the unknown parameters is modiðed with the others kept Ðxed, i.e., the various parameters that best Ðt the observations are determined independently. The drawback of these methods is that they introduce ambiguities in the results because the 475 whole space of parameters cannot be sampled efficiently. Despite being much more involved, inversion techniques Ðrst determine how the emergent spectra respond to perturbations of the atmospheric parameters (Ruiz Cobo & del Toro Iniesta 1994) and then use this information to modify the initial guess model. In order to assure self-consistency, all parameters need to be considered and modiðed simultaneously. The relevance of this approach for a precise and reliable determination of the atmospheric structure has been discussed in detail by del Toro Iniesta & Ruiz Cobo (1996). At present, running and tested inversion techniques that Ðt the whole shape of observed Stokes spectra include the Milne-Eddington (ME) procedure by Skumanich & Lites (1987), SIR (Stokes Inversion based on Response functions) by Ruiz Cobo & del Toro Iniesta (1992), and the inversion code for MIcro-Structured Magnetic Atmospheres (MISMAs) developed by Sa nchez Almeida (1997). A comparison of the Ðrst two techniques has been carried out in a recent paper (Westendorp Plaza et al. 1998), where the reliability of the inferred parameters was also investigated. Neither of these procedures speciðcally account for the geometry of unresolved magnetic elements, but ME inversions of Stokes spectra from plage regions have been completed successfully (Mart nez Pillet, Lites, & Skumanich 1997). Numerical tests on the performance of ME inversions in Ñux-tube scenarios indicate that the retrieved geometric and magnetic parameters agree well with those used to construct the synthetic tubes (Skumanich, Grossmann-Doerth, & Lites 1992). More quantitatively, Westendorp Plaza et al. (1998) have demonstrated that ME inferences of a given parameter are line-of-sight averages of its actual stratiðcation with depth. These satisfactory results imply that the information on the properties of solar magnetic elements gained from ME inversions is an excellent starting point for more sophisticated analyses. SIR, on the other hand, has been applied to a number of problems including sunspots (Collados et al. 1994; Westendorp Plaza et al. 1997), penumbrae (del Toro Iniesta, Tarbell, & Ruiz Cobo 1994), solar granulation (Rodr guez Hidalgo et al. 1995), and solar oscil-

2 476 BELLOT RUBIO ET AL. Vol. 535 lations (Ruiz Cobo, Rodr guez Hidalgo, & Collados 1997). No attempt to investigate facular and network regions with SIR has ever been made because a proper treatment of the geometry of these structures was deemed necessary. Finally, the inversion code by Sa nchez Almeida has proved to be able to reproduce Stokes spectra observed in quiet-sun and plage regions (Sa nchez Almeida & Lites 2000). At present, the sole inversion code implementing the thin Ñux-tube model is that of Keller et al. (1990). This code was prepared to reproduce a set of observables extracted from Stokes V. In the end, 11 model parameters are determined: the temperature and macroturbulent velocity stratiðcations within the tube (represented by Ðve parameters each), and the magnetic Ðeld strength at a given height in the atmosphere. The major limitation of the code is its inability to reproduce the asymmetries of Stokes V because no line-ofsight velocity gradients are considered. In addition, the code uses a prescribed external atmosphere that is not modiðed during the inversion. In spite of these drawbacks, the importance of the technique of Keller et al. (1990) is beyond doubt because of its pioneering approach. Characterizing unresolved magnetic elements from spectropolarimetric measurements is a complex problem because of the large number of unknowns. The increasing efficiency of modern computers, however, makes it possible to extract the wealth of information encoded in the Stokes spectra. With this in mind we have developed a new LTE inversion code for the analysis of solar magnetic elements. It has been prepared to reproduce the shape of observed Stokes proðles by means of a standard least-squares Marquardt algorithm. The signiðcant challenge of our approach is the simultaneous inference, within the thin Ñux-tube scenario, of the whole set of parameters considered relevant for the problem. These include the stratiðcation with depth of temperature, magnetic Ðeld strength, and LOS velocity. Our ultimate goal is to characterize the magnetic, thermal, and dynamic properties of the elusive constituents of plage and network regions; eventually, this will result in model atmospheres capable of explaining the observed Stokes spectra and, more particularly, the amplitude and area asymmetries of Stokes V. Knowledge of the properties of Ñux tubes in photospheric layers is important to understand the physical processes responsible for the heating of the solar chromosphere and corona and may also provide theoretical models of magnetic Ñux emergence with observational constraints. The paper is organized as follows. First, we brieñy review the basics of the inversion technique ( 2). We then summarize the properties of the thin Ñux-tube model adopted for the inversion ( 3). The synthesis of Stokes spectra is explained in 4. In 5, an analysis of the uncertainties of the inferred model parameters is carried out. Here we discuss the various sources of error and try to quantify them in a systematic way. Section 6 reports on several tests aimed at characterizing the performance of the code. In that section we investigate the dependence of the solution on the signalto-noise ratio and on the set of spectral lines used. Finally, we summarize the results of various numerical simulations in INVERSION OF STOKES PROFILES By inverting Stokes proðles we aim at obtaining the physical conditions of the atmospheres in which they originate. This involves the comparison of observed I obs(j) and synthetic I syn(a, j) spectra. The latter are computed with models intended to represent the structure of solar magnetic elements. The values of the parameters that deðne the model are the basic ingredients with which the radiative transfer equation can be solved to yield I syn(a, j). The resulting Stokes spectrum is a (nonlinear) function of these parameters, denoted here by the vector a. In the general case, the N ] p ] r components of a are the p physical quantities varying with depth in the grid of N points used to discretize the atmosphere, and the r single-valued parameters of the model. During the inversion, di erences between observed and synthetic spectra are used to modify a so that a better agreement is reached. The inversion itself proceeds by minimizing a merit function, which is the sum of the squared di erences between the observed and synthetic data weighted by the uncertainties of the observations and by some factors w2. The merit func- tion is deðned as ki s2 4 1 l ; 4 M ; [Ik obs(j ) [ Isyn(a, j )]2 w 2 ki i k i p2, (1) k/1 i/1 ki where index k \ 1,..., 4samples the 4-component vectors containing the observed (obs) and synthetic (syn) Stokes proðles, i \ 1,..., M sample the wavelengths j at which i the spectrum is known, l is the number of degrees of freedom (i.e., the number of observables minus the number of parameters to be inverted), and p are the uncertainties ki of the observations. Usually, the factors w2 are chosen so as ki to give the same relative weight to the Stokes parameters of di erent spectral lines, independently of their amplitudes. The minimization of equation (1) is carried out by means of a Marquardt algorithm (Press et al. 1986), whereby the nonlinear problem is converted to the system of linear equations +s2]a@da \ 0. (2) +s2 stands for the gradient of the merit function, and A@ is a modiðed Hessian matrix containing the second-order derivatives of s2 with respect to the model parameters a, A@ 4 L2s2 (1 ] "d ). (3) jk La La jk j k Here, d is Kronecker delta and " is an arbitrary factor to control jk the convergence in the parameter space. Note that A@ becomes the standard Hessian matrix A when " \ 0. In order to solve equation (2) for the variation da that leads to a better agreement between observed and synthetic Stokes spectra, +s2 and A@ need to be computed. As shown below, these two quantities can be approximated quite straightforwardly by products of response functions (RFs). RFs are the partial derivatives of the emergent Stokes proðles with respect to the atmospheric parameters (see Ruiz Cobo & del Toro Iniesta 1994 and references therein). Let disyn be the variation of the emergent Stokes spectrum induced by changes da in the model parameters a. To Ðrst order, disyn is given by j j disyn(j) \ ; R (j)da *z, (4) aj j j where R (j) represents the RF of Isyn to changes in a, and *z is the aj spacing adopted to discretize the atmosphere. j From equation (1), the variation of s2 after a pertur-

3 No. 1, 2000 INVERSION OF STOKES PROFILES 477 bation disyn can be written as ds2\[ 2 l ; 4 M Iobs(j ) [ Isyn(a, j ) ; k i k i w2 disyn(j ). (5) p2 ki k i k/1 i/1 ki Combining equations (4) and (5), the partial derivative of s2 with respect to a turns out to be q Ls2 \[ 2*z 4 M Iobs(j ) [ Isyn(a, j ) ; ; k i k i w2 R (j ), (6) La l p2 ki aq,k i q k/1 i/1 ki and the second-order derivative of s2, neglecting the dependence of R on a, aq,k j L2s2 ^ 2*z 4 M w2 ; ; ki La La l p2 R aq,k (j i )R aj,k (j i ). (7) q j k/1 i/1 ki The modiðed Hessian matrix A@ can therefore be approximated by products of RFs. According to Press et al. (1986), neglecting the dependence of R on a prevents the development of numerical instabilities. aq,k j The inversion problem, whereby perturbations da to a guess atmosphere a are found, is solved by minimizing s2. MarquardtÏs algorithm transforms this problem into one of solving equation (2), whose ingredients are all known. Hence, the problem is reduced to the inversion of matrix A@. In practice, two difficulties arise when doing so. On the one hand, A@ may have large dimensions, especially if many grid points are used and/or many model parameters are to be found. The inference of N ] p ] r parameters would lead to dim(a@) \ N ] p ] r. The inversion of large matrices is very expensive from the computational point of view. On the other hand, some model parameters may not produce any signiðcant variation of s2. In this case, the system of linear equations (2) is ill conditioned. To handle the Ðrst problem, the perturbations of the model parameters varying with depth are computed only in a few grid points called nodes. For each physical quantity p (e.g., temperature, LOS velocity, etc.), the atmosphere is represented by a di erent set of nodes. Let ap denote the r reduced set of m model parameters corresponding to p at p the nodes (of course, ap \ ap \ a). Once the perturbations r dap have been found, the perturbations of ap in the whole r atmosphere are approximated by interpolation of dap, i.e., r mp dap \ ; f p dap i \ 0,..., N, (8) i ij r,j j/1 where f p represent interpolation coefficients whose values are determined ij by the discretization of the atmosphere and by the speciðc location of the nodes.1 For simplicity, the nodes are evenly distributed throughout the atmosphere. The code allows the user to change the number of nodes after a cycle of iterations has Ðnished. In this way, the solution is approached iteratively with an increasing number of free parameters. The variation of the emergent Stokes spectrum produced by changes in ap is written in terms of the perturbations at the nodes as N N mp disyn \ ; Ra p dap *z \ ; Ra i/1 i i p ; f p dap *z, (9) i/1 i ij r,j j/1 1 For one node, the same perturbation is applied to all the elements of ap; hence f p \ 1,#i. For two and three nodes, linear and parabolic inter- polations are i1 used, respectively. With four or more nodes, cubic-spline interpolation is applied. or, equivalently, mp A B N mp disyn \ ; ; Ra p f p dap *z 4 ; R3 j/1 i/1 i ij r,j aj p dap *z, (10) r,j j/1 where R3 are equivalent RFs at the nodes, obtained as aj linear combinations p of the RFs through the whole atmosphere. Equivalent RFs ensure that all depth points contribute to s2 even though only some of them are actually used in the inversion. Instead of the original +s2 and A@, the gradient + s2 and the modiðed Hessian matrix A@ for the reduced set r of parameters at the nodes are constructed r in terms of equivalent RFs. The inversion is then carried out with fewer parameters (the physical quantities at the nodes), which greatly simpliðes the problem. Implicitly assumed in this strategy is that the depth stratiðcation of the di erences between the current and the actual parameters is smooth, at least to the extent that it can be approximated locally by third-order polynomials. The second problem, that A@ is normally a quasi-singular matrix, is handled by the singular r value decomposition (SVD) method (Press et al. 1986). SVD expresses A@ in terms of its eigenvalues and eliminates those smaller r than a certain threshold to guarantee that det(a@) ^/ 0. A major r disadvantage of this procedure is that the role of some physical quantities is neglected. This is because their e ects on the emergent Stokes spectrum are much less important than those produced by other model parameters. Temperature, having the largest inñuence on the Stokes spectrum, is always favored to the detriment of parameters such as magnetic Ðelds or LOS velocities. To solve this problem, Ruiz Cobo & del Toro Iniesta (1992) developed a modiðed SVD method in which the contribution of each group of parameters (temperature, velocity, etc.) to the eigenvalues of A@ is separated. More speciðcally, the diagonal matrix con- r taining the eigenvalues of A@ is decomposed as a sum of r matrices each relevant to one group of parameters. For these diagonal matrices, elements smaller than the maximum multiplied by a tolerance factor e are eliminated. Hence, all physical quantities are considered for the solution of equation (2), since at least one element representing each group of parameters is kept in A@. r In order to carry out the inversion, the synthetic spectrum Isyn(a, j) needs to be constructed Ðrst. This requires a prior knowledge of the model parameters a. We address this problem in the next section, where the model selected for further treatment is described. 3. THE MODEL Solar magnetic elements are small structures embedded in Ðeld-free surroundings. In this section we give the rationale for adopting the thin Ñux-tube model and identify the free parameters of the model. As pointed out by Auer & Heasley (1978), the absence of line-of-sight velocity gradients leads to symmetric Stokes proðles. Hence, velocities have to be considered in order to reproduce the asymmetries exhibited by Stokes V.Sa nchez Almeida, Collados, & del Toro Iniesta (1988) demonstrated that the area asymmetry can be explained by the combined e ects of a magnetic Ðeld increasing with height and a downñow decreasing with height. In principle there is no obvious reason for the increase of the magnetic Ðeld with height, but this picture cannot be rejected without Ðrst considering the existence of magnetic canopies. Imagine an

4 478 BELLOT RUBIO ET AL. Vol. 535 interlaced atmosphere consisting of two components. One of them is magnetized and lies on top of the second, which is nonmagnetic. Assume further that the LOS velocities are zero in the Ðrst component but nonzero in the second. If analyzed in terms of a continuously varying atmosphere, this particular interlaced atmosphere would be naturally interpreted as magnetic Ðelds increasing with height, which is the net e ect of the jump from the nonmagnetic to the magnetic region. This example illustrates the fact that interlaced atmospheres can be suitable for modeling unresolved magnetic elements. Of course, more complex geometries are to be expected. Among these, the simplest is that of thin Ñux tubes (Defow 1976), i.e., axisymmetric structures whose diameters are smaller than any scale height along the tube. Zero-order thin Ñux tubes are characterized by the horizontal constancy of the physical parameters from the axis to the external, Ðeld-free medium. In particular, the magnetic Ðeld vector is assumed to be parallel to the axis. Such a simpli- Ðed model does not allow for radial variations of the magnetic Ðeld, temperature, or velocity within the magnetic elements. It also does not include horizontal motions resulting from the dynamical interaction with convective Ñows. In spite of these limitations, however, the actual structure of unresolved magnetic elements is thought to be well represented by thin Ñux tubes (e.g., Steiner & Pizzo 1989; Zayer, Solank, & StenÑo 1989). Because of its relative simplicity, this scenario seems adequate for trying to reproduce the observed Stokes spectra. More complex scenarios would be needed only where the thin Ñux-tube model was unable to explain the observations. Recently, Sa nchez Almeida et al. (1996) have proposed the MIcro-Structured Magnetic Atmosphere (MISMA) model as an alternative description of solar magnetic elements. MISMAs are characterized by the existence of structures much smaller than the mean free path of the photons. This scenario has already been used to Ðt plage and quiet-sun spectra, but the assumption of stochastic distributions of magnetic Ðelds and velocities within the same resolution element introduces additional complexity. It is our belief that the natural procedure is to test models having increasing degrees of complexity, hence the implementation of the Ñux-tube model in our code. Thin Ñux tubes are subject to a number of physical constraints including horizontal pressure equilibrium at the boundary layer, hydrostatic equilibrium, and magnetic Ñux conservation. The assumption of hydrostatic equilibrium could be more valid than in SIR inversions of, for example, sunspot penumbrae, where adjacent lines of force act as totally independent atmospheres. Using indices ]1 and [1 to distinguish between magnetic and nonmagnetic atmospheres, the relevant equations are (z) ] P`1 B2(z) 8n \ P ~1 (z), C P (z) \ P exp [ g z k (z@) dz@d B1, B1 0,B1 R T (z@) z0 B1 B(z)r2(z) \ B r2, 0 0 where g is the gravitational acceleration and R is the gas constant. The gas pressure at geometrical height z (positive outward) is represented by P (z), the magnetic Ðeld B1 strength by B(z), the temperature by T (z), the mean B1 P molecular weight by k (z), and the radius of the tube by r(z). We have deðned B1 P 4 P (z ), B 4 B(z ), and r 4 r(z ) at a certain reference 0,B1 layer B1 0 z that 0 is 0 usually placed 0 at 0 the bottom of the atmosphere. 0 The thermodynamic properties of the plasma are described by the equation of state for an ideal gas with variable mean molecular weight k (z) to take into account the partial ionization of the various B1 atomic species. These constraints diminish the number of free model parameters. Given the external temperature stratiðcation and the external gas pressure P at z, hydrostatic equilibrium determines the gas pressure 0,~1 stratiðcation 0 outside the magnetic element. With the magnetic Ðeld strength B at the reference layer and the internal temperature, the 0 stratiðcation of the internal gas pressure is obtained by applying hydrostatic equilibrium. Horizontal pressure balance yields the Ðeld strength stratiðcation within the tube, while magnetic Ñux conservation determines the shape of the magnetic element once its radius r is given. Thus, the structure of a thin Ñux tube is completely 0 speciðed by P, B, r, and the temperature stratiðcations T (z) inside 0,~1 and outside 0 0 the tube. The runs with depth of the B1 LOS velocity v (z), micro v (z), and macroturbulence v (z) B1 mic,b1 mac,b1 have to be considered as well for the computation of synthetic spectra. Since large-scale turbulent motions a ect the magnetic interior and the surroundings in much the same way, a single height-independent macroturbulence v is mac adopted for both atmospheres. Flux tubes spread out with height as required by magnetic Ñux conservation. They can occupy either the whole resolution element or only a fraction f (z) of it. The remainder 1 [ f (z) is Ðlled with a nonmagnetic atmosphere. Except for very high spatial resolution observations, granular and intergranular structures coexist in the vicinity of the magnetic elements. Hence, at least a third (mean) atmosphere is necessary to reproduce the observed Stokes spectra. Such an atmosphere has to be taken into account together with the magnetic interior and the nonmagnetic medium adjacent to the walls of the tubes. At least two reasons argue that this third atmosphere is not necessarily well represented by any standard quiet-sun model. On the one hand, the proportion of granular and intergranular elements does vary from one point to the next. On the other hand, granulation shows abnormal properties in active regions (e.g., Muller 1989; Title et al. 1989; Collados, Rodr guez Hidalgo, & Manso Sainz 1998a, 1998b). The inversion of a third atmosphere intended to represent the granular medium would increase the number of model parameters which, in turn, would demand a larger number of observables. When high spatial resolution is available, however, it can be assumed that the third atmosphere is well represented by the external medium outside but close to the boundaries of the magnetic structures. Therefore, in order to keep the problem tractable we restrict the analysis to high spatial resolution observations and adopt a single nonmagnetic atmosphere to describe the surroundings. Accordingly, in its present form the code deals with only two atmospheres. It is possible that Ñux tubes merge with one another, but the merging is thought to occur well above the region where photospheric lines are formed (see, e.g., the modeling by Steiner & Pizzo 1989, and Bu nte, Solanki, & Steiner 1993). Since we limit ourselves to LTE conditions, such high atmospheric layers are not appropriately considered in our analysis.

5 No. 1, 2000 INVERSION OF STOKES PROFILES 479 To summarize, the full set of model parameters determining the structure of the tube and the radiative transfer is T (z), v (z), v (z), v, P, B, and r. These quantities, B1 B1 together mic,b1 with the mac stray 0,~1 light 0 factor a, 0 are the components of vector a needed to compute the synthetic spectrum I syn(a, j). 4. SYNTHESIS OF STOKES SPECTRA Spectral synthesis is the process whereby the Stokes spectrum emerging from an atmosphere described by the model parameters a is computed. In this section we outline the basics of the radiative transfer through both single and interlaced atmospheres Radiative T ransfer through Single Atmospheres Consider a steady, plane-parallel atmosphere in which radiation is described by the Stokes vector I 4 (I, Q, U, V )T. I represents the intensity, and Q, U, and V the polarization state of the light. The Stokes vector is governed by the radiative transfer equation (RTE) di(z) \[K(z)[I(z) [ S(z)]. (11) dz K(z) is the total absorption matrix (see, e.g., Rees 1987 for an explicit deðnition), and S(z) the source function. The formal solution of the RTE is P z I(z) \ O(z, z )I(z ) ] O(z, z@)k(z@)s(z@)dz@ (12) 0 0 z0 (Landi DeglÏInnocenti & Landi DeglÏInnocenti 1985; Landi DeglÏInnocenti 1987), where O(z, z@) stands for the evolution operator. In particular, if z labels the point from which lim K(z) can be considered zero outward and z the bottom of 0 the atmosphere, the emergent Stokes spectrum is given by I(z ) with O(z, z )I(z ) ] 0. Only seven of the 16 ele- lim lim 0 0 ments of O(z, z@) are independent (Sa nchez Almeida 1992). The evolution operator, however, does not admit an analytical expression except in very few cases (notably in Milne-Eddington atmospheres). This is an unfortunate circumstance as far as the evolution operator is required for the calculation of response functions. According to equation (11), the radiation Ðeld is completely speciðed by the model parameters a, which, together with the relevant atomic parameters, deðne both the absorption matrix and the source function. If LTE conditions hold and the continuum radiation is unpolarized, the source function becomes S \ [B (T ), 0, 0, 0]T. Here, B (T ) represents the Planck function at l frequency l and local temperature T. l Partly because of the large number of spectral syntheses required, inversion techniques are very expensive from the computational point of view. Thus, the adoption of an efficient integrator of the RTE is of interest for diminishing the computer workload. Our inversion code implements a new Hermitian method based on the Taylor expansion of I(z) to fourth order in geometrical height (Bellot Rubio, Ruiz Cobo, & Collados 1998). The Hermitian algorithm is superior to other methods in terms of speed and accuracy, which e ectively helps accelerate the solution of the polarized RTE Radiative Transfer through Vertical T hin Tubes Thin Ñux tubes are characterized by the radial constancy of the physical parameters from the axis to the nonmagnetic surroundings that, for simplicity, are assumed to be horizontally constant as well. Hence, they can be modeled as structures made up of two di erent atmospheres: the magnetic interior and the external medium. At vertical incidence, any ray piercing the tube Ðrst encounters a Ðeld-free region and then evolves in the presence of a magnetic Ðeld. For these rays, the Stokes spectra at any height can be obtained as a linear combination of the solutions of the RTE through each atmosphere considered independently (del Toro Iniesta et al. 1995). Therefore, only two integrations of the RTE are needed to compute the Stokes spectra of all rays traversing the tube. This result greatly reduces the computer time required for the calculation of synthetic spectra, but has some limitations. It cannot be applied, for instance, when there are radial variations of the physical parameters. In this case the integration of the RTE has to be carried out independently for each ray. From now on we assume that the atmospheres are discretized in a grid of points z (i \ 0,..., N) with an even i spacing *z. z and z are, respectively, the lowest and 0 N highest points of the grid. The emergent Stokes spectrum of a ray that pierces the tube at height z (or distance r from j j the axis) will be denoted by I(z ; z ). In particular, I(z ; z ) N j N 0 represents lines of sight traversing the magnetic cylinder of radius r within the Ñux tube, and I(z ; z ) represents the 0 N N Stokes spectrum of lines of sight going through the external atmosphere. The spectrum emerging from the resolution element, I(z ), is computed as a weighted sum of the spectra of di er- N ent rays whose distances to the axis range from 0 to r, plus N the contribution of the nonmagnetic surroundings that lie within the resolution element, i.e., N I(z ) \ nr2 I(z ; z ) ] 2n ; I(zN ; z )r *r N 0 N 0 j j j j/1 ] n(r2 [r2)i(z ; z ), (13) res N N N where r and r are the radii of the Ñux tube at z and z, 0 N 0 N respectively; r is the radius of the resolution element; and res 2nr *r is the area on the solar disk sampled by the jth ray. If the j resolution j element does not cover the entire Ñux tube (i.e., if r [ r ), the summation in equation (13) is truncated at r \ N r. Obviously, res the last term of equation (13) van- ishes j in this res case. The evaluation of equation (13) is straightforward once the model parameters a are speciðed. The synthetic Stokes spectrum to be compared with the actual proðles is Isyn(a) \ (1 [ a)i(z N ) ] ai str, (14) where I represents the stray light proðle. The last ingredients needed str for the inversion are the RFs of I(z ) to the various model parameters. These were derived by N Bellot Rubio, Ruiz Cobo, & Collados et al. (1996, 1997a; see also Bellot Rubio 1998). 5. CONFIDENCE LIMITS ON THE RETRIEVED MODEL ATMOSPHERES The s2 minimization carried out by the inversion code yields the set of optimum model parameters a that produce

6 480 BELLOT RUBIO ET AL. Vol. 535 the best Ðt to the observed Stokes spectrum. The question naturally arises as to how reliable the derived parameters are. From the very beginning we may distinguish three sources of error: (1) possible inadequacies of the adopted assumptions; (2) propagation of measurement errors; and (3) approximations and numerical limitations of the inversion algorithm. In practical applications, the breakdown of hypotheses is by far the most important source of error. The adoption of a given scenario determines how the inversion is performed in so far as various constraints are used. If such constraints do not properly embody the physics of solar magnetic elements, the e ects they induce on the synthetic Stokes spectrum are likely to be compensated by spurious changes in the model parameters. The problem worsens either if the inversion procedure does not take all model parameters into account at the same time or if some of them are Ðxed a priori. Our code performs a simultaneous determination of the whole set of model parameters to avoid this additional increase of the error. The extent to which the inadequacy of the assumptions inñuences the results, however, is not easily quantiðable. We have tried to keep such uncertainties as small as possible by selecting reasonable constraints that we now discuss brieñy. 1. The adoption of the thin Ñux-tube scenario is somewhat arbitrary but reasonable in view of its simplicity. Flux tubes include most of the physics thought to play a role in small-scale magnetic elements. Other models cannot be ruled out a priori, but the ability or inability of the simpler Ñux-tube model to reproduce the observations still needs to be ascertained, especially since mass Ñows have always been neglected. 2. Departures from LTE conditions are minimized by choosing spectral lines that are weakly susceptible to NLTE e ects. It is clear, however, that a proper treatment of these e ects is necessary for the analysis of high atmospheric layers. The problem of NLTE inversion of Stokes proðles is currently being addressed by Socas-Navarro, Ruiz Cobo, & Trujillo Bueno (2000). Our inversion code has been prepared to admit a straightforward implementation of their NLTE routines. 3. The merging of adjacent tubes is thought to occur high in the atmosphere, so it cannot inñuence the Stokes spectrum of the lines we consider. The analysis is carried out under the assumption that Ñux tubes do not interact with each other. This may not apply in strong plages where a dense clustering of magnetic elements is expected. 4. High spatial resolution spectra are used in an attempt to cope with our present inability to infer the parameters of more than two atmospheres from the observations. The spatial resolution currently available, however, is not enough to isolate individual magnetic elements, as a consequence of which the derived model parameters probably represent a statistical average of di erent tubes in di erent evolutionary stages. This problem will be overcome in future with data of higher spatial and temporal resolution. 5. Highly inclined tubes are not allowed. These are identiðed by inspecting the emergent linear polarization, the entire spectrum being rejected if either Stokes Q or U is nonnegligible. An extension of the code to deal with inclined tubes is planned for the near future. Such an extension is by no means straightforward because of the crucial role that geometry plays for features outside disk center. In particular, the RFs derived by Bellot Rubio et al. (1996, 1997a) would need to be reformulated from scratch. 6. The hypothesis of radial constancy of the physical quantities within the tube is perhaps the most severe constraint on the analysis. We feel, however, that this is a reasonable approximation. Departures from such a simpliðed scenario cannot be accounted for by our model, but they seem to be negligible in photospheric layers according to recent numerical calculations (e.g., Steiner et al. 1998). Another kind of uncertainty stems from measurement errors. The noisier the observed Stokes spectrum, the wider the range of variation of the model parameters that provide a Ðt to the data. It is customary to deðne the uncertainty of parameter a, p, in terms of the variation *s2 induced when a is substituted j aj by a ] p. The standard procedure for the j determination of conðdence j aj limits on the retrieved model parameters is explained in detail by Press et al. (1986). By taking advantage of the information contained in the diagonal elements of the inverse of the Hessian matrix, uncertainties that account for correlations among the model parameters are derived. The standard formula reads p2\2[a~1] s2 /n, (15) aj jj min where s2 is the value of s2 resulting from the Ðt, and n the min number of observables. This formula is valid under the assumption that the minimum of the merit function has been reached on exit. When this is not the case, Sa nchez Almeida (1997) suggests the use of the modiðed error estimate p2\2[a~1] s2 /m, (16) aj jj min m being the number of free model parameters. Since m is always smaller than n, these errors turn out to be larger than the standard ones by the constant factor (n/m)1@2. Whatever the error estimate, the uncertainties p are proportional to the inverse of the RFs to changes aj in a. j Therefore, parameters that have little inñuence on the emergent Stokes spectrum show the largest uncertainties. This is the case for most parameters outside the region where the spectral lines used are formed. Uncertainties expressed by p yield the range of variation of the parameters, considered aj independently, within which the quality of the Ðt (i.e., the s2 value) is maintained. Of course, it is always desirable that the error bars embrace the correct model parameters, even if the inversion results in inferred parameters that do not match the actual ones. However, it is similarly undesirable that the error be overestimated, since uncertainties that are too large may hide important physical processes. In order to test the adequacy of equations (15) and (16), we have carried out the inversion of a number of Stokes I and V proðles (with varying signalto-noise ratios) emerging from a prescribed reference atmosphere. Di erent initializations were used for this purpose. As many as 775 model parameters have been inferred and their uncertainties computed according to equations (15) and (16). Since the actual parameters are known, it is possible to compare the di erences between the estimated and correct values of the parameters with the uncertainties resulting from equations (15) and (16). This analysis is displayed in Figure 1, where circles represent standard errors, and crosses modiðed errors. Figure 1a depicts all the estimated uncertainties. A cursory glance at that Ðgure reveals the large dispersion of the estimates: neither of the two

7 No. 1, 2000 INVERSION OF STOKES PROFILES 481 FIG. 1.ÈComparison of the uncertainties estimated from eqs. (15) (circles) and (16) (crosses) with the di erences between actual and inferred parameters resulting from the inversion of a number of synthetic Stokes proðles. These di erences are identiðed as the error of the retrieved parameters. Relative quantities (i.e., normalized to the parameter value) are shown. (a) Individual estimated uncertainties vs. actual error. (b) Average of the estimated uncertainties vs actual error. formulae is able to reproduce the exact value of the di erence between inferred and actual parameters. If this were so, all the points would lie over the straight dash-dotted line. Note that errors above this line are overestimated, whereas those below are underestimated. The general trends of Figure 1a are more clearly seen in Figure 1b, where averages of the estimated uncertainties corresponding to given intervals of actual errors are depicted. Two main conclusions can be drawn from Figure 1b. First, that equation (15, circles) provides better estimates for the uncertainties than equation (16, crosses) does, which follows from the fact that they are closer to the straight line representing the one-to-one correspondence. This result can be understood by noting that only successful inversions were considered for analysis. In these inversions, the Ðnal s2 value was likely to be the true minimum, thus verifying the hypothesis under which equation (15) is valid. The neglect of unsuccessful inversions does not modify our conclusions, since in practical applications they are always rejected. Second, Figure 1b indicates that, on average, the uncertainties associated with small di erences between inferred and actual parameters tend to be overestimated, whereas those associated with large di erences are somewhat underestimated. The adoption of equation (15) is thus preferable, since it yields uncertainties closer to the straight line in the range of interest (i.e., the interval of actual relative errors from 10~2 to 0.5). When the relative error is smaller than, say, 10~3, overestimation of the uncertainty is of no concern because the error bars are already small enough so as to permit the derivation of statistically signiðcant conclusions. For very large relative errors, say in excess of 1.0, underestimation of the uncertainties is not a crucial problem because the errors are so large that nothing can be said about the corresponding parameters. In view of these considerations, we adopt the standard procedure (eq. [15]) for the computation of error bars. The last source of uncertainty stems from the approximations of the numerical algorithm. In this respect, it is worth mentioning here that no information can be extracted if it does not produce measurable e ects on the Stokes proðles. The information provided by the inversion is therefore limited to the height range at which the analyzed spectral lines are formed. A careful selection of the spectral lines to be inverted is thus highly desirable, not only to probe wider atmospheric regions but also better to constrain the inferred models. The main approximations of the numerical algorithm are related to the problem of inverting matrix A@. In particular, the use of cubic splines for interpolating the perturbations at the nodes may result in unwanted spatial oscillations of the physical quantities. Successful inversions do not normally introduce oscillations (they are either absent or negligible), but sometimes they become large enough to ruin the analysis. This problem is unavoidable unless other interpolation methods are adopted. Linear interpolation, the simplest choice, is less preferable than cubic splines because it cannot account for the curvature that the stratiðcation of the physical quantities usually has between nodes. Neglect of these curvatures may cause any inversion technique to fail in Ðnding the correct solution. Tests are being carried out to determine whether cubic splines under tension may remedy the problem of undulations. At present, a further limitation of the algorithm is the need to choose the number of nodes by hand. Experience reveals that a proper selection of nodes is crucial for the success of the inversion. However, information on the optimum number of nodes is already provided by the RFs. The larger the RF to a given parameter, the larger the sensitivity of the Stokes spectrum to that parameter and the larger the amount of information that can be obtained on it. This outlines a procedure for the automatic selection of nodes whereby the amplitude of the corresponding RFs determines the optimum number of nodes for each physical quantity. The implementation of this procedure is currently in progress. 6. NUMERICAL TESTS Testing the code with synthetic data is mandatory for assessing the reliability of the parameters inferred from the inversion of actual spectra. With this in mind we have simulated the Stokes spectra emerging from a number of reference atmospheres. After adding di erent noises to the synthetic proðles they were o ered for inversion. This procedure allows us to compare the retrieved atmospheres with the original ones. In this section we present the results of

8 482 BELLOT RUBIO ET AL. Vol. 535 some representative experiments. Further numerical tests are discussed in detail by Bellot Rubio (1998) Reference Atmosphere and Spectral L ines The synthetic proðles were generated from a particular Ñux-tube model consisting of two atmospheres: the magnetic interior and the nonmagnetic surroundings. For the axis of the tube we adopt the Holweger-Mu ller atmosphere (Holweger & Mu ller 1974) homogeneously cooled by [1000 K at equal geometrical heights to account for the lack of convective heating (Spruit 1976). The external medium is constructed from the intergranular atmosphere of Rodr guez Hidalgo et al. (1995). Denoting the temperature and LOS velocity of this intergranular model by T (z) and v (z), respectively, our external atmosphere is described i by i T (z) \ T (z) ] T ] T z, ~1 i 0 1 v (z) \ v (z [ 100), ~1 i where T \ 600 K and T \ 1 K km~1. The addition of the term T 0 z implies a gradient 1 of temperature that is absent in 1 the intergranular atmosphere of Rodr guez Hidalgo et al. Moreover, the LOS velocity stratiðcation is shifted toward higher layers by 100 km with respect to the intergranular model. The atmospheres are discretized in a grid of 37 points extending from z \[130 km to z \ 734 km. The 0 N radius of the tube at the base of the photosphere is r \ 50 0 km, the resolution element having 800 km in diameter. The Ðeld strength and the external gas pressure at z are 2000 G 0 and 2.69 ] 105 dyn cm~2. Additionally, we adopt a constant microturbulence of 0.6 km s~1 and a macroturbulence of1kms~1 in both atmospheres. The internal LOS velocities are given by (z) \ 0.2v (z) ] 0.2 (km s~1). The v`1 ~1 stray light factor is taken to be a \ 0.2. The stray light proðle is computed from the Harvard Smithsonian Reference Atmosphere (HSRA; Gingerich et al. 1971), i.e., I \ str (I,0,0,0)T. We stress that these models do not rep- HSRA resent any particular realistic scenario since our main purpose is to show some examples of how the retrieved atmospheric parameters compare with the original input. The radiative transfer is carried out for the Fe I lines listed in Table 1. We note that the inversion code is capable of dealing with a larger number of spectral lines. However, in most practical situations only a limited set of lines is available, so the present numerical tests are restricted to the following cases: the pair of Fe I lines at and A, and the full set of four lines. The Ðrst case simulates observations of the Advanced Stokes Polarimeter (ASP; see TABLE 1 ATOMIC PARAMETERS OF THE FE ILINES USED FOR TESTING THE INVERSION CODE j (A ) Transition s (ev) log gf g 0 e eff D È5D 3.21 [ P È5P 2.28 [ P È5D 3.64 [ P È5D 3.69 [ NOTE.ÈCentral wavelengths (j ) are from Nave et al s indicates the excitation potential 0 of the lower level. log gf represents e the logarithm of the degeneracy of the lower level times the oscillator strength of the line. g is the e ective Lande factor. eff Elmore et al. 1992), a state-of-the-art instrument managed by the High Altitude Observatory whose typical mode of operation is the recording of the Stokes spectra of Fe I and A. The second case is intended to simulate observations from a number of spectral regions. Telescopes like the German VTT in Teide Observatory can record up to four spectral regions simultaneously, thus enlarging the set of lines available for analysis Initial Guess Model and Inversion Conditions White noise leading to signal-to-noise ratios (S/N) of 1000 and 2000 in the continuum of Stokes I has been added to the synthetic spectra emerging from our reference model. The resulting proðles have been inverted from a number of initial guess atmospheres. Since the retrieved Ñux-tube structure is largely independent of the choice of these atmospheres, we illustrate here only the results obtained from a particularly unfavorable initialization. Normally, the closer the initial guess to the actual model, the faster the convergence to the correct solution. When dealing with real data, one often uses standard models of the solar photosphere to initialize the inversion in the hope that the actual (unknown) model is similar. However, one might wonder whether the inversion algorithm is robust enough to obtain correct atmospheres even when unphysical initializations are fed into the procedure. Our starting guess model has been constructed to investigate this issue. More speciðcally, we use a depthindependent temperature of 5000 K within the tube. The temperature of the external medium is taken to be that of the HSRA atmosphere plus 500 K. For both components the LOS velocity Ðeld has been set constant at v (z) \ 2.0 B1 km s~1. The initial Ñux-tube model has a radius r \ 20 km, 0 a magnetic Ðeld strength B \ 1000 G, and a external gas 0 pressure of 2.0 ] 105 dyn cm~2 at the base of the photosphere. For the macroturbulence we adopt 2 km s~1, the microturbulent velocity being initialized at the constant value of 0.6 km s~1 both inside and outside the tube. The stray light factor is taken to be a \ 0.5. The inversion of the proðles has been carried out in four iterative cycles, the number of nodes being kept constant in each cycle. From one cycle to the next, the number of nodes was increased so as to permit more Ñexibility to the solution. This is particularly important in the case of velocity, for example, since the stratiðcations of v are character- ized by steep variations with depth and B1 the initial models have constant velocities. As many as 42 free parameters are sought in the last iteration cycle. This gives an idea of the complexity of the problem: Marquardt has to Ðnd the minimum of s2 in a hypersurface of 42 dimensions Results We Ðrst present the results of the inversion of the four spectral lines. Figure 2 depicts the simulated observations with noise added at the level of S/N \ 2000 (dots), the best- Ðt proðles (solid lines), and the spectra emerging from the initial guess atmosphere (dashed lines). Residuals are shown in the lower panels of each subðgure. The rms di erence between simulated and synthetic spectra turns out to be 5 ] 10~4 in units of the continuum intensity. Note that the huge di erences between the simulated proðles and those emerging from the initial guess atmosphere are completely