THE ASTROPHYSICAL JOURNAL, 530:977È993, 2000 February 20 ( The American Astronomical Society. All rights reserved. Printed in U.S.A.

THE ASTROPHYSICAL JOURNAL, 530:977È993, 2000 February 20 ( 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A. NON-LTE INVERSION OF STOKES PROFILES INDUCED BY THE ZEEMAN EFFECT H. SOCAS-NAVARRO,1 J. TRUJILLO BUENO,1,2 AND B. RUIZ COBO1 Received 1999 July 29; accepted 1999 October 6 ABSTRACT This paper presents a new diagnostic tool for the inference of the thermal, dynamic, and magnetic properties of the solar chromosphere. It consists of a non-lte inversion code of Stokes proðles induced by the Zeeman e ect in magnetized stellar atmospheres. This code is the generalization, to the non-lte Stokes transfer case, of the inversion code for unpolarized line proðles of Socas-Navarro, Ruiz Cobo, & Trujillo Bueno. It is based upon a full non-lte multilevel treatment of Zeeman line transfer in which the thermal, magnetic, and dynamic properties of the atmospheric model are adjusted automatically by means of nonlinear least-squaresèðtting techniques until a best Ðt to the observed Stokes proðles is obtained. Our non-lte inversion approach is based on the concept of response functions, which measure the emergent Stokes proðlesï Ðrst-order reaction to changes in the atmospheric parameters. We generalize our Ðxed departure coefficients (FDC) approximation in order to allow fast computation of such response functions in the present non-lte Zeeman line transfer context. We present several numerical tests showing the reliability of our inversion method for retrieving the information about the thermodynamics and the magnetic Ðeld vector that is contained in the polarization state of the chosen spectral lines. We also explore the limitations of the inversion code by applying it to simulated observations where the physical hypotheses on which it is based on are not met. Finally, we apply our non-lte Stokes inversion code to real spectropolarimetric observations of a sunspot observed in the IR triplet lines of Ca II. As a result, a new mean model of the sunspot chromosphere is provided. Subject headings: line: proðles È methods: numerical È polarization È radiative transfer È stars: chromospheres È Sun: magnetic Ðelds 1. INTRODUCTION The development of spectral diagnosis methods that allow us to extract the maximum possible information from spectropolarimetric observations lies at the basis of solar and stellar spectroscopy. The goal is to learn about the physical conditions in stellar atmospheres by means of the application of inference methods that have the potential of yielding reliable atmospheric models capable of accounting for as many observed spectral features as possible. There is a minimum of physical consistency that the resulting models should have. This shall, in principle, be possible if we are able to introduce physically plausible constraints in our diagnosis methods. Clearly, the Ðnal goal is always to achieve a thorough understanding of the physical processes that lead to the observed spectral features. This requires us, additionally, to compare the results of magnetohydrodynamic simulations with the observations themselves. Although much progress has been made in recent years concerning the inference of the physical properties of the solar and stellar photospheres by means of inversion methods ÏÏ based on the local thermodynamic equilibrium (LTE) approximation (see the review by del Toro Iniesta & Ruiz Cobo 1996; see also Sa nchez Almeida 1997), much remains to be done concerning the diagnosis of spectral features formed ÏÏ in stellar chromospheres. By inversion methods,ïï we mean numerical algorithms that help us to go from the observed data to making inferences about the stellar atmosphere under study. The present paper brings a new contribution in this complicated research Ðeld (see, e.g., the review about chromospheric diagnostics and dynamics by Kneer & Von Uexku ll 1999). It deals with the development of a non-lte inversion code of Stokes proðles that 1 Instituto de Astrof sica de Canarias, E-38200, La Laguna (Tenerife), Spain; hsocas=iac.es, jtb=iac.es, brc=iac.es. 2 Consejo Superior de Investigaciones Cient Ðcas, Spain. 977 are induced by the Zeeman e ect in magnetized stellar atmospheres. This new code is based upon a full non-lte multilevel treatment of Zeeman line transfer in which the thermal, magnetic, and dynamic properties of the atmospheric model are adjusted by means of nonlinear leastsquaresèðtting techniques until a best Ðt to the observed proðles is obtained. We believe that this new diagnostic tool will Ðnd application in the investigation of chromospheric magnetic Ðelds (e.g., for the exploration of dynamical phenomena in sunspot chromospheres). Unfortunately, the situation is rather complicated because of several facts. First, the physics of line formation in the solar (stellar) chromosphere cannot be described using the LTE approximation. Thus, one needs to carry out efficiently non-lte multilevel radiative transfer numerical calculations, which is not an easy task mainly when the spatial three-dimensionality of the stellar chromospheric plasma (Fabiani Bendicho & Trujillo Bueno 1999) and/or the polarization in spectral lines (Trujillo Bueno & Landi DeglÏInnocenti 1996) is to be accounted for in a fully selfconsistent way. Second, both observations (e.g., Lites, Rutten, & Kalkofen 1993; Deubner 1998) and modeling (e.g., Carlsson & Stein 1998) indicate that the natural state of the quiet ÏÏ solar chromosphere is very dynamic, with waves leading to shocks and temperature Ñuctuations as high as several thousand kelvins. As demonstrated by Kneer (1980), for this type of dynamic scenario, even the currently made assumption of a statistical steady state is questionable because in the outer chromospheric layers ionization equilibrium is adjusted slowly compared with the dynamical timescales. Given the above and other extra complications, and the limited temporal and spatial resolution of spectro- (polarimetric) observations, it becomes clear that we have to progress carefully step by step. In this respect, the paper by

978 SOCAS-NAVARRO, TRUJILLO BUENO, & RUIZ COBO Vol. 530 Socas-Navarro, Ruiz Cobo, & Trujillo Bueno (1998, hereafter Paper I) considered the simpler problem of the non-lte inversion of intensity proðles, i.e., without polarization information and without taking into account magnetic Ðelds. In the present paper we go a step further and consider the case of polarization in spectral lines due to the Zeeman splitting produced by magnetic Ðelds stronger than approximately 100 G. For Ðeld strengths larger than this typical value, we can assume, as we do here, that quantum interferences (or coherences) among the Zeeman sublevels are negligible (see Landi DeglÏInnocenti 1983). As in Paper I, we consider a one-component, onedimensional plane-parallel atmosphere and make use of the complete angle and frequency redistribution approximation (CRD; see Mihalas 1978). Thus, with these approximations, and assuming also statistical equilibrium, the present paper is dedicated to describing in some detail our approach to the problem of non-lte inversion of Stokes polarization proðles. As mentioned above, we assume that such Stokes proðles are due to the Zeeman e ect. Future papers will consider the non-lte inversion problem of polarization signals due to scattering processes and to the Hanle e ect (see Landi DeglÏInnocenti 1985; StenÑo 1994; Faurobert- Scholl 1996; Manso Sainz & Trujillo Bueno 1999). We also have plans to address the issue of the non-lte inversion, assuming model atmospheres made of two or more components, either optically thick or optically thin, using a deterministic or stochastic description. Our approach is based on the combination of very efficient multilevel transfer methods for non-lte radiative transfer (see Socas-Navarro & Trujillo Bueno 1997) with a nonlinear least-squares minimization procedure based on the Levenberg-Marquardt algorithm (see Press et al. 1986). The key point here is the efficient calculation of the derivatives of the merit function with respect to each free parameter. Like the LTE inversion method of Ruiz Cobo & del Toro Iniesta (1992), our approach is based on the concept of response functions (RFs), which measure the Ðrst-order response of the emergent Stokes proðles to changes in the atmospheric conditions. Since we are not assuming LTE, an important point here has been the development of a suitable approximation that allows us to rapidly compute non-lte RFs and the required derivatives of the merit function. As in Paper I, we call it the Ðxed departure coefficients (FDC) approximation. We describe it here in more detail, showing also its generalization to the present case of non-lte Zeeman line transfer. This paper is organized as follows. Section 2 describes the way by which we solve the non-lte multilevel Zeeman line transfer problem and the method we apply in order to compute the emergent synthetic Stokes proðles that correspond to the current atmospheric model. This is the model produced, at each iteration, by the least-squares minimization algorithm. Section 3 is mainly devoted to the issue of non-lte RFs and to describing the above-mentioned FDC approximation. Section 4 summarizes the structure of our non-lte inversion code. Section 5 demonstrates that this inversion code works when it is applied to simulated polarization proðles corresponding to some chromospheric lines of interest. We then consider in 6 some examples of the limitations of our inversion code in its present stage of development. Section 7 shows an application to real spectropolarimetric observations of a sunspot umbra. These are simultaneous Stokes I and V observations of two of the infrared triplet lines of Ca II that we made using the German Gregory Coude Telescope at the Spanish Observatorio del Teide of the Instituto de Astrof sica de Canarias. Finally, 8 presents our conclusions, with a forecast of future research. 2. THE SYNTHESIS PROBLEM Solving the synthesis problem consists in calculating, for the spectral lines of interest, the Stokes parameters emergent from a given known model atmosphere. In our case, this known ÏÏ model atmosphere is simply the one corresponding to the current iterative step of the least-squares minimization algorithm. This synthesis involves the determination of the atomic level populations that are consistent with the radiation Ðeld generated in the particular model atmosphere by the chemical species considered. We introduce a number of simplifying assumptions to make the problem manageable. As in Paper I, we assume statistical equilibrium, CRD, and the one-dimensional plane-parallel approximation. Magnetic atmospheres are treated assuming that all the Zeeman sublevels pertaining to each given atomic level are equally populated and that there are no quantum interferences among them. In other words, we consider the case of non-lte Zeeman line transfer without atomic polarization (see Trujillo Bueno & Landi DeglÏInnocenti 1996 for details). With this simpliðcation, the unknowns of the problem are the same as those corresponding to the standard unpolarized multilevel transfer case; i.e., the overall population of each atomic level characterized by its total angular momentum. The only novelty is that, in the statistical equilibrium equations, instead of having each radiative transitionïs well-known J1-quantity (which is an average over frequencies and directions of the speciðc intensity weighted by the Voigt proðle), we now have a (which is an average over frequencies J1pol -quantity and directions of the Stokes parameters, each one of them weighted by its corresponding associated proðle). As pointed out by Trujillo Bueno & Landi DeglÏInnocenti (1996), this is a result that can be obtained (and understood) by neglecting atomic polarization e ects in the general statistical equilibrium equations presented by Landi DeglÏInnocenti, LandolÐ, & Landi DeglÏInnocenti (1976). Thus, as demonstrated by Trujillo Bueno & Landi DeglÏInnocenti (1996), the numerical solution of the multilevel Zeeman line transfer problem without atomic polarization can indeed be done with the same operator-splitting methods currently used for solving unpolarized multilevel transfer problems (see Socas-Navarro & Trujillo Bueno 1997 and references therein). In the unpolarized case, it suf- Ðces to perform a formal solution of the standard scalar transfer equation to calculate J1 and the diagonal of the "-operator. However, in the presence of a magnetic Ðeld, one should now perform, at each iterative step and for each radiative transition of the chosen atomic model, a formal solution of the full Stokes vector transfer equation, in order to calculate J1 and the diagonal of the "-operator (which depends now on pol the Zeeman splitting). Although the above is something that can indeed be done with present-day computers and numerical methods, it is, nevertheless, computationally expensive and, in any case, not really necessary. There exists a very good approximation (the polarization-free ÏÏ approximation of Trujillo Bueno & Landi DeglÏInnocenti 1996) that gives a fairly good account of the inñuence of a magnetic Ðeld on the atomic level populations without having to solve the full

No. 2, 2000 NON-LTE INVERSION OF STOKES PROFILES 979 Stokes vector transfer equation. With the polarization-free approximation, the solution of the non-lte multilevel Zeeman line transfer problem (without atomic polarization) goes practically as simply as with the standard unpolarized multilevel case. The only di erence is that one has to solve the scalar transfer equation for the speciðc intensity, using, instead of the standard Voigt line shape, the proðle /, which appears as the diagonal element of the 4 ] 4 line I absorption matrix. It is mainly through the shape of this / proðle (associated with each radiative line transition of the I chosen atomic model) that a magnetic Ðeld makes its e ect on the atomic level populations. However, as demonstrated by Bruls & Trujillo Bueno (1996) by means of Zeeman line transfer calculations that use realistic atmospheric models and multilevel atoms, such an e ect of the magnetic Ðeld on the statistical equilibrium is, in practice, very small, even in the presence of magnetic Ðeld gradients. This is because the atomic populations in real atoms are dominated by strong UV lines (only weakly split) and continua, and most lines with large Zeeman splittings (in the red and the infrared) are relatively weak and unable to produce signiðcant changes in the statistical equilibrium. Thus, in practice, it is justiðable to use the simpler Ðeld-free method (Rees 1969), which consists of using populations calculated with a standard nonmagnetic non-lte code and then simply applying a formal solution of the full Stokes vector transfer equation in order to obtain the emergent Stokes parameters that are due to the Zeeman e ect. In this manner, the departure coefficients of the atomic levels are independent of the magnetic Ðeld, and the non-lte radiative transfer is solved as simply as in the unpolarized case. In summary, the method used in this paper for the synthesis of Stokes proðles induced by the Zeeman e ect is the following: after evaluating the departure coefficients using a very efficient nonmagnetic multilevel transfer code (see Socas-Navarro & Trujillo Bueno 1997), the atomic levels are split into their corresponding Zeeman sublevels, and Ðnally a formal Stokes solution is carried out in order to calculate the four emergent Stokes proðles of the spectral lines selected for the inversion. We should now brieñy comment on the integration method that we use for solving the Stokes vector transfer equation and obtaining the synthetic Stokes parameters. We do this by applying a new method, which is an improvement on the diagonal element lambda operator (DELO) method of Rees, Murphy, & Durrant (1989, hereafter RMD89). Using the notation of RMD89, the vector radiative transfer equation along the line of sight is di dq \ I [ S, (1) where S \ S@ [ K@I, (2) with S@, K@, and dq as given by RMD89. The DELO method of RMD89 assumes that the e ective source function S varies linearly in the interval (q, q ). Since the Stokes vector I is already known at k`1 q k (assuming that the ray propagates from large to small k`1 values of k), the formal solution of equation (1) can be integrated analytically between q and q to obtain the fol- lowing expression for I(q ): k`1 k k I(q ) \ P ] Q I(q ), (3) k k k k`1 where P and Q are a known vector and a matrix, respectively. k k Instead of using the linear approximation described above, we expand the e ective source function S in the following way. The second term in equation (2), K@I, is still assumed to be linear between q and q, but the source function S@ is approximated by k`1 a parabola k that passes through q, q, and q. Note that, while S@ is known a priori through k`1 the k whole k~1 atmosphere, the term K@I is not. That is the reason why we have to assume a linear behavior for the latter. After some algebra, it can be seen that equation (3) is still valid but only with a di erent coefficient P, which is now given by k P \ [1 ] (F [ G )K@]~1 k k k k ] (( k`1 S k`1 @ ] ( k S k @ ] ( k~1 S k~1 @ ), (4) where the (-coefficients are similar to those given in the paper by Kunasz & Auer (1988). This improved DELO method is such that it reduces to the standard shortcharacteristics method of Kunasz & Auer (1988) when the Zeeman splitting is zero (i.e., when K@ \ 0). The advantage of this improved formal solution method is that it is signiðcantly more accurate than the standard DELO method. For example, note that for the case of negligible Zeeman splitting, the new method has parabolic accuracy, while DELO simply has linear accuracy (see Trujillo Bueno 1998). This improved DELO formal solver will be presented in more detail in a forthcoming paper, demonstrating also its usefulness for full non-lte Zeeman line transfer calculations. Before turning to the next section, we should emphasize that the previous discussion about the atomic level populations in the context of non-lte Zeeman line transfer refers to the particular case in which atomic polarization is neglected, i.e., to the case in which the populations of the Zeeman sublevels associated with each atomic level are assumed to be equal. This would indeed be the case if the rates of elastic collisions that destroy the atomic polarization were sufficiently large in the regions of line formation. However, we should point out that these rates do not seem to be important enough in the quiet ÏÏ solar chromosphere (see Landi DeglÏInnocenti 1998; Trujillo Bueno 1999). Atomic polarization has two possible manifestations: atomic orientation and alignment (see, e.g., the discussion by Trujillo Bueno & Manso Sainz 1999). One has atomic orientation when the populations of the Zeeman sublevels (pertaining to the same level) with magnetic quantum numbers M and [M are di erent. For this to occur, one needs a magnetic Ðeld (which splits the sublevels) and a macroscopic velocity Ðeld gradient such that, as a result of the radiative transitions due to the ensuing Doppler-shifted radiation Ðeld, the sublevels with magnetic quantum turn out to be more (or less) populated than those with number M (see Trujillo Bueno et al. 1993). Although the solar and stellar atmospheric plasma has both magnetic Ðelds and macroscopic mass motions, there is a physical argument that suggests that the errors we make by neglecting atomic orientation e ects should actually be small. This is because the level populations in realistic atomic models are controlled mainly by strong ultraviolet (UV) line transitions and continua. Since the Zeeman splitting scales with the wavelength and these UV lines are strong and broad, we

980 SOCAS-NAVARRO, TRUJILLO BUENO, & RUIZ COBO Vol. 530 believe that it is unlikely that the errors we make by neglecting atomic orientation are signiðcant. The other manifestation of atomic polarization is the alignment of the atomic levels, which happens when the populations of the sublevels with o M o D o M@ o are di erent. This certainly occurs in a stellar atmosphere, even if it is static and unmagnetized, since it suffices by having atoms illuminated by an anisotropic and/or polarized radiation Ðeld. In fact, the non-lte Zeeman line transfer problem cannot be handled rigorously without properly taking into account the inñuence of atomic alignment. From the twolevel model atom calculations of Bommier & Landi DeglÏInnocenti (1996) for isothermal atmospheres, we expect the inñuence of atomic alignment on the emergent Stokes proðles to be signiðcant only for weakly split lines and concerning only the linear polarization proðles Q and U. In conclusion, a word of warning is appropriate. We cannot safely exclude the e ects of atomic polarization until some ongoing research regarding the extent to which atomic orientation and alignment can modify the non-lte Zeeman polarization signals is completed. 3. NON-LTE RESPONSE FUNCTIONS Our non-lte inversion technique proceeds through a non-linear least-squares minimization of the di erence between the observed (Iobs) and the synthetic (Isyn) proðles. The synthetic spectrum is a function of the model parameters (x ) from which the model atmosphere is constructed. j The merit function to be minimized is deðned as s2\ 1 4 Nj [Iobs(j ) [ Isyn(j )]2 ; ; k i k i, (5) 4N p2(j ) j k/1 i/1 k i where i is the wavelength index, k stands for the components of the vector I (i.e., it stands for the four Stokes parameters), N is the number of points in the wavelength j grid, and p are appropriate weights that can be used to k Ðne-tune the inversion, e.g., giving more emphasis to those data that one considers to be the most reliable or compensating for the di erent amplitudes of the Stokes parameters. In their work, Ruiz Cobo & del Toro Iniesta (1992) assumed LTE. They pointed out that the required derivatives of the s2 function are given, except for some factors, by the RFs (see, e.g., Landi DeglÏInnocenti & Landi DeglÏInnocenti 1977). These have a clear (and useful) physical meaning, since they represent the Ðrst-order reaction of the emergent Stokes proðles to a small perturbation in a given physical parameter of the model atmosphere at a given height. If the LTE approximation is used, it is relatively straightforward to compute efficiently such response functions. However, in the present non-lte case, the problem becomes much more difficult because of the complicated (nonlinear and nonlocal) dependence of the line opacity and source function on the parameters of the model atmosphere. If a given model parameter (say x ) is slightly changed in a small geometrical interval *s@ centered j at the spatial point s@, there is certainly a response in the emergent Stokes vector (i.e., a reaction of I at the surface point s). Formally, we may write this response simply as di(s) \ di(s) dx (s@)*s@ dx j (s@)*s@ \ R(x j, s@)dx (s@)*s@, (6) j j where R(x, s@) is the response function. As a result, the ensuing change j in the s2 function reads ds2\[ 1 4 Nj [Iobs(j ) [ Isyn(j )] ; ; k i k i disyn(j ), (7) 2N p2(j ) k i j k/1 i/1 k i which shows that the changes in the merit function depend on the response functions. To gain some physical insight into the issue of how to calculate response functions without assuming LTE, it is Ðrst convenient to write down the formal solution of the linearized Stokes vector transfer equation. The Ðrst-order reaction of the emergent Stokes vector when the emission vector j and the absorption matrix K are perturbed is given by P s di(s) \ O(s, s@)[dj(s@) [ dk(s@)i(s@)]ds@, (8) s0 where s is the boundary of the integration domain, O(s, s@) is the evolution 0 operator between the points s and s@ (see Landi DeglÏInnocenti & Landi DeglÏInnocenti 1985), and dj(s@) and dk(s@) are perturbations at the depth-point s@ on the emission vector and absorption matrix, respectively. If a perturbation on a given model parameter (say x ) is j made at a particular spatial atmospheric point, the emission vector j and the absorption matrix K are modiðed. If LTE is assumed, then j and K react P only locally, and we can write s di(s) \ R(xj, s@)dx (s@)ds@, (9) j s0 with an analytical expression for the response function R given by CA LjB B D R(x, s@) \ O(s, s@) [ ALK I(s@). (10) j Lx Lx s{ s{ With this expression for the response function, one can compute very rapidly the required derivatives of the merit function. This is because, assuming LTE, one has analytical expressions for the partial derivatives of the emission vector and the absorption matrix that enter equation (10), and because the evolution operator O(s, s@) and the Stokes vector I(s@) can be computed efficiently while the formal solution of the Stokes vector transfer equation is carried out. If LTE is not assumed, a perturbation in the variable x at s produces a nonzero reaction of j and K not only at s, but j in i general at all the spatial atmospheric points s@. Formally, i we can express this nonlocal response by writing df (s@) df (s@) \ dx (s ) dx j (s i ), (11) j i with f being any element of the emission vector or the absorption matrix. Thus, formally, we can write the response function as follows: R(x, s ) \ di(s) j i P dx (s )*s j i i s \ O(s, s@) C dj(s@) dx(s ) [ dk(s@) I(s@)D ds@. (12) dx(s ) s0 i i To give a feeling for the degree of nonlocality of this reaction, we show in Figure 1, for the unpolarized nonmagnetized case, the global response of the source function and the opacity of the Ca II K line and one of the IR triplet lines to perturbations on the temperature made at several

No. 2, 2000 NON-LTE INVERSION OF STOKES PROFILES 981 FIG. 1.ÈResponse of the line source function (left) and line opacity (right) of Ca II K(top) and Ca II j8542 (bottom) to temperature perturbations at di erent depth-points in the atmosphere. The points where the perturbation is applied are marked by vertical dotted lines. The response to perturbations at log (q 500 ) \[6 has been multiplied by 5 in the upper panels and by 10 in the lower panels to make them visible in the plots. heights in model C of Vernazza, Avrett, & Loeser (1981, hereafter VAL-C). As seen in the Ðgure, these responses are the highest at the point where the perturbation is applied. Note also that they are highly local far away from the surface layers, but the nonlocality is nonnegligible near the surface where the response is the weakest. Motivated by our desire to calculate the required derivatives of the merit function efficiently, we now make the following approximation: df (s@) df (s@) B dx (s ) dx (s ) d(s@ [ s i ) ; (13) j i j i i.e., we assume that f reacts at s@ only if the perturbation on x is made at s \ s@. With this local approximation, the expres- sion for the i response function simpliðes to R(x j, s i ) \ di(s) dx j (s i )*s i \ O(s, s i ) C dj(s i ) dx j (s i ) [ dk(s i ) dx j (s i ) I(s i )D. (14) This expression should give a sufficiently good approximation, even for chromospheric non-lte lines. However,

982 SOCAS-NAVARRO, TRUJILLO BUENO, & RUIZ COBO Vol. 530 in order to be able to compute the required derivatives of the merit function rapidly, we still need to introduce a further approximation, namely, one that allows the analytical calculation of df (s )/dx (s ) (with f being any element of either j or K). To this end, i we j i make use of the FDC approximation of Paper I, which can easily be generalized to the present case of Stokes transfer. With the FDC approximation, the variation of j and K at s due to a perturbation on x at s is evaluated according to the i following steps: j i 1. Using the current model atmosphere, Ðnd the selfconsistent solution for the atomic level populations n of each level i. Compute the departure coefficients b \ n /n* i of the lower and upper levels of the line transitions i chosen i i for the inversion. 2. For each one of these line transitions, calculate the derivatives of the line opacity and of the line source function as done in equations (7) and (10) of Paper I, i.e., by assuming that the departure coefficients do not react to the perturbation on x. 3. With this information, j compute analytically the reaction of j and K to the perturbation in x and evaluate an j approximate value for the response function according to equation (14). Figure 2 shows an example of the goodness of this approximate way of calculating non-lte response functions for the Ca II K line. The Ðgure shows, for the four Stokes parameters, and assuming perturbations in the temperature, a comparison between accurate non-lte response functions calculated numerically (solid lines) and the FDC response functions. 4. STRUCTURE OF THE INVERSION CODE This section summarizes how our non-lte inversion code works. We denote the model parameters of the starting guess by the vector x. This is the vector containing the free parameters that can be modiðed by the inversion algorithm to Ðnd the model atmosphere that leads to synthetic Stokes proðles that best Ðt the observations. For this starting guess atmosphere, we solve the statistical equilibrium equations jointly with the scalar radiative transfer equations to Ðnd the non-lte (NLTE) atomic level populations that are consistent with the radiation Ðeld in the atmosphere (neglecting the e ects of the magnetic Ðeld, i.e., using the Ðeld-free approximation). With the populations obtained we compute the absorption matrix and the emission vector, and carry out a Stokes formal solution for the lines present in the observed spectrum using the improved DELO method. In this step, the approximate FDC RFs of the synthetic proðles to the model parameters are calculated. The synthetic Stokes proðles obtained are compared with the observations, and the function s2 is evaluated. The Ðrst and second derivatives of s2 are computed using the previously obtained RFs. Then, the Levenberg-Marquardt method is used to derive the corrections dx to the model parameters from s2 and its derivatives. This method requires the solution of a linear system of equations, which may be singular either if the RFs to a particular parameter are too small (i.e., if the proðles are insensitive to that particular parameter) or if two rows in the Hessian matrix are very similar to each other (i.e., if two model parameters produce the same e ect on the emergent proðles). In order to avoid such singularities, the singular value decomposition technique (see, e.g., Press et al. 1986) is applied before solving the linear system of equations. This allows us to identify which model parameters are responsible for the singularities, and to eliminate them from the inversion scheme. These parameters are left unchanged and assigned an inðnite error bar. The new model atmosphere is, then, FIG. 2.ÈComparison between strict, numerically evaluated (solid lines) and approximate FDC (dashed lines) response functions at *j \ 0.50 A line center of Ca II K in the model umbra inferred in this paper and presented in Table 2. from the

No. 2, 2000 NON-LTE INVERSION OF STOKES PROFILES 983 FIG. 3.ÈInversion test with simulated ASP data containing the Fe I lines at 6301 and 6302 A and the Mg I lines b and b. Dotted line: Reference model. 1 2 Solid line: Recovered model. Dashed line: Starting guess. Reference macroturbulence: 2.00 km s~1. Recovered: 1.98 ^ 0.01 km s~1. Starting guess: 0.50 km s~1. constructed from the corrections dx, and the procedure is iterated until no signiðcant decrease is obtained in s2 by successive iterations. A typical FDC-based inversion with the Ca II model atom used in this paper takes roughly 10 minutes in a modern UltraSparc 5 workstation. Computations with the Mg I model atom take more time, roughly 30 minutes, because of its complexity (13 levels and 26 transitions, as opposed to six levels and Ðve transitions considered in the Ca II atom). If the resulting Ðt to the observed proðles is not yet satisfactory, the inversion may be improved using accurate RFs calculated numerically via full non-lte computations. In most practical applications, it suffices to compute only numerically derived RFs to temperature and use FDC RFs for the other parameters. Note that the Ðeld-free assumption (which states that the atomic populations do not depend on the magnetic Ðeld in the atmosphere) implies that the departure coefficients do not react to changes in the magnetic Ðeld vector. Therefore, with the Ðeld-free approximation, FDC is exact for the RFs to the magnetic Ðeld. The error bar dx associated with the model parameter x is evaluated, as explained j in Paper I, using a variation of the j formula of Sa nchez Almeida (1997): (dx j )2\ 4N j N par s j 2(a~1) jj, (15) where N is the number of free parameters, a is the Hessian matrix, par and s2 is a merit function weighted by the j RFs. If we use R (j ) to denote the RFs of the proðle to kj i

984 SOCAS-NAVARRO, TRUJILLO BUENO, & RUIZ COBO Vol. 530 FIG. 4.ÈSynthetic and reference proðles emerging from the model atmosphere shown in Fig. 3. Dots: Reference proðles. Solid line: Synthetic proðles. The proðles are normalized to the quiet-sun continuum intensity. Residuals are shown below each panel. perturbations in x, then s2 is given by j j s2\ 1 4 ; Nj R2 (j )[Iobs(j ) [ Isyn(j )]2 ; i/1 kj i k i k i. (16) j 4N ; Nj p2(j )R2 (j ) j k/1 i/1 k i kj i We point out that the error bars are calculated at each of the nodes chosen for the inversion (see Ruiz Cobo & del Toro Iniesta 1992 for details). 5. INVERSION TESTS It was shown in Paper I that, in practical problems, our non-lte inversion method is able to recover the depth dependence of the thermodynamical variables within the limitations of the available observations and the basic simplifying assumptions imposed (mainly CRD and one-

No. 2, 2000 NON-LTE INVERSION OF STOKES PROFILES 985 FIG. 5.ÈInversion test with a simulated Stokes I and V spectrum of the Ca II lines at 8498 and 8542 A. Dotted line: Reference model. Solid line: Recovered model. Dashed line: Starting guess. Reference macroturbulence: 2.00 km s~1. Recovered: 1.98 ^ 0.01 km s~1. Starting guess: 0.50 km s~1. dimensional plane-parallel geometry). Now we introduce more free parameters, namely, the run of the magnetic Ðeld vector through the atmosphere, and the Ðrst question that should be addressed is to what extent the additional information contained in the proðles of Stokes Q, U, and V is enough to univocally determine these new unknowns. In order to answer this question, we carried out a number of tests, using simulated observations as input for the algorithm. For the Ðrst test in this section, we simulate observations that can be suitably performed with the Advanced Stokes Polarimeter (ASP; see Elmore et al. 1992). We choose the sunspot umbra model atmosphere of Obridko & Staude (1988) as our reference atmosphere and consider it the real Sun.ÏÏ This model does not include magnetic Ðeld or macroscopic velocity, so we added a depth-dependent magnetic Ðeld vector with decreasing strength and increasing inclination as we move outward in the atmosphere and also a variable line-of-sight velocity. The sign of the velocities is deðned such that positive values correspond to redshifts, while negative velocities represent blueshifts. With this reference model, we synthesize the Stokes proðles of the Fe I lines at 6301.5 and 6302.5 A, assuming LTE and the Mg I b and b lines in non-lte using a multilevel model atom. 1 These 2 synthetic proðles, with random noise, are given to the inversion code. The results of the inversion are shown in Figure 3. We can see that the recovered microturbulence has very large error bars, which means that this quantity is poorly determined. This is due to the lack of sensitivity of the lines to this parameter. As for the other parameters, we can see that the agreement is almost perfect up to log (q ) \[4. 500 Above that point, the error bars become larger. Note that this good result is obtained despite the chosen starting guess (dashed line), which is clearly unrealistic and very far away from the reference atmosphere. In particular, it can be seen that initializing the magnetic Ðeld with a very small value does not present any problem for the algorithm in Ðnding a solution. In Figure 4 we show the Stokes proðles of the Mg I lines emergent from the recovered model and the simulated observations. The lines have been matched within the noise of the observed proðles, which means that all the information contained in the spectral lines has been successfully retrieved. Therefore, a better inversion (i.e., smaller error bars) would be possible only using better observations (less noise, more spectral resolution, more lines, etc.). Tests with di erent initializations resulted in retrieved models, which are within the error bars obtained. Many current spectropolarimeters do not allow for the simultaneous measurement of the whole Stokes vector, and it is common to Ðnd polarimetric observations of Stokes I and V only. What can we expect to obtain from these observations? Is it still possible to obtain good estimates of at least the magnetic Ðeld strength and inclination? Figure 5 shows a typical example where we simulate observations of Stokes I and V for the Ca II lines at 8498 and 8542 A. The microturbulence is not plotted because, as above, it is almost completely undetermined. The main di erence with respect to the previous test is a lower magnetic sensitivity, mainly due to the fact that we no longer count on the information provided by the linear polarization proðles. Another important di erence is that these lines are sensitive to higher atmospheric layers.

986 SOCAS-NAVARRO, TRUJILLO BUENO, & RUIZ COBO Vol. 530 6. LIMITATIONS OF THE INVERSION CODE In this section we present numerical tests where the hypotheses assumed in our inversion code do not hold. First, 6.1 deals with the e ects of partial redistribution (PRD) in the formation physics of the Ca II resonance lines. Then, 6.2 considers the presence of optically thick unresolved components in the reference atmosphere. Finally, 6.3 explores the uncertainties introduced in the inversion by the use of inaccurate input data, such as oscillator strengths, chemical abundances, etc. 6.1. E ects of Partial Redistribution One of the simplifying hypotheses that we assume in the treatment of the non-lte radiative transfer problem is CRD. Many interesting chromospheric lines are accurately described with a CRD formalism (e.g., the Ca II infrared triplet, as shown by Uitenbroek 1989). These are, in principle, the Ðrst target that one should consider for the application of our inversion method. Unfortunately, some important transitions, such as the Ca II H and K resonance lines, do not belong to this category, since their formation physics involves PRD e ects. However, the diagnostic potential of these lines is so obvious that one becomes compelled to use the code with them. The question that immediately arises is: how reliable will the results of such an inversion be? To answer this question, we carried out a test using PRD synthetic intensity proðles of the Ca II H and K lines as input to our CRD inversion code. The reference atmosphere is model C of Fontenla, Avrett, & Loeser (1990). The recovered temperatures, line-of-sight velocities, and line proðles are shown in Figure 6. Note that the simulated observations can be Ðtted assuming CRD. The recovered micro- and macroturbulence are not reliable. Since these parameters have the overall e ect of smearing out the emergent proðle, they are used to compensate for PRD e ects. This results in systematically large values for both micro- and macroturbulence, which have nothing to do with their actual reference values. The macroscopic velocity, however, can be accurately recovered. Simulations of the Ca II H and K lines (see, e.g., Uitenbroek 1989) show that the PRD proðles usually have darker cores than those obtained assuming CRD. This e ect is compensated by our CRD code recovering a cooler chromosphere. It therefore seems that it is still possible to derive some information from our CRD inversion of these lines, just keeping in mind that the recovered line-of-sight velocity is reliable and the temperature is somewhat lower than its actual value. In any case, the generalization of our non-lte inversion code to PRD is relatively straightforward, and we are currently working in this direction. 6.2. E ects of Unresolved Components The solar atmosphere is very inhomogeneous, showing variations on a wide variety of scales. In this section we analyze the e ects on the inferred model atmospheres of di erent unresolved atmospheric components, coexisting within the resolution element of the observed spectrum. Two di erent numerical simulations are presented, in which the reference proðles are synthesized in a two-component atmosphere and then inverted using our inversion algorithm. In the Ðrst test, the reference atmosphere consists of a magnetic structure with a hot upñow and a cool downñow. The temperature di erence between them is a constant value and each component covers half of the resolution element. The second test considers a combination of a hot chromosphere with a cool atmospheric component, which does not present a chromospheric temperature rise. The assumed Ðlling factor here is also equal to 50%. Our results indicate that, even when the two atmospheric components are very di erent, the inferred atmosphere can be regarded as a good estimate of the mean ÏÏ model, i.e., the model resulting from averaging both atmospheres weighted by their corresponding Ðlling factors. 6.2.1. Magnetic Structure with Hot and Cool Opposite Flows We start by considering an atmosphere with opposite Ñows coexisting within the resolution element. The Ðrst component, which we will refer to as the hot ÏÏ component, consists of the VAL-C model with a constant magnetic Ðeld vector of 2 kg, inclined 20 with respect to the vertical, and a macroscopic line-of-sight velocity of [1 km s~1 (recall that the minus sign represents velocities directed toward the observer). For the second component, which we will denote by cool,ïï we take the hot model and subtract 500 K from the temperature at all the depth-points, reduce the magnetic Ðeld strength to 1.5 kg, and change the sign of the velocity, so that it is now an inward Ñow of 1 km s~1. The Stokes proðles of both components are synthesized and averaged, and we added random noise simulating a signal-to-noise ratio of 1000. This scenario is intended to simulate a situation where we have either very low temporal resolution or optically thick spatial inhomogeneities with hot upñows and cool downñows. We again simulate ASP observations (see 5), which are presented to the inversion code. The proðles are accurately matched. Therefore, with the available set of data, the reference two-component atmosphere is indistinguishable from the inferred single-component model that is depicted in Figure 7. First consider the variables that are identical in both components. The recovered macroturbulence (2.32 km s~1) exceeds the sought value (2 km s~1) by an amount that is clearly larger than its error bar (0.05 km s~1). This excess broadens the lines in order to compensate for the fact that we are mixing two di erent proðles separated in wavelength by 2 km s~1. The microturbulence is poorly determined, as indicated by the large error bars corresponding to this quantity. This is due to the lack of sensitivity of the lines to this parameter. As for the magnetic inclination and azimuth, we can see that they are reliable. If we now turn to the parameters that are di erent in the hot and cool components, we see that the temperature, magnetic Ðeld strength, and line-of-sight velocity lie between the reference values. The recovered model is not the exact average, but instead, there is a slight o set toward the hot component. This is exactly what we would have expected, since this component yields higher values of intensity and larger amplitudes in the other Stokes proðles, thus contributing more weight to the average proðle. 6.2.2. Unresolved Components with Di erent T hermal StratiÐcations Recent hydrodynamic simulations by Carlsson & Stein (1998) emphasize that the quiet solar atmosphere is a dynamical medium in which the whole thermal structure is constantly changing in time. It is also pointed out in this work that semiempirical mean models based on the Ðtting

No. 2, 2000 NON-LTE INVERSION OF STOKES PROFILES 987 FIG. 6.ÈPRD e ects on a CRD inversion of the Ca II H line (left panels) and the K line (right panels). Symbols for the Ðrst four panels are deðned as follows: dotted line, reference model; solid line, recovered model; dashed line, starting guess. Symbols for the two panels at the bottom are the following: open circles, PRD reference proðles; solid line, CRD emergent proðles from the recovered model. of strong resonance UV lines and continua can be misleading because of the strong nonlinear dependence of the thermal source function on temperature in this region. In this manner, the best-ðt model to the mean proðle would sample the hottest unresolved components rather than the mean atmosphere in the resolution element. In the infrared side of the spectrum, however, this argument is weakened, since the thermal source function depends almost linearly on the electron temperature. In Figure 8 we show an inversion test where the simulated observations are the Ca II infrared lines emergent from a two-component atmosphere with dramatically di erent temperature structures. The Ðrst component (hot chromosphere) is, again, the VAL-C model, and the second (without chromospheric temperature rise) is the quiet-sun model of Holweger & Mu ller (1974), extrapolated to higher atmospheric layers. Both components contribute a Ðlling factor of 50% to the total spectrum. The infrared line pro- Ðles are accurately matched (see Fig. 9), which again means that there is not enough information to (univocally) separate both atmospheric components. Note that, although the inversion of the infrared triplet yields a model that is very close to the mean model, the Ca II K line is not well reproduced by this model (see Fig. 9). This supports the idea that the infrared lines might present some advantages with respect to the ultraviolet resonance lines for the chromospheric diagnostics, in the sense that they provide a better description of the mean atmosphere. The ultraviolet

FIG. 7.ÈInversion of simulated ASP data emergent from a two-component model atmosphere. Dotted line: Reference hot and cool components. Solid line: Recovered model. Dashed line: Starting guess. Reference macroturbulence: 2.00 km s~1. Recovered: 2.32 ^ 0.05 km s~1. Starting guess: 0.50 km s~1.

NON-LTE INVERSION OF STOKES PROFILES 989 FIG. 8.ÈInversion test with the Ca II infrared triplet emergent from a two-component model atmosphere. Dotted line: Reference hot and cool components. Solid line: Recovered model. Dashed line: Starting guess. Dash-dotted line: Mean model. Note that the inferred atmosphere is very close to the mean model. lines, on the other hand, tend to weight the hot components present in the atmosphere more, as discussed above. 6.3. E ects of Inaccurate Atomic Data or Chemical Abundances The reliability of a model obtained via inversion techniques is, in general, critically dependent on the accuracy of the input parameters used for the spectral synthesis. Chemical abundances for the various elements, oscillator strengths of the spectral lines, or collisional rates for the transitions under study are often considered known data, but the e ects of the typical uncertainties in these values or the approximations employed to calculate them should be explored carefully. In order to have an estimate of the possible errors introduced into our inversions by these uncertainties, we carried out several simulations where the synthesis and inversion of the reference proðles are computed using slightly di erent values for the input parameters. We found that small errors in the abundance of the element considered do not signiðcantly a ect the recovered atmospheres. The reason is that the cores of the strong chromospheric lines that we are dealing with in this work are saturated, which means that the emergent intensity is not sensitive to changes in these parameters. As for the wings, their variations are compensated by recovering a slightly di erent photospheric temperature. Since their response functions in the photosphere are large, a very small temperature di erence is enough to Ðt the proðle. In this manner, an error of 0.1 dex in the abundance yields an error of about 10È50 K in the photospheric temperature. The background opacities are inñuenced by changes in the chemical abundances, but this is not important for strong lines because the line contribution dominates both the opacity and the source function of the transitions. Finally, the relation between gas pressure and electronic pressure (which is given by the ionization equilibrium)

990 SOCAS-NAVARRO, TRUJILLO BUENO, & RUIZ COBO Vol. 530 FIG. 9.ÈSynthetic proðles emergent from the two-component atmosphere used as reference for the test in Fig. 8 (dots), and from the inferred singlecomponent model atmosphere (solid line). depends also on the chemical abundances. Our tests show that small variations in the recovered atmosphere (mainly in the temperature) are able to compensate for the errors introduced in the synthetic proðle by these small changes in the electron density. FIG. 10.ÈInversion of the Mg I lines b and b, using di erent values for the collisional parameter C. Only the 1 temperature 2 stratiðcation is depicted, since the other parameters are equally well recovered regardless the value of C. Dotted line: Reference temperature. Solid lines: Recovered models for the di erent values of C. If we are dealing with only one spectral line then, as in the case of uncertainties in the chemical abundance of that element, the errors in log (gf ) are not important. The argument is, again, the line core saturation and the high temperature sensitivity of the line wings, which are able to compensate for the error with a small temperature di erence. In the case of a multiline inversion, however, the situation is di erent because random errors in the log (gf )ofthe di erent lines cannot be compensated simultaneously for all the lines. The inversion algorithm then Ðnds the optimum solution. In this synthetic spectrum, some of the lines will systematically lie above the observed lines and some others will lie below. When this happens, it is a clear indication that the atomic parameters used for the lines in the inversion should be revised. The approximations for the collisional rates used in the non-lte calculations are also important. An example of the e ects of variations in the collisional rates is shown in Figure 10, where we depict the reference model and the result of inverting the Mg I lines multiplying the collisional rates by an arbitrary parameter C. We can see that values of the collisional parameter that are larger than unity yield cooler recovered chromospheres, while smaller values result in the recovery of higher temperatures. 7. APPLICATION TO REAL OBSERVATIONS In this section we apply our non-lte inversion code to observations of Stokes I and V from a sunspot umbra. The