Frédéric André

Vladimir P Solovjov

Brent W Webb

Mathieu Galtier

Philippe Dubuisson

Frédéric Andre
email: frederic.andre@insa-lyon.fr.

SPECTRAL MAPPING METHOD BASED ON INTERVALS OF COMONOTONICITY FOR MODELLING OF RADIATIVE TRANSFER IN NON-UNIFORM GASEOUS MEDIA

Keywords: gas radiation, non-uniform, spectral mapping method (SMM), intervals of comonotonicity, comonotonic spectra, Ck, CKD. 

come

Spectral mapping method based on intervals of comonotonicity for modelling of radiative transfer in non-uniform gaseous media

NOMENCLATURE g

cumulative k-distribution -Eq. ( 1) k absorption coefficient (cm -1 ) u abscissa at the origin for the lines used to split the g-g plot (Sections 2 and 3) u scaling coefficient (Section 4)

Greek symbols

  spectral band width (cm -1 )

INTRODUCTION

Extension of approximate models of gas radiation from uniform to non-uniform paths is among the most difficult problems in gas radiation modeling [START_REF] Sj | Band model theory of radiation transport[END_REF]. Although many methods are accurate enough for radiative heat transfer applications, most of them do not permit to achieve a precision sufficient for narrow and wide band spectroscopic (environmental, etc.) applications.

In these cases, advanced techniques to improve approximate models of gas radiation for general non-uniform applications are required. These techniques can be organized in two main categories: Multiple-Line-Group methods [START_REF] Sj | Band model theory of radiation transport[END_REF] and Spectral Mapping techniques.

Multiple-Line-Group (MLG) approaches are among the oldest techniques. They first appeared in 1962 as part of Wyatt's quasi-random band model [START_REF] Wyatt Pj | Quasi-random model of band absorption[END_REF], and consist of an explicit grouping of spectral lines according to values of their linestrengths. The idea of combining spectral lines with similar lower state energy levels appeared in the 70s, as a way to ameliorate the treatment of highly non-isothermal paths. Early methods based on this idea are described in Refs. [START_REF] Ludwig Cb | Handbook of infrared radiation from combustion gases[END_REF][START_REF] Vitkin Ei | New method of calculating the infrared emission of nonuniform volumes of molecular gases[END_REF][START_REF] Khodyko Yv | Methods of calculating moleculargas radiation on the basis of spectral-composition modelling[END_REF][START_REF] Khodyko Yv | Multigroup method for the calculation of selective IR radiation transfer in nonhomogeneous media[END_REF].

Recent appellations for these approaches are the fictitious gases [START_REF] Di Leon R | A fictive gas-method for accurate computation of lowresolution IR gas transmissivities: application to the 4.3 µm CO2 band[END_REF] and the multi-scale method described in Baradwaj and Modest [START_REF] Bharadwaj Sp | A multiscale Malkmus model for treatment of inhomogeneous gas paths[END_REF]. The method presented in Ref. [START_REF] Bharadwaj Sp | A multiscale Malkmus model for treatment of inhomogeneous gas paths[END_REF] mostly reformulates in different words the approach proposed a few years earlier in Ref. [START_REF] Soufiani | A fictitious-gas based statistical narrow-band model for IR long-range sensing of H2O at high temperature[END_REF]. The output of a MLG treatment is a set of groups of spectral lines whose centers are assumed to be statistically independent, from one group to another.

Spectral Mapping Methods (SMM) seek to avoid the use of simplifying assumptions to treat non-uniform gaseous paths (constant absorption coefficient, scaling, etc.) by iteratively splitting the spectral interval associated with the model's definition into sub-intervals over which these non-uniform treatments are not assumptions but actual properties of gas spectra. The output of a SMM treatment is a set of spectral intervals over which a particular relationship between high resolution spectra in distinct states (scaling, correlation / comonotonicity, etc.) can be assumed to be exact. Any SMM thus leads to some simplifications in the treatment of path non-uniformities. Application of SMM requires high resolution spectral data, which explains why these techniques are more recent than MLG. The first SMM was in fact proposed by West and Crisp in 1990 [START_REF] West R | Mapping transformations for broadband atmospheric radiation calculations[END_REF]. It consists in building sub-intervals, or bins, in such a way that gas spectra are constant over the bins at each spatial location along non-uniform paths. This technique was improved recently in Refs. [START_REF] Bennartz R | A modified k-distribution approach applied to narrow band water vapor and oxygen absorption estimates in the near infrared[END_REF][START_REF] Doppler | k-bin and k-IR: k-distribution methods without correlation approximation for non-fixed instrument response function and extension to the thermal infraredapplications to satellite remote sensing[END_REF]. Other methods produce intervals over which gas spectra can be treated as scaled. Modest and co-workers' multi-group (MG) [START_REF] Zhang | Multi-group full-spectrum k-distribution database for water vapour mixtures in radiative transfer calculations[END_REF] (not to be confused with Ludwig's [START_REF] Ludwig Cb | Handbook of infrared radiation from combustion gases[END_REF] or Khodyko / Vitkin [START_REF] Vitkin Ei | New method of calculating the infrared emission of nonuniform volumes of molecular gases[END_REF][START_REF] Khodyko Yv | Methods of calculating moleculargas radiation on the basis of spectral-composition modelling[END_REF][START_REF] Khodyko Yv | Multigroup method for the calculation of selective IR radiation transfer in nonhomogeneous media[END_REF] multi-group methods which are both founded on MLG) and multi-spectral techniques [START_REF] Andre F | The multispectral gas radiation modeling: a new theoretical framework based on a multidimensional approach to k-distribution methods[END_REF][START_REF] Andre F | An exact formulation of k-distribution methods in non-uniform gaseous media and its approximate treatment within the Multi-Spectral framework[END_REF][START_REF] Andre F | A multi-spectral reordering technique for the full spectrum SLMB modeling of radiative heat transfer in nonuniform gaseous mixtures[END_REF][START_REF] Andre F | The multi-spectral reordering (MSR) technique for the narrow band modeling of the radiative properties of non-uniform gaseous paths: the correlated/uncorrelated approximations[END_REF] fall in this category and differ principally by the way the spectral intervals are built. In two recent papers [START_REF] Hu | Improved MSMGFSK models apply to gas radiation heat transfer calculation of exhaust system of TBCC[END_REF][START_REF] Hu | Improved spectral absorption coefficient grouping strategy of wide band k-distribution model used for calculation of infrared remote sensing of hot exhaust systems[END_REF], efforts were made by Hu and Wang to improve the grouping scheme used in the Multi-Scale Multi-Group (MSMG) method of Modest by attempting to replace constraints on scaling coefficients, as used in the original version, by pragmatic criteria based on a correlated view. The necessity to treat differently single species and mixtures is closely related to the initial choice made by the authors to apply the MSMG method for their problem.

When formulated and applied properly, both MLG and SMM have the potential to improve significantly the treatment of path non-uniformities. No combinations of the two techniques, which are by construction redundant, are in this case required.

This work proposes a SMM technique that allows the building of spectral intervals over which gas spectra in distinct states are rigorously linked through a strictly increasing function.

Accordingly, over these intervals that will be from now on referred to as intervals of comonotonicity, the two spectra at the distinct states can be related through monotonically increasing functions: spectra are called comonotonic [START_REF] Nelsen R | An introduction to Copulas -Second Edition[END_REF], and the Ck / CKD approaches are exact. Notice that Ck and CKD acronyms represent the same method (the so-called Correlated approximation applied within the frame of the k-distribution technique) but one of them is mostly encountered in Mechanical Engineering (Ck) whereas the other one is more usual in Atmospheric Sciences (CKD).

The present method improves k-distribution methods by directly addressing the blurring effect of non-uniform k-distribution approximations: this effect is encountered both for high temperature (as considered here for illustration) and atmospheric applications. The simplification of the treatment of path non-uniformities that is provided by the present SMM results from the construction of spectral intervals over which this blurring effect can be effectively eliminated. The step-by-step process to construct intervals of rigorous comonotonicity, and thus "correlation" or "zero-blurring", is described. It is applied here to narrow spectral intervals but its extension to any band width is possible since it only requires the specification of the cumulative k-distributions of the gas in distinct states: any particularity of the problem under study (width of the band, possible influence of optical filters and / or nonconstant blackbody function) is already included in the calculation of the cumulative kdistribution. Application of the method to highly non-uniform situations, one of which is representative of the classically challenging IR plume signature calculation, illustrates the relevance of the method. The present grouping scheme can be used alone, as proposed here, or together with other techniques for which it can provide a relevant initial guess at the earliest stage of their iterative process.

BUILDING INTERVALS OF COMONOTONICITY

Let us consider two gas spectra at two distinct thermophysical states for which the spectral absorption coefficients in the two states are 1   and 2   , respectively. The cumulative k- distribution functions that correspond to these two spectra are:

    1 12 i i i i g k H k d , i ,            (1)
where H is the Heaviside step function. The problem is formulated here within the frame of narrow band approaches but can be extended to any other band, up to the full spectrum, by replacing d  in Eq. ( 1) by any weighting function of the form

  d    where   0   and   1 d      .
We start our analysis by plotting the absorption spectra for the two thermophysical states not as functions of wavenumbers, as in usual LBL representations, but with respect to the cumulative distribution function of the absorption coefficient in State 1,  

1 1 g   .
State 1 is a 1 % H2O -99 % N2 mixture at 300 K ; State 2 is 10 % H2O -90 % N2 at 1500 K. The 25 cm -1 narrow band considered for the following figures is centered at 3400 cm -1 , in the 2.7 µm band of H2O. This spectral region is widely used in IR plume signature studies [START_REF] Soufiani | A fictitious-gas based statistical narrow-band model for IR long-range sensing of H2O at high temperature[END_REF][START_REF] Riviere Ph | Correlated-k fictitious gas model for H2O infrared radiation in the Voigt regime[END_REF].

For this purpose, we start with the high resolution (10 -2 cm -1 ) spectra in the two states, provided as vectors  

        12 1 1 1 1 1 1 , ,.., N g g g       and         12 2 2 2 2 2 2 , ,.., N g g g      
by application of Eq. ( 1). If one sets

        12 1 1 1 1 1 1 , ,.., N g g g      
as abscissa axis and plots  

12 1 1 1 , ,.., N       and   12 2 2 2 , ,.., N   
   with respect to this vector, one receives Figure 1.

After the change of abscissa axis from wavenumbers to   1 1 g   , the gas spectrum in State 1 is monotonically increasing (this is the main principle of the k-distribution method) whereas the absorption spectrum in State 2 is not (in the present case but this observation is rather general) an increasing function of  

1 1 g   .
This departure from monotonicity is the main source of error in the Ck / CKD approaches.

Our aim is to build spectral intervals over which gas spectra are comonotonic viz. to extract subsets of points from the cloud of circles (absorption coefficient in State 2 reorganized with respect to   For this purpose, we will mostly follow the same steps as in Ref. [START_REF] Andre F | An exact formulation of k-distribution methods in non-uniform gaseous media and its approximate treatment within the Multi-Spectral framework[END_REF] where the objective was to construct sub-intervals over which gas spectra are scaled. However, the starting point is slightly different. Indeed, the main difference between the two approaches (scaled in [START_REF] Andre F | An exact formulation of k-distribution methods in non-uniform gaseous media and its approximate treatment within the Multi-Spectral framework[END_REF] or correlated / comonotonic here) is that we will replace the C-curves of Ref. [START_REF] Andre F | An exact formulation of k-distribution methods in non-uniform gaseous media and its approximate treatment within the Multi-Spectral framework[END_REF] (2-dimensional parametric curve 12 12 k , k

   in the (k1, k2)-plane) by a g-g plot (2-dimensional parametric plot of     12 1 1 2 2 g g , g g   
in the (g1, g2)-plane). This kind of representation is not new in the radiative heat transfer literature and can be found for instance in Ref. [START_REF] Fu Q | On the correlated-k distribution method for radiative transfer in nonhomogeneous atmospheres[END_REF] to illustrate the blurring effect of k-distribution methods viz. to show their possible departure from the so-called "correlated" assumption. An example of g-g plot is given in Figure 2 corresponding to the same set of LBL data as for Figure 1. This figure was obtained by setting

        12 1 1 1 1 1 1 , ,.., N g g g      
as abscissa axis and plotting Figure 3 represents a simplified model of g-g plot. If gas spectra were actually "correlated", their cumulative distribution would take equal values at all wavenumbers and all the points

       
    12 1 1 2 2 g g , g g   
would be aligned on the straight diagonal line D0. As gas spectra are not correlated / comonotonic in the case considered, points of coordinates

    12 1 1 2 2 g g , g g   
can be found in various regions of the unit square. The building of subintervals of comonotonicity consists in splitting the unit square in a pertinent way so as to define these intervals. In other words, we seek to identify using Fig. 2 wavenumber intervals where the two spectra are related through a strictly increasing function, viz. over which they share the same monotonicity by increasing or decreasing simultaneously.

    21 21 u g g u         (2)
Consequently, for small increments , all the values of the

      21 21 , g u g u          (3)
or equivalently:

        2 1 1 1 1 2 1 2 1
, where

uu g u g k g u g k                   (4)
Eq. ( 4) shows that the two absorption spectra 12

        1 , 1, 2 i i i i u g k u H k d i u                  (5)
take equal values for any wavenumber inside

  u  
and follow rigorously:

      12 12 , g u g u u                     (6)
This relationship arises directly from the equalities Eqs. [START_REF] Ludwig Cb | Handbook of infrared radiation from combustion gases[END_REF][START_REF] Vitkin Ei | New method of calculating the infrared emission of nonuniform volumes of molecular gases[END_REF] and the definition of comonotonicity, viz. two variables are comonotonic if and only if they can be linked to each other through a strictly increasing function in which case their rank / cumulative distribution functions are equal [START_REF] Nelsen R | An introduction to Copulas -Second Edition[END_REF]. Intervals   u   thus correspond to intervals of comonotonicity, i.e., spectral intervals over which gas spectra are comonotonic.

Based on the same principle, the unit square can be discretized into diagonal belts at fixed values of u. All the points that belong simultaneously to the g-g plot and a given belt can then be grouped to define a spectral interval over which gas spectra are comonotonic and, consequently, over which the Ck / CKD method is exact. By varying the parameter u between -1 and 1 one reconstructs the unit square and thus the set of wavenumbers (the narrow band considered in this work)   .

APPLICATION

The results provided in this section are based on the same LBL dataset as described in Ref. [START_REF] Andre F | An exact formulation of k-distribution methods in non-uniform gaseous media and its approximate treatment within the Multi-Spectral framework[END_REF].

They use spectral line parameters taken from HITEMP2010 [START_REF] Ls | HITEMP, the high-temperature molecular spectroscopic database[END_REF] as inputs. The two states considered in this section are the same as for the previous figures: State 1 is a 1 % H2O -99 % N2 mixture at 300 K ; State 2 is 10 % H2O -90 % N2 at 1500 K. As noticed earlier, spectra were restricted to the 25 cm -1 spectral band centered at 3400 cm -1 for Figures 1,2,4,5.

Building intervals of comonotonicity from the sets of vectors described in the previous section up to radiative heat transfer calculations requires several steps:

A / the first step consists of splitting the g-g plot of Figure 2 into different diagonal belts, each of which corresponds to a distinct value of the parameter u. The splitting process was done by: 1/ evaluating the distribution function F of the spectral variable    

    21 2 1 1 ii u g g u       .
In this section, N = 25 belts were constructed by this method. Each belt contains roughly the same number of points (100 ± 2 spectral values of the absorption coefficients). Results (only a few subsets are plotted for legibility) are depicted in Figure 4.

COMPARISON WITH A MAPPING TECHNIQUE BASED ON INTERVALS OF SCALING

As emphasized in the introduction, there exist many mapping methods in the literature to improve approximate models of gas radiation in non-uniform situations. Among them, techniques based on intervals of scaling are probably the simplest. They are all built on the same simple idea, explained below.

At high spectral resolution, gas spectra are rigorously scaled. However, the scaling property cannot, in general, be extended to spectral bands since, for any two absorption spectra

      1 F u H u u d u u             P (7)
This distribution function can be used to define intervals of scaling. Indeed, they can be built by application of the following steps:

1/ the unit interval [0,1] is divided into N + 1 equally spaced values 11 , 1, 1; 0; 1 iN X i N X X      ; 2/   ii F u X 
is solved, where F is now given by Eq. ( 7) and represents the distribution function of scaling coefficients, which provides



 is roughly a constant (in the same way that when applied directly to gas spectra the method provides gray gases). Absorption spectra can thus be treated as scaled (their ratio is a constant) over the corresponding sets of wavenumbers.

It can be noticed that for large N, a constant value of the ratio of the two spectra is a fair approximation inside each of the subintervals provided by this method so that a mean scaling coefficient can be realistically introduced. For small values of N, however, the use of the Ck / CKD method can correct errors due to non-constant scaling coefficients and should be preferred, as suggested in Ref. [START_REF] Zhang | Multi-group full-spectrum k-distribution database for water vapour mixtures in radiative transfer calculations[END_REF]. This is the method selected in this section since the choice N=2 was made. Indeed, since the two methods that we want to compare (based on intervals of scaling or comonotonicity) converge toward the exact solution at infinite values of N, their comparison only has relevance for small values of N. Notice also that the two methods based on intervals of scaling and comonotonicity share in this case almost the same RTE solver. Only the intervals used to construct conditional (viz. restricted to subintervals, as introduced in Eq.

(5) for the comonotonic case) k-distribution functions differ.

This method to construct intervals of scaling is not exactly the same as used, for instance, by

Zhang and Modest in the multi-group method [START_REF] Zhang | Multi-group full-spectrum k-distribution database for water vapour mixtures in radiative transfer calculations[END_REF] or in Refs. [START_REF] Andre F | The multispectral gas radiation modeling: a new theoretical framework based on a multidimensional approach to k-distribution methods[END_REF][START_REF] Andre F | An exact formulation of k-distribution methods in non-uniform gaseous media and its approximate treatment within the Multi-Spectral framework[END_REF][START_REF] Andre F | A multi-spectral reordering technique for the full spectrum SLMB modeling of radiative heat transfer in nonuniform gaseous mixtures[END_REF][START_REF] Andre F | The multi-spectral reordering (MSR) technique for the narrow band modeling of the radiative properties of non-uniform gaseous paths: the correlated/uncorrelated approximations[END_REF] for the multispectral approach, both of which use some criterion on spectral scaling coefficients over sequences of thermophysical states to construct intervals of scaling. Here, only two states are treated.

However, these techniques are founded on the same principle (aggregate intervals in such a way that the ratio of any two spectra is a constant) and should provide, in theory, exactly the same results as those provided by the present method.

The two spectral mapping techniques (scaling / comonotonicity intervals) are compared and assessed against LBL calculation in a case taken from Ref. [START_REF] Zhang | Multi-group full-spectrum k-distribution database for water vapour mixtures in radiative transfer calculations[END_REF]. Results of a Ck-12.5 model and the spectral mapping method proposed by Zhang and Modest in [START_REF] Zhang | Multi-group full-spectrum k-distribution database for water vapour mixtures in radiative transfer calculations[END_REF], based on the arbitrary cut-off on scaling coefficients ucut-off = 10, are also provided for completeness. The two cells contain the same gas, 20% H2O, 80% N2 at atmospheric pressure. The hot layer is at 2000 K and its length is 50 cm, and the cold layer is at 300 K and its length varies. Calculations are made over narrow bands and then integrated over the full spectrum to provide total intensities which are used for comparisons with LBL calculations. These intensities are evaluated at the cold side. Results are shown in Figure 8.

Figure 8. Comparison of several spectral mapping methods

From the results shown in Figure 8, the following conclusions can be drawn:

1/ Splitting the 25 cm -1 narrow band into two contiguous subintervals (yielding the Ck-12.5 solid circles) does not improve the approximation significantly compared to the Ck-25 method (solid squares).

2/ Using intervals of scaling or comonotonicity clearly yields a high gain in terms of accuracy, with errors reduced in the present case by a factor 4 when either the grouping scheme based on scaling or comonotonicity described in this work are used. At identical computational costs, spectral mapping methods have the potential to improve significantly the accuracy of the kdistribution methods compared to a crude splitting of the band into narrower sets. In the present case, the model based on intervals of comonotonicity has a slightly higher accuracy than the model based on intervals of scaling. The spectral scaling method proposed by Zhang and

Modest also improves the accuracy of narrow band calculations, but can clearly not compete in the case considered here with the two other grouping schemes described in this paper.

Finally, Figure 9 depicts mean Spearman's coefficients calculated over narrow bands.

Spearman's coefficient is a measure of dependence [START_REF] Nelsen R | An introduction to Copulas -Second Edition[END_REF] Over a narrow band   , and for two thermophysical states as defined in section 2, Spearman's coefficient can be calculated as [START_REF]Effective scaling factors in non-uniform gas radiation modeling[END_REF]:

      1 2 1 2 SP 1 2 1 , 12 3 g g d                    (8)
One can readily confirm that when spectra follow rigorously the implicit equation associated with so-called "correlated" models, this coefficient is actually 1. In the modern statistical literature [START_REF] Nelsen R | An introduction to Copulas -Second Edition[END_REF], a Spearman's coefficient equal to 1 characterizes comonotonic variables.

Over sub-intervals, one can similarly define group Spearman's coefficients by simply replacing the full band   by the interval

  i u  
, where i u is the quantity assumed to be fixed over the spectral interval and that can either represent a constraint on scaling or comonotonicity. It is then possible to calculate an average value by weighting the Spearman's coefficients over the groups by the fraction of the interval associated with each group. In the case of two groups, as considered in this section, one then obtains:

                    12 1 2 1 2 1 2 SP SP,1 SP,2 1 2 1 2 SP, 1 2 , , , 1 , 12 3 i
i i i i u uu g u g u d u                                                  (9)
These averaged Spearman's coefficients   12 SP ,     are plotted in Figure 9 for the same case as for Figure 8 and for the two grouping schemes discussed in this paper (Modest and Zhang's technique is not considered). For the Ck model, Spearman's coefficients were calculated from Eq. ( 8): low and even negative values are observed. This explains why the Ck / CKD method provides poor predictions in this high temperature gradient case, since the so-called "correlation" assumption is clearly inappropriate in this situation. Using the two grouping techniques described in this paper leads to higher values of averaged Spearman's coefficients Notice also that averaged Spearman's coefficients are slightly higher in the comonotonic case than with scaling, which proves that the method to construct intervals of comonotonocity actually improves the monotonic relationship between spectral values over the groups. Furthermore, increasing the number of groups provides averaged Spearman's coefficients that approach unity: the relatively small values observed here are due to the choice of only two groups. Finally, observing distinct values for the sets of Spearman's coefficients proves that the two techniques, based on intervals of scaling or comonotonicity, yield overlapping though distinct sets of wavenumbers.

CONCLUSION AND PERSPECTIVES

The method described in the present note to construct intervals of comonotonicity: 1/ is simple to understand and use, 2/ does not require an iterative process in the case of two-cell problems (its extension to more general situations is kept as future work), and 3/ can be applied over any band width, from narrow bands up to the full spectrum. It was shown to provide accurate predictions of radiative intensities in situations representative of IR plume signature configurations, a problem widely recognized as among the most complicated in band model theory. Comparisons with a Ck / CKD model based on intervals of scaling were also described and shown to provide a similar accuracy.

In addition, the method presented in this paper can be applied to atmospheric problems requiring accurate treatment of gaseous absorption, such as for radiative forcing calculations or remote sensing in absorbing bands. In this paper, the proposed method has been applied using two thermophysical states including temperatures of 300 and 1500 / 2000 K. However, atmospheric applications cover a narrower range of temperature. In this respect, Fu and Liou [START_REF] Fu Q | On the correlated-k distribution method for radiative transfer in nonhomogeneous atmospheres[END_REF] have presented g-g plots for atmospheric conditions (temperature range between 245 and 300 K) and shown that the temperature effect produces more blurring. The degree of blurring is then weaker for atmospheric applications, with points less scattered around the diagonal line in the g-g plot. The building of intervals of comonotonicity should then be easier to define, leading to reduced calculations times. The approach presented in this study is then a promising method to improve the k-distribution and to reduce errors associated with the Ck / CKD assumptions between atmospheric levels.

Finally, one can notice that no application of the Godson-Weinreb-Neuendorffer's (GWN) method [START_REF] Sj | Band model theory of radiation transport[END_REF] or use of effective scaling factors (see Ref. [START_REF]Effective scaling factors in non-uniform gas radiation modeling[END_REF], in which a detailed analysis can be found) was considered in this paper. Such comparisons are scheduled as future work together with a comprehensive comparison of the two present techniques, extended to more than two cells, with the Mixture l-Distribution approach (MLD) viz. the l-distribution method [START_REF]The l-distribution method for modeling non-gray absorption in uniform and non-uniform gaseous media[END_REF] applied to intervals of scaling.

  is the set of wavenumbers used in the LBL calculation, and calculate two new vectors

  Figure 1 that increase with respect to g1.

Figure 1 .

 1 Figure 1. Example of two spectra reordered with respect to the same (related to State 1)

.

  For the sake of legibility, only the points associated with the discrete wavenumbers used in the LBL calculation of12 and   are depicted.

Figure 2 .

 2 Figure 2. Example of g-g plot (same case as Figure 1)

Figure 3 .

 3 Figure 3. Schematic representation of a g-g plot

  and   are linked to each other through a strictly increasing function u  , that depends on u, at all spectral locations   u   . In this case, the cumulative-k distributions restricted to the interval   u   which are defined for the two states 1 and 2 as:

Figure 4 . 2 g

 42 Figure 4. Discretization of the g-g plot into diagonal belts

Figure 5 .

 5 Figure 5. Subsets of comonotonic absorption coefficients (groups)

Figure 6 .Figure 7 .

 67 Figure 6. Radiative intensity at the exit of the cold path

  why the radiative transfer results of Figure8are more in accordance with LBL calculations when these grouping schemes are used than with the narrow band Ck / CKD.

Figure 9 .

 9 Figure 9. Comparison of averaged Spearman's coefficients for several spectral methods

  and characterizes the non-linear dependence between sets of values / random variables. The Ck / CKD model assumes this coefficient to be 1. A null value of Spearman's coefficient indicates complete statistical independence. Negative values can be observed when high values in a sequence are associated with low values in the other sequence, in which case variables are called counter-monotonic.

AKNOWLEDGEMENTS

This work has been supported by the Programme National de Télédétection Spatiale (PNTS, http://www.insu.cnrs.fr/pnts grant N° PNTS-2017-04).


Vladimir P Solovjov

Frédéric André

Denis Lemonnier

Brent W Webb

Frederic Andre

The rank correlated SLW

Keywords: Gas radiation, rank correlation of gas absorption spectra, Generalized SLW Method, SLW reference approach, Rank Correlated SLW model

INTRODUCTION

Global methods in modelling of radiation transfer in high-temperature gases such as SLW (Spectral Line Weighted-sum-of-gray-gases), ADF (Absorption Distribution Function), FSK (Full Spectrum k-distribution), and MBWSGG (Moment-Based Weighted-Sum-of-Gray-Gases) methods, and the hybrid forms of the SLW model with other spectral models [START_REF] Denison | A spectral line based weighted-sum-of-gray-gases model for arbitrary RTE solvers[END_REF][START_REF] Denison | The spectral line based weighted-sum-of-gray-gases model in nonisothermal non-homogeneous media[END_REF][START_REF] Solovjov | Spectral Line-Based Weighted-Sum-of-Gray-Gases Modeling of Radiative Transfer in Multicomponent Mixtures of Hot Gases[END_REF][START_REF] Solovjov | Multilayer modeling of radiative transfer by SLW and CW methods in non-isothermal gaseous media[END_REF][START_REF] Pearson | Efficient representation of the absorption line blackbody distribution function for H2O, CO2, and CO at variable temperature, mole fraction, and total pressure[END_REF][START_REF] Solovjov | The Generalized SLW model[END_REF][START_REF] Andre | The spectral-line moment-based (SLMB) modeling of the wide band and global blackbody-weighted transmission function and cumulative distribution function of the absorption coefficient in uniform media[END_REF][START_REF] Pierrot | A fictitious-gas-based absorption distribution function global model for radiative transfer in hot gases[END_REF][START_REF] Modest | Radiative Heat Transfer[END_REF][START_REF] Modest | The full-spectrum correlated-k distribution and its relationship to the weighted-sum-of-gray-gases method[END_REF][START_REF] Cai | Improved full-spectrum k-distribution implementation for inhomogeneous media using a narrow-band database[END_REF][START_REF] Chu | Relationship between the spectral line based weighted-sum-ofgray-gases model and the full spectrum k-distribution model[END_REF] are computationally the most efficient methods in modelling of gas radiation. These methods can provide high accuracy of prediction of radiation transfer, often comparable to that of the benchmark line-by-line (LBL) method but at significantly lower computational cost. Despite the seeming difference between SLW, ADF, and FSK methods, they all are based on the same fundamental principle in modelling of the gas absorption spectrum, and it has been shown that all of them can be described as particular cases of the recently published Generalized SLW Model [START_REF] Solovjov | The Generalized SLW model[END_REF]. The primary challenge of global methods is to extend the spectral modeling from uniform (isothermal, homogeneous) to nonuniform (non-isothermal, nonhomogeneous) media, for which some additional assumptions regarding the gas absorption spectrum (e.g., assumption of correlated or scaled spectra) must be made. The existing approaches are based on assumptions that can yield large errors in the case of high spatial temperature gradients in the medium. Simply put, the reference approach to handling nonisothermal, nonhomogeneous media involves defining a reference state and correcting all local states relative to that reference state. The main drawback of the reference approach is the absence of a consistent recommendation for the choice of the gas reference state. Despite the interest in the reference approach in global methods, not all possible variations of the method have been discovered. The Generalized SLW Model provides a backdrop to reveal more possibilities for new versions of the reference approach.

The objective of the present paper is to develop and present all possible versions of the SLW reference approach, and investigate and compare their characteristics. The main result is the new Rank Correlated SLW Model which distinguishes itself from other approaches in that it is the only method i) which does not require the specification of a gas reference state, and ii) which preserves the emission term in the spectrally integrated Radiative Transfer Equation. This method is introduced in Section 3.5.I to follow, and its detailed outline is presented in Section 3.7. Rigorous theoretical justification of properties for the Rank Correlated SLW model is given in Appendices 1 -3. It can be shown that the development presented for the Rank Correlated SLW model in nonuniform media is based on the more general co-monotonic spectral model of gas radiation in nonuniform media employing an arbitrary probability distribution, and is a particular case of its application.

THE SLW METHOD

Spectral Radiative Transfer Equation

Propagation of radiation in absorbing and emitting gaseous media along a pathlength s in a direction Ω is characterized by the spectral radiation intensity ( )

I s,
∂ = - +     ∂ Ω Ω ( 1
)
where η is the wavenumber, gas thermodynamic state at temperature ( ) T s , total pressure ( ) p s , and mole fraction ( ) Y s . High resolution spectral databases such as HITRAN and HITEMP are used to compile the spectral gas absorption cross-section ( )

C η φ .
The Line-by-Line (LBL) method solves the spectral RTE for each wavenumber η , following which the total intensity of radiation may be obtained by spectral integration over all wavenumbers. The narrow band approximation requires solution of the RTE at hundreds of thousands of spectral points i η in the numerical integration: ( ) ( )

0 I s, I s, d η η ∞ = ∫ Ω Ω ( ) i i i I s, η ∆η ≈ ∑ Ω (2)
The SLW model is a particular case of the so-called global family of methods in modeling gas radiation which, in contrast to the Line-by-Line method, starts first with spectral integration of the RTE, and then solves it directly for the spectrally integrated intensity ( ) I s,Ω for a finite number of discrete values of the absorption cross-section (rather than wavenumber). To determine total radiative properties it may be said that the SLW method integrates over absorption cross-section rather than wavenumber.

Characterization of the gas absorption cross-section -ALBDF and inverse ALBDF

A brief tutorial on the Generalized SLW model is presented here to provide context for the reference approaches identified in sections to follow. The spectral integration of the RTE in the SLW model is made efficient by application of the direct and inverse Absorption Line Blackbody Distribution Function (ALBDF).

E T H C C d E T η η φ η ∞ = - ∫ (3)
Here, H is the Heaviside unit-step function. The ALBDF is a strictly increasing function varying between 0 and 1 with respect to the variable C, and therefore, it is invertible. The inverse ALBDF ( )

g b
C F ,T ,T of the variable F is defined such that ( )

g b g b C F C, ,T , ,T C φ φ   =   , and
( )

g b g b F C F , ,T , ,T F φ φ   =   (4)
The inverse ALBDF can be interpreted as a reordered gas absorption cross-section. The ALBDF is calculated in advance from the high-resolution gas absorption cross-section, and has recently been presented in tabulated form or mathematical correlations [START_REF] Pearson | Efficient representation of the absorption line blackbody distribution function for H2O, CO2, and CO at variable temperature, mole fraction, and total pressure[END_REF][START_REF] Solovjov | The Generalized SLW model[END_REF] using the most up-to-date spectral database, HITEMP 2010. The values of the inverse ALBDF may be obtained by inversion of the ALBDF tabulated data, or by polynomial interpolation of mathematical correlations with the help of the method of moments [START_REF] Solovjov | The Generalized SLW model[END_REF].

The SLW Method -spectral integration of RTE in uniform media

Consider a uniform gaseous medium at a specified thermodynamic state

{ } g T ,Y , p φ =
. The spectral gas absorption coefficient at this state is given as

( ) ( ) NYC η η κ φ φ =
, where

( )

C η φ is the gas spectral absorption cross-section.

SLW histogram spectrum

As shown in Fig. 1, the continuous line absorption coefficient ( )

C η φ is modelled in the SLW
method by a histogram spectrum in the following way. First, define a set of supplemental absorption cross-sections ref j C  spaced between chosen limiting values min C and max C which effectively span the entire range of gas absorption spectrum. As an example, logarithmic spacing has been previously used as ( )

min max min j n ref j C C C C =  , , ,..., j 0 1 n =
, where n is the number of gray gases in the model. The gray gas absorption cross-sections Cj can be chosen arbitrarily in the interval , and the clear gas is defined as 0 0 κ = . The supplemental cross-sections define the gray gas and the clear gas spectral intervals:

j 1 j j C C C -< <   ,
( ) { } j j 1 j : C C C η ∆ η φ - = < <   , ( ) { } 0 0 : C C η ∆ η φ = < 
Then the SLW histogram spectrum is defined as

( ) j η κ φ κ = , j η ∆ ∈ , j 0,1,...,n =
Spectral integration of the RTE Integration of the spectral RTE over the gray gas spectral intervals j ∆ , j 0,1,...,n = yields the following expression:

( ) ( ) ( ) ( ) ( ) b 0 0 0 I s, d s I s, d s I T s d s η η η η η η κ η κ η ∞ ∞ ∞ ∂ = - +     ∂ ∫ ∫ ∫ Ω Ω (5)
Because the spectral gray gas intervals are fixed for all spatial locations in a uniform medium, the order of differentiation and integration on the left-hand side of the equation can be interchanged. That yields the gray gas RTEs:

( ) ( ) ( ) j j j j j b I s, I s, a I T s η κ κ ∂ = - + ∂ Ω Ω (6)
where ( ) ( )

j j I s, I s, d η ∆ η = ∫ Ω Ω (7)
is defined as the gray gas intensity, and ( ) ( )

j j g b j1 g b a F C , ,T T F C , ,T T φ φ φ φ - = = = - = =   (8)
are the gray gas weights. After solution of the gray gas RTEs, the total intensity can be found by a summation over all gray gases:

( ) ( ) ( ) j n n j j 0 j 0 I s, I s, d I s, η ∆ η = = = = ∑ ∑ ∫ Ω Ω Ω (9)

The SLW Method -spectral integration of RTE in non-uniform media

In non-uniform (nonisothermal/nonhomogeneous) media, the gas absorption cross-section ( )

C η φ
varies with spatial location s if the thermodynamic state ( ) s φ depends on location. In this case the intersection of the fixed supplemental absorption cross-sections j C  with the absorption crosssections ( )

1 1 s φ φ = and
( )

j 1 j 2 s s ∆ ∆ ≠ .
Therefore, in the spectral integration of the RTE, interchange of differentiation with respect to the spatial variable s and integration with respect to the wavenumber η over the gray gas intervals

( ) ( ) ( ) { } j i , j i , j s a s ,b s ∆   = 
 will produce the so-called Leibnitz terms [START_REF] Denison | The spectral line based weighted-sum-of-gray-gases model in nonisothermal non-homogeneous media[END_REF] due to variation of the limits of integration:

( ) ( ) j s I s, d s η ∆ η ∂ ∂ ∫ Ω ( ) ( ) j s I s, d s η ∆ η ∂ = ∂ ∫ Ω ( ) ( ) ( ) ( ) Leibnitz terms i , j i , j i , j i , j i db s da s I s, b s , I s, a s , ds ds η η η η         + = - =           ∑ Ω Ω   (10)
It is impossible to account for these numerous Leibnitz terms, but their neglect can yield a significant error in prediction of radiation transfer in gas media with high non-uniformity. To eliminate Leibnitz terms from the spectrally integrated RTE, the gray gas intervals must be fixed for all spatial locations, ( )

j j s const ∆ ∆ = = : ( ) j I s, d s η ∆ η ∂ ∂ ∫ Ω ( ) j I s, d Leibnitz terms s η ∆ η ∂ = + ∂ ∫ Ω (11)
The gray gas intervals may be fixed by choosing different supplemental absorption cross-sections ( ) j C s  at different locations in such a way that they define the same spectral intervals j const ∆ = for all locations, as illustrated in Fig. 2. In the SLW model, this is ensured by invoking the reference approach.

The SLW reference approach -"ideal spectrum" assumption

The SLW reference approach is based on the assumption of "ideal behavior" of the gas absorption cross-section in non-uniform media defined as follows. For an arbitrary prescribed reference value T will be the same as illustrated in Fig. 3:

( ) ( ) loc ref loc ref ref ref F C , ,T F C , ,T φ φ = ( 12
)
This implicit equation is used in the SLW reference approach to find the local supplemental absorption cross-sections loc j C  which maintain the same spectral intervals, and, therefore, eliminate the Leibnitz terms in the gray gas RTEs. An iterative technique, such as bisection or Newton-Raphson methods, has been used for solution of this implicit equation.

It should be noted that this global method approach is similar to the correlated k-distribution method used for narrow bands in non-uniform media [START_REF] Goody | Atmospheric Radiation[END_REF]. The "ideal behavior" assumption for spectra is valid for the case of scaled spectra when the absorption cross-section can be expressed as a simple product

( ) ( ) ( ) C f h η φ φ η =
, separating the dependence on thermodynamic state φ and wavenumber η (Milne-Eddington approximation).

Outline of the SLW Reference Approach [2]

The SLW reference approach thus follows the steps below: 1. Choose a reference state

{ } ref ref ref ref T ,Y , p φ =
by some appropriate weighting of the properties over the domain (e.g., simple volumetric average). 2. Choose a set of reference supplemental cross-sections ref j C  (log-spaced or optimized). 3. Find the corresponding local supplemental cross-sections loc j C  by solving the implicit equations:

( ) ( ) loc ref j loc ref j ref ref F C , ,T F C , ,T φ φ =   4.
Calculate the local gray gas absorption coefficients from the supplemental cross-sections:

( ) loc j loc loc j s N Y C κ = ( ) ( ) 1 loc loc j j N s Y s C C - =   5.
Calculate the local weights of the gray gas absorption coefficients:

( ) loc j j a s a = ( ) ( ) 1 , , , , ref ref j ref b loc j ref b loc F C T T F C T T φ φ - = = - =   6.
Solve the gray gas RTE in the non-isothermal, non-homogeneous medium using the gas property parameters found in steps 1-5:

( ) ( ) ( ) ( ) ( ) ( ) j j j j j b I s s I s s a s I T s s κ κ ∂ = - +     ∂
The ADF reference approach [START_REF] Pierrot | A fictitious-gas-based absorption distribution function global model for radiative transfer in hot gases[END_REF] differs from the original SLW reference approach by the way in which the gray gas weights are calculated. Instead of calculating the weights at the reference supplemental cross-sections ref j C  , the ADF method calculates the weights with the help of the previously determined local supplemental cross-sections.

( ) loc j j a s a = ( ) ( ) 1 , , , , loc loc j loc b loc j loc b loc F C T T F C T T φ φ - = = - =   (13)
The two methods differ in their ability to handle exactly the transmissivity and emissivity of a gas layer as is discussed in [START_REF] Pierrot | A fictitious-gas-based absorption distribution function global model for radiative transfer in hot gases[END_REF].

THE RANK CORRELATED SLW MODEL

Concept of "correlated" gas absorption spectrum currently adopted in global methods

The concept of "ideal spectrum" discussed in [START_REF] Denison | The spectral line based weighted-sum-of-gray-gases model in nonisothermal non-homogeneous media[END_REF] was reformulated in [9, p. 660] to a concept of "correlated" spectrum as the relationship between absorption coefficient . This may be expressed mathematically as [START_REF] Modest | Radiative Heat Transfer[END_REF] ( )

, η κ η φ ( ) k ,k η η φ * = (14) If the ratio ( ) k ,k k η η η φ *
is constant, then the absorption coefficient is said to be scaled, and the ratio ( ) ( )

0 u , k ,k k η η η φ φ φ * =
is called the scaling function, which depends only on the local and reference states. If this ratio depends on the reference absorption coefficient ( ) 0 , η κ η φ , then the spectra are said to be correlated. These definitions are summarized in [START_REF] Modest | Radiative Heat Transfer[END_REF] as ( )

, η κ η φ ( ) ( ) ( ) * 0 k ,k correlated u , k scaled η η η φ φ φ η   =    (15)
It is suggested here that the definition given of the assumption of correlated spectrum is, in fact, not really an assumption. Due to the presence of η in ( ) , η κ η φ the relationship can be established point by point with respect to wavenumber η for any arbitrary absorption spectrum regardless of whether the spectrum is related or not. The concept of spectral "correlation" presented in [START_REF] Modest | Radiative Heat Transfer[END_REF] limits the possibility of further improvement of global modelling of gas radiation in non-uniform medium.

In this work the term "correlated spectrum" will be understood in the sense of "ideal spectrum" as defined in [START_REF] Denison | The spectral line based weighted-sum-of-gray-gases model in nonisothermal non-homogeneous media[END_REF]. It is important to note that all existing definitions of spectral correlation are essentially based on a requirement to specify the reference absorption spectrum ( ) 0 , η κ η φ at some chosen gas thermodynamic state 0 φ . All existing reference approaches of the SLW, ADF, and FSK methods involve the gas absorption coefficient at the reference state as their starting point.

The rank correlated gas absorption spectrum

Further development of the SLW model in non-uniform media can be achieved by modification of the assumption of "ideal spectrum" or "correlated spectrum" discussed above to a less restrictive assumption of rank correlation regarding the relationship between absorption spectra at different thermodynamic states. The assumption of the rank correlated spectrum is defined here as follows: Let 1 φ and 2 φ represent two arbitrary gas thermodynamic states, and let

( ) ( ) { } 1 1 ' 1 : C C η η ∆ η φ φ = < and ( ) ( ) { }
( ) ( ) { } 1 1 b : C C F , ,T η η φ φ < ( ) ( ) { } 2 2 b : C C F , ,T η η φ φ = < ∆ = (16)
These wavenumber intervals are identical. The rigorous proof of this property of rank correlated spectrum is given in Appendix 1. Because of this property, the inverse ALBDF ( )

m b C F , ,T φ , 1, 2,... m =
can be defined as the rank correlated or co-monotonic reordered continuous absorption cross-sections as depicted in Fig. 5.

The thermodynamic states m φ can be any local thermodynamic states in the medium. It is important to note that no reference thermodynamic state is involved in this definition of rank correlated reordered absorption cross-sections. Thus, in the SLW model, this approach will yield piecewise constant rank correlated reordered gray gas absorption cross-sections, expressed as

( ) ( ) ( ) j m b j 1 j C F , ,T H F F H F F φ -   - - -     , where (
)
H F is the Heaviside unit-step function. The rank correlated reordered absorption cross-sections are shown schematically in Fig. 6.

As stated previously, the concept of rank correlated reordered absorption cross-section ( )

m b C F , ,T φ
considered in this work is a particular case of a more general approach to co-monotonic global spectral modelling of gas radiation in non-uniform media based on arbitrary probability measures, where it can be shown that if two arbitrary constants 1 2 k ,k 0 > , one of two conditions exist:

( ) { } ( ) { } 1 1 2 2 : k : k η η η κ φ η κ φ < ⊆ < or ( ) { } ( ) { } 1 1 2 2 : k : k η η η κ φ η κ φ < ⊇ < (17)
Consequently, the corresponding spectral absorption coefficients the set of all wavenumbers η such that ( )

1 1 C C η φ < is a subset of all wavenumbers η for which ( ) 2 2
C C η φ < , or vice versa. Mathematically, this may be stated

( ) { } ( ) { } 1 1 2 2 : C C : C C η η η φ η φ < ⊆ < or ( ) { } ( ) { } 1 1 2 2 : C C : C C η η η φ η φ < ⊇ < . ( 18
)
For example, if one assumes that the first inclusion

( ) { } ( ) { } 1 1 2 2 : C C : C C η η η φ η φ < ⊆ <
is true, then the relationship between the ALBDFs at two states in SLW terms yields:

( )

1 1 b F C , ,T φ ( ) ( ) ( ) ( ) b b 1 1 b b 0 I T H C C d I T η η φ η ∞ = - ∫ ( ) ( ) ( ) { } 1 1 b b b b :C C I T d I T η η η φ η < = ∫ ( ) ( ) ( ) { } 2 2 b b b b :C C I T d I T η η η φ η < ≤ ∫ ( ) 2 2 b F C , ,T φ = (19)
Therefore, ( )

1 1 b F C , ,T φ ( ) 2 2 b F C , ,T φ ≤ . Further, in this case, ( ) ( ) ( )
( )

1 1 2 2 H C C H C C η η φ φ - - ( ) ( ) 1 1 H C C η φ = - (20)
Therefore, the joint distribution function of absorption cross-sections at two different thermodynamic states 1 φ and 2 φ in this case is equal to the minimum of the two functions ( )

1 1 b F C , ,T φ and ( ) 2 2 b F C , ,T φ : ( ) 1 2 1 2 b F C ,C ; , ,T φ φ ( ) ( ) ( ) ( ) ( ) ( ) b b 1 1 2 2 b b 0 I T H C C H C C d I T η η η φ φ η ∞ = - - ∫ ( ) ( ) ( ) ( ) b b 1 1 b b 0 I T H C C d I T η η φ η ∞ = - ∫ ( ) 1 1 b F C , ,T φ = ( ) ( ) { } 1 1 b 2 2 b min F C , ,T ,F C , ,T φ φ = (21)
The same result, ( )

1 2 1 2 b F C ,C ; , ,T φ φ ( ) ( ) { } 1 1 b 2 2 b min F C , ,T ,F C , ,T φ φ = , can be obtained if the second condition ( ) { } ( ) { } 1 1 2 2 : C C : C C η η η φ η φ < ⊇ < is true.
It can be demonstrated that in this case, the gas spectra at states 1 φ and 2 φ are related through the equation

( ) ( ) ( ) ( ) 1 1 b 2 2 b F C , ,T F C , ,T η η φ φ φ φ =
for any wavenumber η , and thus, for any fixed value ( )

0,1 ξ ∈ it may be stated ( ) ( ) ( ) ( ) 1 b 1 b 2 b 2 b F C , ,T , ,T F C , ,T , ,T ξ ξ φ φ ξ φ φ = = . ( 22
)
This relationship does not require the definition of any reference state because it is valid for any two states, and preservation of the intervals of integration is guaranteed. Therefore, construction of the Rank Correlated SLW model with the help of the inverse ALBDF (reordered absorption coefficient) at any local state is given by the equation

( ) ( ) , , j loc j loc b C C F T φ φ = ( 23
)
without the need of specification of the reference state. This fact will be applied in Section 3.4 for the construction of the Rank Correlated SLW model which starts with subdivision of the F-variable into fixed j j F ,F  , j 0,1,...,n = values. The rank correlated gray gas reordered absorption coefficients loc j C can be found at any local gas thermodynamic state without need of the reference spectrum. This approach preserves ranking of the local gray gas absorption coefficient loc j C at all local states, and maintains the gray gas spectral intervals of integration j ∆ fixed at all states. The rigorous proof of the preservation of spectral intervals of integration by the Rank Correlated SLW model is given in Appendix 3. This simply means that according to the assumption of rank correlation of the gas spectrum, only a discretization of the interval of variations of the variable

( ) 0,1 F ∈
is required to define 1) the gray gas absorption coefficients in any state and, as a consequence, 2) the spectral intervals over which the gas spectra are assumed to be constant; they are considered to be the same for any state and do not depend on the choice of any reference thermophysical condition.

Comments on reference gas thermodynamic state and reference partition into gray gases

The SLW reference approach is based on introduction of different reference quantities. It is for this reason that it is important to clarify their definition and meaning here.

( ) { } ref ref j j 1 ref j : C C C η ∆ η φ - = < <   .
In the reference approach these spectral intervals should be held constant for any spatial location of the gas medium. Partition of the C-variable can be done in a variety of ways, but it is usually logarithmically evenly spaced between chosen values of min C and max C which effectively cover the entire range of gas absorption spectrum.

Reference partition into gray gases can be performed starting either by partition of the C-variable or by partition of the F-variable: i)

If partition into ref j C  is chosen then partition into ref j F  is calculated as ( ) ref ref j j g ref b ref F F C , ,T T φ φ = = =   and
( )

ref ref j j g ref b ref F F C , ,T T φ φ = = = . ii) Alternatively, reference partition of the F-variable into arbitrary fixed values ref j F  and ref j F
can be performed without a need of the gas thermodynamic state. Subdivision into gray gases may be then performed with the local absorption cross-sections as ( )

loc ref j j g loc b ref C C F , ,T T φ φ = = =  
, preserving the same gray gas spectral intervals

( ) { } loc loc j j 1 loc j : C C C η ∆ η φ - = < <  
for any spatial location. As noted previously, the choice of the reference blackbody source temperature b ref T T = is arbitrary and it is needed only to maintain rank correlation of the reordered absorption cross-sections at different locations. The corresponding partition into reference supplemental cross-sections can be made with the inverse ALBDF as ( )

ref ref j j g ref b ref C C F , ,T T φ φ = = =   .
It is important to note the difference between the choice of the gas reference temperature ref The main drawback of current reference approaches is that predictions can be quite sensitive to the choice of the reference state. There are a number of proposed methods for defining the reference state, including volumetric averaging, Planck emission (T 4 ) averaging, the maximum temperature of the medium, temperature of an equivalent isothermal medium, a temperature which yields the same emissivity, etc. However, these choices are not universal, have questionable theoretical justification, and they cannot be applied in all cases. Consequently, no systematic recommendation can be given.

A very detailed analysis of the factors which influence non-correlatedness of the absorption coefficient of real gases, and of the importance of the choice of the reference state is presented in [START_REF] Modest | Radiative Heat Transfer[END_REF] with a conclusion that the choice of ref φ is very important and should be optimized for each unique problem. At the same time it is again noted that the choice of the reference blackbody temperature

Construction of the Rank Correlated Generalized SLW Reference Approach

The development of possible reference approaches begins with the Generalized SLW Method, starting with subdivision of either C or F variables as follows:

1) The C-variable is subdivided into supplemental cross-sections ref j C  spaced between chosen values min C and max C which effectively cover the entire range of the gas absorption spectrum. For example, logarithmic spacing has been previously used as ( )

min max min j n ref j C C C C =  , , ,..., j 0 1 n =
, where n is the number of gray gases in the model. 2) Because F varies between 0 and 1 and the subdivision increment of the F variable plays the role of weights of gray gas absorption coefficients in the spectral integration, a reasonable subdivision may be made with integral quadratures, the positive abscissa j x 0 > and corresponding weights j w , , ,..., j 1 2 N = of the Gaussian-Legendre quadratures for integration over the interval [ ]

, 0 1 into the interval min max , F F     : ref 0 min F F =  , ( ) j ref j min max min k k 1 F F F F w = = + - ∑  ,
( )

ref j min j max min F F x F F = + ⋅ - , , ,..., j 1 2 N =
Generally speaking, the SLW reference approach has the possibility of two gas thermodynamic states ( ) , ,

g ref b ref F C T T φ φ = = , ( ) , , g ref b ref C F T T φ φ = = ( ) , , g loc b ref F C T T φ φ = = , ( ) , , g loc b ref C F T T φ φ = = ( ) , , g ref b loc F C T T φ φ = = , ( ) , , g ref b loc C F T T φ φ = = ( ) , , g loc b loc F C T T φ φ = = , ( ) , , g loc b loc C F T T φ φ = =
Two methods are used to establish dependence between reference and local spectra ( ) = . Depending on the choice of the reference discretization into gray gases, the choice of the blackbody temperature, and the choice of the local or reference supplemental cross-sections for calculation of the local gray gas weights, there are thus eight possible versions of the SLW reference approach.

g ref C η φ φ = and
Simply put, the objective of the SLW reference approach is to find: 1) the local supplemental cross-sections loc j C  which define the local gray gas absorption coefficient

loc loc loc loc j loc loc j 1 j loc loc j N Y C C N Y C κ - = =   , and
2) the supplemental values loc j F  , which define the local weights of gray gas absorption coefficients 7 demonstrates the principles of the reference approach confirming conservation of the spectral intervals of integration at different states. Figure 8

loc loc loc j j j 1 a F F - = -   . Figure

Outline of all possible versions of the SLW reference approach

The possible versions of the reference approach may be classified generally as i) those that treat the spectrum relationship for the blackbody source temperature at the reference temperature Tb = Tref (designated Method I.x.x), and ii) those that treat the spectrum relationship for the blackbody source temperature at the local temperature, Tb = Tloc (designated Method II.x.x). The steps in treating the spectral relationship are summarized below for all possible approaches.

loc loc j loc loc loc loc j 1 loc loc loc loc F C F T T F C F T T φ φ φ φ - = -   = loc loc j j 1 F F - -   ( ) ( ) , , , , ref ref j ref loc j 1 ref loc F C T F C T φ φ - = -   ( ) ( ) ( ) , , , , , , , ,
The calculation of the weights in this method is identical to that of Method II.1.1. Therefore, this approach produces results equivalent to Method II.1.1 (and the two equivalent methods may be hereafter referred to as II.1.1-2).

Method II.2.1

This reference approach method is also new. As seen from Figs. 8 and9, this method requires three steps for calculation of the local absorption cross-sections, instead of multiple steps for the previous methods.

1) Define the partition of the F-variable into reference values

j C  ( ) , , ref j ref ref C F T φ =  , loc j F  ( ) , , ref j ref loc F C T φ =  ref j C ( ) , , ref j ref ref C F T φ = , loc j F ( ) , , ref j ref loc F C T φ = loc j C  ( ) , , loc j loc ref C F T φ =  , loc j C ( ) , ,
j C  ( ) , , ref j ref ref C F T φ =  , loc j F  ( ) , , ref j ref loc F C T φ =  ref j C ( ) , , ref j ref ref C F T φ = , loc j F ( ) , , ref j ref loc F C T φ = loc j C  ( ) , ,
loc loc j loc loc loc loc j 1 loc loc loc loc F C F T T F C F T T φ φ φ φ -     = -     loc loc j j 1 F F - = -   ( ) ( ) , , , , ref ref j ref loc j ref loc F C T F C T φ φ = - a ( ) ( ) , , , , , , , ,
which is equivalent to step 4) of Method II.2.1. Therefore, this approach is equivalent to Method II.2.1 (and the two equivalent methods may be hereafter referred to as II.2.1-2).

Spectral integration of the source term of the RTE

It is instructive to investigate the spectral integration of the emission source term in the RTE for the reference approach methods presented here. Spectral integration of the emission source term for the medium at location s with thermodynamic state ( )

g

(

) ( )

N loc ref ref j j ref b loc j 1 ref b loc j 1 F C , ,T T F C , ,T T κ φ φ - =   = - =   ∑   ( ) P ref loc ,T κ φ = (26a)
For methods I.1.2 and I.2.2

( ) ( )

N loc loc loc j j loc b loc j 1 loc b loc j 1 F C , ,T T F C , ,T T κ φ φ - =   = - =   ∑   ( ) P loc loc ,T κ φ = (26b)
Therefore, it may be seen that only Methods I.1.2 and I.2.2 correctly calculate the emission source term as ( ) ( )

b loc P loc loc I T ,T κ φ
, thus preserving the local Planck mean absorption coefficient. No other approach preserves the spectrally integrated emission term at the local state, and without that conservation there is the potential for significant inaccuracy in modelling the source term in the spectrally integrated RTE where deviation of the local gas temperature from the reference temperature exists.

The differences in computational efficiency among the different methods should also be noted here. The number of steps needed to calculate the local gray gas absorption coefficients by the different methods is summarized as follows: Methods I. Consider the graphical relationships illustrated in Fig. 10. The methodology for determining the gray gas absorption coefficients and corresponding gray gas weights for the RTE are as follows:

1) Define the partition of the F-variable into reference values such that

min max , F F F   ∈   : ref 0 min F F =  , ( )
j ref j min max min k k 1 F F F F w = = + - ∑  , ( ) ref j min j max min F F x F F = + - , , ,..., j 1 2 n = ( 27
)
where j x 0 > are the positive abscissa and wj, , ,..., j 1 2 N = , are the corresponding weights of Gaussian-Legendre quadratures for integration over the interval [ ]

( ) ( ) ( ) ( ) ( ) ( ) j j j j j b I s s I s s a s I T s s κ κ ∂ = - +     ∂
It should be noted that the current outline is presented for a single gas. The case of a mixture of gases with non-scattering soot particles can be treated as a single gas using the approach outlined in [START_REF] Solovjov | The Generalized SLW model[END_REF].

Continuous limit of the Rank Correlated SLW model (Method I.2.2)

With an increase in the number of gray gases the Generalized SLW model has been shown to approach the continuous exact limit [START_REF] Solovjov | The Generalized SLW model[END_REF]. To develop the limiting case, continuous exact limit of the Rank Correlated SLW model, Method I.2.2, one chooses ref j F  as the discrete independent variable, following which the gray gas intensity is defined as

( ) j I s,Ω ( ) ref ref j j I F ,s, F ∆ = Ω  ( 28
)
where

ref ref ref j j j 1 F F F ∆ - = - 
 (see Fig. 11 for supporting graphical interpretation). Then the gray gas RTEs can be written in finite increment form ( )

ref j I F ,s, s ∂ ∂ Ω  ( ) ( ) loc j loc ref loc j j j b loc ref j F I F ,s, I T F ∆ κ κ ∆ = - + Ω  ( ) ( ) ( ) ( ) loc loc j loc loc j 1 loc loc loc ref loc j j j b loc ref ref j j 1 F C , ,T F C , ,T I F ,s, I T F F φ φ κ κ - - - = - + - Ω      ( ) ( ) ( ) ( ) ref ref j loc ref loc loc j 1 loc ref loc loc loc ref loc j j j b loc ref j F C F , ,T , ,T F C F , ,T , ,T I F ,s, I T F φ φ φ φ κ κ ∆ -     -     = - + Ω    (29)
Here, the gray gas absorption coefficients and their weights are calculated as outlined in the foregoing sections for Method I.2.2. In the limit, with an increase of number of gray gases, the increment

ref j F 0 ∆ → ;
the discrete supplemental values approach the continuous variables

ref ref j 1 j j F F F ξ -→ → →   , ( ) loc j κ κ ξ →
; and the ratio of finite increments becomes a derivative of the ALBDF with respect to a continuous variable ξ :

( ) I s ξ ∂ ∂ ( ) ( ) ( ) ( ) ( ) b loc b loc b loc ,s I ,s I T s F C , ,T , ,T T κ ξ ξ κ ξ ξ φ φ ξ ∂ = - + =         ∂ ( 30
)
After solving for ( ) I ξ , the total intensity can then be calculated by integration with respect to ξ ( )

n ref ref j j j 0 0 I I d I F ,s, F η η ∆ ∞ = = = ∑ ∫ Ω  → ( ) 1 0 I I d ξ ξ = ∫ (31)
This equation is a continuous limit of the Generalized SLW model in non-uniform media for Method I.2.2. The significance of Eq. ( 30) is that the scaling factor (the derivative) in the emission term includes no reference state, i.e., the term depends only on the local state φloc (and arbitrary blackbody source temperature Tb). It should also be noted that the function C(ξ,φloc) can be defined over an arbitrary measure, entirely eliminating the need of the blackbody source temperature Tb.

VALIDATION

This section illustrates the use of the reference approaches presented above in example problems designed to test their accuracy over a range of problem parameters.

Example 1 Hot layer -cold layer system

The first example considered here is commonly used as a rigorous standard test case for comparison of the performance of different spectral models in nonisothermal gas media [START_REF] Pierrot | A fictitious-gas-based absorption distribution function global model for radiative transfer in hot gases[END_REF][START_REF] Cai | Improved full-spectrum k-distribution implementation for inhomogeneous media using a narrow-band database[END_REF]. Consider a onedimensional system of two adjacent gas layers with different fixed values of gas temperature,

hot T 2000K = and cold T 300K =
, as shown in Fig. 12a. The system is bounded by walls which are assumed to be black and cold. The gas in both layers is a homogenous mixture of water vapor with mole fraction . . The objective is to predict the directional total radiative flux at the exit of the cold layer ( ) q L as a function of the thickness of the cold layer, Lcold. This example is known to be challenging for spectral modelling in gases because of the strong temperature gradient between the two layers. The SLW model analytical solution for the directional total radiative flux in the two-layer system bounded by black cold walls has been presented previously as [START_REF] Solovjov | Multilayer modeling of radiative transfer by SLW and CW methods in non-isothermal gaseous media[END_REF] (

n 1 1 2 1 2 2 2 4 4 4 4 j 1 3 j 1 j 2 j 1 j 2 3 j 2 j 2 j 0 q L 2 a T E L L a T a T E L a T 2 σ κ κ κ =   = - + + - +   ∑ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
where

1 hot T T = , 2 cold T T = , 1 hot L L = = 0.5 m, 2 cold L L = , 1 2
L L L = + , and the parenthetical superscripts refer to the corresponding layer. In this equation, ( ) k j κ and ( ) k j a are the local values of the gray gas absorption coefficient and their local weights, respectively, which must be found according the reference approach selected. Tabulated data for the ALBDF [START_REF] Pearson | Efficient representation of the absorption line blackbody distribution function for H2O, CO2, and CO at variable temperature, mole fraction, and total pressure[END_REF] were used in the calculations (rather than mathematical correlations). Line-by-line benchmark predictions were generated using the same database as that from which the ALBDF data were generated. Figure 12b shows predictions of the normalized total flux at the exit of the cold layer

( ) ( ) ( ) 4 b hot hot
q L E T q L T σ = obtained using the different reference approaches presented previously. In Methods I.1.1 and I.2.1 the local weights are calculated with the reference supplemental cross-sections ref j C  . Figure 12b shows that predictions for these two methods are nearly identical, deviating significantly from the line-by-line solution, with increasing error as the thickness of the cold layer is increased. Recall that neither of these methods preserve the Planck mean absorption coefficient, P κ , in the spectral integration of the RTE. Predictions using Methods I.1.2 and I.2.2 (both of which preserve P κ ), in which the local weights are calculated based on local supplemental cross-sections loc j C  , are considerably more accurate. As seen in Fig. 12b for this example, all approaches, Methods II.1.1-2 and II.2.1-2, yield identical results.

Example 2 Isothermal dual layer system

This example is intended to confirm that if the gas spectra at different states are rigorously scaled (i.e., in the calculation of the absorption coefficient ( )

NYC T η η κ =
only the mole fraction changes), then all versions of the reference approach presented here should produce accurate predictions in non-uniform medium.

Consider a system of two adjacent plane-parallel gas layers filled with CO2 at the same temperature T = 1000K with different but fixed values of gas mole fraction, .

1 Y 0 4 = (thick layer) and .

1

Y 0 1 = (thin layer). The system is bounded by black and cold walls. Because the two layers differ only in mole fraction of the same gas, the gas absorption cross-sections according to the spectral model adopted are exactly the same for both layers (air-broadening and self-broadening are assumed to be the same for CO2), and the absorption coefficient will be scaled only by the mole fraction. Therefore, all versions of the reference approach should theoretically yield the same prediction of the exiting radiative flux, and should match the line-by-line result. The solution of this problem is obtained by the same analytical method used in Example 1.

Figure 13a shows the prediction of the normalized total flux at the exit of the thin layer

( ) ( ) ( ) 4 b q L E T q L T σ =
as a function of the thickness of the thinner layer 2 L obtained using the different reference approaches. Figure 13b shows the normalized relative error. For this example featuring perfectly scaled spectra, all approaches yield nearly identical, accurate results. The relative error is generally on the order of 0.3% or less except at low L2 for methods II.1.1-2, II.2.-1, and I.1.2. Although the relative error in prediction of the exiting flux for all methods is very low as expected for this example, the Rank Correlated SLW reference approach, Method I.2.2, (which does not require the specification of a reference state) exhibits noticeably better accuracy. The likely reason for the lower error of Method I.2.2 is that this method requires the evaluation of only two ALBDF distribution functions, unlike the other approaches which require the evaluation of three distribution functions. This example demonstrates that indeed, if spectra are perfectly scaled then all proposed reference approaches yield accurate prediction of radiative transfer in non-uniform medium.

Example 3 Non-isothermal, non-homogeneous layer of H2O

Consider now the prediction of the divergence of total net radiative flux in a plane-parallel layer of thickness . = L 1 0 m filled with a gas at total pressure . = p 1 0 atm with a parabolic temperature profile, ( ) ( )

, = - + 2 T
x 4000 x L x L 800 K , and imposed wall temperatures ( ) = T 0 800 K and

( ) =
T L 800 K . The layer is filled with water vapor of spatially variable mole fraction given by ( ) ( )

. . = - + 2 2 H O Y
x 0 8x L x L 0 12 . Predictions of the total divergence of the net radiative flux using the Generalized SLW reference approaches Methods I.1.1 and I.2.2 and the LBL solution, obtained with the same spectral database [START_REF] Pearson | Efficient representation of the absorption line blackbody distribution function for H2O, CO2, and CO at variable temperature, mole fraction, and total pressure[END_REF], are shown in Fig. 14. Method I.1.1 is the original SLW reference approach, and Method I.2.2 is the simpler, more efficient Rank Correlated SLW model (Method I.2.2) which does not require specification of a reference state and preserves the emission term in the RTE. For this example both the original (I.1.1) and the improved reference approach (I.2.2) yield good accuracy relative to the line-by-line benchmark prediction, which is not unexpected in a problem with smooth, continuous variation of temperature and concentration with moderate local gradients.

Example 4 Triangular temperature profile H2O

The Rank Correlated SLW model requires the specification of only the blackbody source temperature Tb (which has been previously stated is only a matter of mathematical convenience), whereas the other SLW reference approaches need a reference state which includes both Tg=Tref and Tb=Tref. Their roles are different: the reference state defines the reference absorption coefficient which must then be scaled to any other local state; the source temperature defines the Planck blackbody distribution which is used simply for reordering of the absorption cross-section to any local state. The same measure (and therefore, the same source temperature Tb) should be used for reordering at all local states in the medium.

As has been stated previously, theoretically, the choice of the blackbody source temperature Tb should be unimportant. To demonstrate the sensitivity of the choice of Tref and Tb the following example known to be a challenging problem from the research reported in ref. [START_REF] Pierrot | A fictitious-gas-based absorption distribution function global model for radiative transfer in hot gases[END_REF] is considered. Consider a plane-parallel layer of thickness . L 0 3 m = filled with a gas at total pressure . = p 1 0 atm with the triangular temperature profile shown in Fig. 15a, and imposed black walls at temperature ( ) ( )

T 0 T L 500 K = =
. The layer is filled with water vapor at uniform mole fraction ( ) . where Tmax is the maximum temperature of the layer. As can be seen in Fig. 15b, prediction of the total divergence of the net radiative flux is very sensitive to the reference temperature for the original SLW reference approach (Method I.1.1). By contrast, Fig. 15c shows that predictions for the Rank Correlated SLW model (Method I.2.2) are virtually insensitive to the choice of reference blackbody source temperature. Further, the Rank Correlated SLW model produces significantly more accurate predictions.

Example 5 Non-isothermal, homogeneous layer of H2O

Subdivision into gray gases in the SLW model is usually performed using logarithmically spaced supplemental absorption cross-sections. This subdivision generally provides good accuracy with 8 -12 gray gases. The new Rank Correlated SLW model can use fixed subdivision of the F-variable somewhat more efficiently numerical integration. For example, positive Gauss-Legendre quadratures have been used with good results and significant computational economy. Therefore, subdivision into gray gases can be more efficient with the Rank Correlated SLW model.

The next example compares prediction by the SLW reference approach and the Rank Correlated SLW model of the radiation transfer in a homogeneous layer of water vapor with significant spatial temperature variation. The focus in this example is on the dependence of simulation results on the sensitivity of predictions in both methods to the number of gray gases used in the model. Consider prediction of the divergence of total net radiative flux in a plane-parallel layer of thickness . L 2 0 m = filled with a water vapor at total pressure . = p 1 0 atm with a sinusoidal temperature profile, ( ) ( )

. cos , T x 1000 0 500

x L K π = +

, and black walls at temperature ( )

T 0 1500 K = and
( )

T L 500 K = .
The layer is filled with water vapor of spatially uniform mole fraction ( ) . The results show that the SLW reference approach (Method I.1.1) is quite sensitive to the number of gray gases used in the simulation, and significant error is observed as the number of gray gases is reduced. The method shows very good accuracy for n > 5 in this problem. By contrast, the Rank Correlated SLW model (Method I.2.2) is insensitive to the number of gray gases used in the prediction, and the predictions show good accuracy with as few as 3 gray gases. The improved accuracy and insensitivity to the number of gray gases for the Rank Correlated SLW model is a critical finding, further underlining the advantages to this new method.

CONCLUSIONS

A range of possibilities for treatment of gas radiative transfer in non-uniform media is explored using the framework of the Generalized SLW model. It is shown that in total, eight different reference approaches are possible, whereas only three have been previously reported in the literature. The development reveals a version of the Generalized SLW reference approach (termed the Rank Correlated SLW model, Method I.2.2) that does not require specification of a gas reference thermodynamic state, and which preserves the spectrally integrated emission (Planck mean absorption coefficient) in the RTE. Construction of the Rank Correlated SLW model requires only two ALBDFs, and subdivision into gray gases can be performed using any accurate integration scheme, e.g., Legendre-Gaussian quadratures. Consequently, this new reference approach appears to have significant advantages over previously published approaches in non-homogeneous gas media. Predictions for sample problems reveal the method's accuracy and robustness.

Appendix 1. Preservation of spectral intervals of integration in the rank correlated method.

This appendix demonstrates theoretically that the Rank Correlated SLW model preserves the spectral intervals of integration, which is critical to ensure that there are no Leibnitz terms in the spectrally integrated RTE, Eqs. [START_REF] Modest | The full-spectrum correlated-k distribution and its relationship to the weighted-sum-of-gray-gases method[END_REF] and [START_REF] Cai | Improved full-spectrum k-distribution implementation for inhomogeneous media using a narrow-band database[END_REF]. Let

( ) { } 1 1 1 : C C η ∆ η φ = <
. Let η′ be any wavenumber such that ( )

1 1 C C η φ ′ = .
At least one such value η′ should exist, because from the definition of the distribution function, ( ) ( )

1 b 1 0 0 C , ,T max C η η ξ φ φ < <∞ ≤ < for any ( ) 0,1 ξ ∈
. Consequently, consistent with the assumption of rank correlated spectra, ( )

2 2 C C η φ ′ = is such that ( ) { } ( ) { } 2 2 2 1 1 1 : C C : C C η η ∆ η φ ∆ η φ = < = = < (A1.2)
The values of the ALBDF at T should be equal because they represent the same fraction of the Planck blackbody emissive power:

( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 2 b b b b b 1 1 b b b b b 1 1 F C , ,T E T d E T d F C , ,T E T E T η η ∆ ∆ φ η η φ ξ = = = = ∫ ∫ (A1.3)
The value of C2 satisfying this equation is unique, because ( )

2 b F C, ,T φ is a strictly increasing function of C. Therefore, ( ) 2
( ) ( ) { } ( ) ( ) { } 1 1 b 1 2 2 b 2 : C C , ,T C : C C , ,T C η η η φ ξ φ η φ ξ φ < = = < = (A1.4)
The implication of this relation is that the Rank Correlated SLW model, Method I.2.2, preserves the gray gas spectral intervals for any gas thermodynamic state, and therefore, spectral integration of the RTE does not generate Leibnitz terms.

( )

min max min j n ref j C C C C =  , , ,
C -  such that ( ) ( ) , , , , ref loc j ref b loc j loc b loc F C T T F C T T φ φ = = =   (A2.2a) ( ) ( ) , , , , ref loc j 1 ref b loc j 1 loc b loc F C T T F C T T φ φ - - = = =   (A2.2b)
Therefore, the gray gas intervals at the local state are the same as the gray gas intervals at the reference state

( ) { } : loc loc loc ref j j 1 loc j j C C C η η φ - ∆ = < < = ∆   (A2.3)
Consequently, this approach preserves the spectral intervals of integration for all locations.

Appendix 3. Co-monotonicity of rank correlated absorption cross-sections.

This appendix demonstrates rigorously that if spectra at two different thermodynamic states

( ) { } ( ) { } 1 1 2 2 : C C : C C η η η φ η φ < ⊆ < (A3.1a) or ( ) { } ( ) { } 1 1 2 2 : C C : C C η η η φ η φ < ⊇ < (A3.1b)
First, the obvious fact may be formulated that for an absorption cross-section ( )

C η φ at an arbitrary state φ , if 1 2 S S < , then ( ) { } ( ) { } 1 2 : C S : C S η η η φ η φ < ⊆ <
. Now let η′ be any wavenumber such that ( )

1 1 C C η φ ′ = .
At least one such value η′ should exist, because min,1

1 max,1 C C C < < . If one defines wavenumber intervals ( ) { } 1 1 1 : C C η ∆ η φ = <
, then according to the assumption of rank correlated spectra, ( )

1 X C η φ ′ = is such that ( ) { } ( ) { } 2 2 1 1 1 : C X : C C η η ∆ η φ ∆ η φ = < = = < (A3.2)
The prescribed value 2 C is, according to established fact, either

( ) { } ( ) { } ( ) { } 1 1 2 2 2 : C C : C X : C C η η η η φ η φ η φ < = < ⊆ < (A3.3) or 2 X C > , from which it follows that ( ) { } ( ) { } ( ) { } 1 1 2 2 2 : C C : C X : C C η η η η φ η φ η φ < = < ⊇ < (A3.4)

  the spectral Planck blackbody intensity at the local gas temperature ( ) T s , and the local spectral absorption coefficient is defined as vector notation for the definition of the local

  cross-section C at a given gas thermodynamic state that lies in the part of the spectrum where the gas absorption cross-section ( ) g C η φ is below the prescribed value C :

  for example, as a geometric mean j

  thermodynamic state. The implication of this observation is that the spectral intervals therefore, the values of the ALBDF at loc C and ref C calculated by integration over this interval with the same blackbody source temperature b

=

  at an arbitrary state φ and the absorption coefficient ( ) ( ) at some chosen reference state 0 φ in the following way: At every wavenumber η where the reference absorption spectrum ( ) has one and the same value k , the local absorption spectrum

2 C 2 C

 22 η φ at the same arbitrary value of wavenumber η′ . If the intervals η φ (neither of which is designed as a reference spectrum) are said to be rank correlated. This definition is visualized in Fig.4. Rank correlation establishes the relationship between the gas absorption cross-sections at two different gas thermodynamic states, and it does not require specification of any reference thermodynamic state ref φ or reference absorption cross-section states φ for the same fixed source blackbody source temperature Tb. According to the assumption of rank correlated spectrum, for a fixed value of the F-variable and fixed blackbody source temperature b T , the wavenumber intervals are defined as follows

1 C η φ and ( ) 2 C

 12 comonotonic (for common monotonicity).An equivalent relationship in terms of absorption cross-sections and the ALBDF can be established for SLW modelling. In Appendix 3, it is shown that if gas spectra at two different thermodynamic states ( ) η φ are rank correlated, then for any two prescribed values 1 C and 2 C , either

=

  gas. The reference mole fractions of gas species m ,ref Y and the reference soot volume fraction ref v f should be added in the case of gas mixtures with soot. The reference state defines the reference absorption crossand the corresponding reference absorption coefficient. The ALBDF calculated at the reference conditions is ( ) temperature not necessarily equal to the temperature defining the reference state. This temperature fixes the reference Planck blackbody intensity ( ) b ref I T η which defines a reference measure ( ) b ref I T d η η , and, as stated correctly by Modest, is "only a mathematical convenience and its choice does not affect the accuracy of calculations [9, p. 665]." Consider the reference partition of the C-variable. The set of the reference supplemental crosssections ref j C  defines the reference values of absorption cross-and corresponding partition into gray gas spectral intervals

  Planck function is only a mathematical convenience.

, 1 1 -

 1 . Consequently, the discretization into values ref j F  which define the gray gas intervals, and into values ref j F which define the absorption cross-sections, is performed by transformation of the Gaussian quadratures from [ ]

  total there are four possible ALBDFs, F(C), and corresponding inverse ALBDFs, C(F):

1 )T = , or 2 )

 12 Dependence with respect to reference blackbody source temperature b ref T Dependence with respect to local blackbody source temperature b loc T T

  reveals four possible ALBDFs generated using combinations of local and reference states ( above. These four ALBDFs graphed qualitatively in the four quadrants of the figure are generated with the possible combinations of reference and local variables, and each has a corresponding inverse ALBDF. Figure2reflects all steps in the possible reference approaches and outlines the difference between the approaches. Referencing the four quadrants of Fig.8, classification and notations for the different versions of the SLW reference approach are summarized in the panels of Fig.9, designated as methods I.1.1 -I.2.2 and II.1.1 -II.2.2.

3. 5 . 3 ) 5 ) 1 . 2 3 ) 5 ) 2 . 1 2 ) 4 ) 2 . 2 2 ) 4 ) 2 3 ) 4 )

 535123521242224234 I Spectrum relationship at the reference blackbody source temperature Tb = Tref Method I.1.1 This method was the first published SLW reference approach, presented in[START_REF] Denison | The spectral line based weighted-sum-of-gray-gases model in nonisothermal non-homogeneous media[END_REF]. 1) Start with subdivision of the C-variable into supplemental cross-sections ref j Calculate the local supplemental absorption cross-sections with the inverse ALBDF Local weights are then calculated using the ALBDF at the reference cross-sections ref j This method is a modification of the SLW reference approach[START_REF] Solovjov | Spectral Line-Based Weighted-Sum-of-Gray-Gases Modeling of Radiative Transfer in Multicomponent Mixtures of Hot Gases[END_REF] and the ADF method[START_REF] Pierrot | A fictitious-gas-based absorption distribution function global model for radiative transfer in hot gases[END_REF]. It differs from the original SLW model (Model I.1.1) only by the manner in which the local gray gas weights are calculated. 1) Start with subdivision of the C-variable into supplemental cross-sections ref j Calculate the local supplemental absorption cross-sections with the inverse ALBDF Local weights are calculated with the ALBDF at the local cross-sections loc j In this new version of the reference approach the assumption of rank correlation allows calculation of the local absorption cross-sections without the help of a reference state. The method requires the generation of three ALBDFs. 1) Define the partition of the F-variable into reference values ref j Calculate the local and the reference partition of the C-variable with the help of the inverse ALBDFs Local weights are calculated using the ALBDF at the reference cross-sections ref j This is a new method termed the Rank Correlated SLW model, distinguished by the fact that there is no need to specify a reference state or a reference absorption spectrum. It requires only two ALBDF distribution functions. It will be shown that this method is superior to all other versions of the SLW reference approach. 1) Define the partition of the F-variable into reference values Find the local partition of the C-variable using the inverse ALBDF Local weights are calculated with the help of the ALBDF at the local cross-sections loc j Because of the advantages this method presents a more detailed theoretical development is given in Section 3.7. 3.5.II Spectrum relationship at the local blackbody source temperature Tb = Tloc Method II.1.1 Only two ALBDFs are generated for this new approach. 1) Start with subdivision of the C-variable into supplemental cross-sections ref j This new method differs from Method II.1.1 only by the manner in which the local gray gas weights are calculated. 1) Start with subdivision of the C-variable into supplemental cross-sections ref j Calculate the local supplemental cross-sections loc j The weights are calculated with the help of the local supplemental cross-sections loc j

2 )

 2 Calculate the local and reference value supplemental absorption cross-sections loc j C  , ref j C  and absorption cross-sections loc j C , ref j C using the inverse ALBDF and the direct ALBDF

  ref

4 ) 2 . 2 2 )

 4222 Local weights are calculated with the help of the ALBDF at the reference cross-sections ref j This method is technically similar to the improved FSK implementation reported in[START_REF] Cai | Improved full-spectrum k-distribution implementation for inhomogeneous media using a narrow-band database[END_REF]. It differs formally from Method II.2.1 only by the manner in which the weights are calculated. However, in fact, it yields the same local weights as Method II.2.1. 1) Define the partition of the F-variable into reference values Calculate the local supplemental cross-sections loc j C  and absorption cross-sections loc j C using the inverse ALBDF and the direct ALBDF

  ref

4 )=

 4 Local weights are calculated with the help of the ALBDF at the local cross-sections loc j In fact, these weights are already predetermined by the values calculated at the reference crosssections ref j C  , which makes this approach equivalent to Method II.2.1. Substitution of ( ) into this equation calculated at step 2), yields loc j

  Planck mean absorption coefficient [9] evaluated at the local state g loc φ φ = and the blackbody source temperature is the local temperature b loc T T = . Therefore, for correct calculation of the source term in the RTE the local gas temperature g T and the emission source temperature b T must be the same local temperature of the medium the spectral integration of the emission term by the SLW model over the gray gas intervals yields ( term in the RTE is thus the product of the total blackbody intensity at the local temperature and a modified Planck mean absorption coefficient calculated with the same local gas and blackbody source temperature. Therefore, any valid model should calculate the gray gas weights with the ALBDF at the local gas thermodynamic state loc φ and the local source temperature b loc T T = (i.e., the method should preserve the local Planck mean absorption coefficient). SLW modelling of the Planck mean absorption coefficient for the different reference approaches described above yields the following expressions: For methods I.1.1, I.2.1, II.1.1-2, and II.2.1-2

2 ) 3 ) 4 ) 5 )

 2345 Determine the local partition of the C-variable using the inverse ALBDF: Calculate the local gray gas absorption coefficients: Calculate the local gray gas weights with the help of the ALBDF at the local cross-sections loc j The parameters thus found define the local gray gas RTE in non-uniform medium:

  by-line benchmark predictions are made, along with predictions using the original SLW reference approach (Method I.1.1) and the Rank Correlated SLW model (Method I.2.2). The same grid resolution (81 equally spaced points) was used in both the line-by-line and the SLW model predictions for both methods.Predictions of the divergence of the total net radiative flux with the original SLW reference approach (Method I.1.1) and the Rank Correlated SLW model (Method I.2.2) are shown in Figs. 15b and 15c, respectively, along with the line-by-line predictions. The values for Tb and Tref (where required) are specified to be identical for these simulations, and for comparison two different values are investigated: i) b ref ave T T T = =where Tave is the volumetrically averaged temperature of the layer, and ii)

  shown in Fig.16for the original SLW reference approach (Method I.1.1) and the Rank Correlated SLW model (Method I.2.2), along with the line-by-line benchmark.

2 C

 2 η φ be rank correlated absorption cross-sections at states φ1 and φ2. Correspondingly, by the definition of rank correlation, may be stated as follows. Fix an arbitrary number ξ in the interval between 0 and 1,

=

  with the same blackbody source temperature b

  desired relationship Eq. (A1.2) holds:

1 )C  and loc j 1

 11 These intervals are fixed because they are defined by fixed reference quantities. The assumption of a correlated spectrum yields that for the absorption cross-section ( ) loc C η φ at any local state φloc there exist local supplemental cross-sections loc j

2 C 2 C

 22 η φ are rank correlated, then for any two prescribed values 1 C and 2 C , either the set of all wavenumbers η such that is a subset of the set of all wavenumbers η for which η φ are rank correlated, then for any two prescribed values C1 and C2 such that min,

Figure 1 .

 1 Figure 1. Geometric interpretation of the SLW model in uniform medium

Figure 2 .Figure 3 .Figure 4 .C η φ at two different thermodynamic states 1 φ

 2341 Figure 2. Gray gas wavenumber intervals

Figure 5 .

 5 Figure 5. Rank correlated reordered absorption cross-section.

Figure 6 .

 6 Figure 6. Rank correlated reordered absorption cross-sections ( ) j loc b C F , ,T φ  of the SLW model.

Figure 7 .Figure 8 .Figure 9 .Figure 10 .

 78910 Figure 7. Visualization of the Generalized SLW reference approach based on the rank correlated spectrum assumption, which maintain fixed wavenumber intervals for spectral integration.

Figure 11 .FFigure 12

 1112 Figure 11. Finite increment form and continuous limit of the Rank Correlated SLW model (Method I.2.2).

Figure 13 .

 13 Figure 13. a) Normalized total wall radiative flux, and b) relative error of prediction of exiting flux, %, for Example 2.

Figure 14 .

 14 Figure 14. Predicted total divergence of the net radiative flux for Example 3.

Figure 15

 15 Figure 15. a) Gas temperature profile for Example 4, and predicted total divergence of the net radiative flux for b) Method I.1.1 (original SLW model reference method), and c) Method I.2.2 (Rank Correlated SLW model).

Figure 16 .

 16 Figure 16. Predicted total divergence of the net radiative flux for Example 5: a) Method I.1.1 (original SLW reference method), and b) Method I.2.2 (Rank Correlated SLW model).

Outline of the Rank Correlated SLW model in non-uniform medium (Method I.2.2)

  Because the Rank Correlated SLW model (Method I.2.2) has significant advantages over other reference approaches the detailed development of the method is summarized in greater detail here.

	2.1 and I.2.2:	one step
	Methods I.1.1, I.1.2, II.1.1, and II.1.2: two steps
	Methods II.2.1 and II.2.2:	three steps
	3.7

Acknowledgement

This work was partially supported by Institut National des Sciences Appliquées de Lyon, France.