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k-distribution of transmission function and theory of Dirichlet series
Journal of Quantitative Spectroscopy & Radiative Transfer 66 (2000) 243}262
k-distribution of transmission function and theory of Dirichlet series S.D. Tvorogov*, L.I. Nesmelova, O.B. Rodimova Institute of Atmospheric Optics, Siberian Branch of the Russian Academy of Sciences, Tomsk, 634055, Russia
Abstract A representation of the transmission function by a series of exponents is discussed in the context of the Dirichlet series theory. A rigorous expression for the distribution function of the absorption coe$cient is obtained for a homogeneous medium. In this case it is a formalization of the ordering procedure for absorption coe$cients. A rigorous extension of this expression to the case of a nonhomogeneous medium, overlapping spectra and integrals with the source function is also performed. The resulting relations can be used to derive approximate formulas. Thus, such a formula whose accuracy can be evaluated is written for a nonhomogeneous medium and in relation to this formula, the meaning of the correlation of k-distributions is discussed. It is shown that the number of terms in the series of exponents used for the transmission function can be greatly reduced by means of the foregoing results. As a general conclusion, it follows that transmission functions should be represented by series of exponents using the distribution function for absorption coe$cients rather than the distribution function density for absorption coe$cients. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction A representation of the transmission function by a series of exponents or the k-distribution, the term generally accepted in atmospheric optics, is widely covered in the literature [1}21]. It is also known as the Dirichlet series in the mathematical language. The method is treated as a tool intended for reducing calculations in radiation codes used in climate models and geophysical applications [7,22,23]. Conventional problems of atmospheric optics, such as calculations of characteristics of a medium integrated over the frequency spectrum in the cases of overlapping
* Corresponding author. 0022-4073/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 1 2 1 - 1
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bands [3] and nonhomogeneous media [3,10], and estimation of radiation #uxes (the source function), are all directly related to this subject. The problem of this kind of expansion appears to be straightforward. Indeed, we are dealing with integrals over the frequency of well-known functions. A purely computational problem is caused by a `palisadea of a great number of spectral lines, and replacement of this `capriciousa integrand with a monotone function allows one to approximate the corresponding integrals by a sum with a small number of terms. The method itself has been known for a few decades. The idea suggested as early as 1936 has gradually assumed features of a formal scheme and has given rise to a number of versions and speci"c algorithms. It should be noted, however, that these results were obtained at the `physicala level of rigor. At the same time special mathematical literature (mainly in Russian) has covered in detail [24}26] the Dirichlet series which actually involve series of exponents used in spectroscopy (some results are given in Supplement for convenience of reference). We believe that combining exact formal results with physical aspects will provide an adequate mathematical climate for the development of appropriate methods and algorithms. Strictly speaking, it is this combination and what it may give for the problems discussed which are the main objective of the paper. It goes without saying that such a formalization is by no means a basis for an improvement in each individual algorithm available. More likely, it can help develop new algorithms and give an insight into a niche occupied by a speci"c algorithm and its advantages. It should be noted that, since a fairly long list of expedients is already available and some formulas given below can be found in the literature, the discussion that follows will sometimes take the form of a review. At this point, however, we can underline some peculiar features illustrating the main goal of the paper. It is important that two mathematically equivalent forms (2a) and (2b) of the expression for the transmission function based on the Laplace transform, in terms of the distribution function of the absorption coe$cients g(s) and in terms of the distribution function density of the absorption coe$cients f (s), possess distinctly di!erent qualities when used to construct computational algorithms. When expressions (2a) and (2b) are considered as the basis for expansion (6a) and (6b) into a series of exponents the dependence on particular values of the absorption coe$cients is centered on the coe$cients of the exponents in the case of (6a) and on the exponents in the case of (6b). In either case rigorous formulas are derived for this dependence. Computationally, a comparison of the two ways of expansion points clearly to the advantages of Eqs. (2b) and (6b). Relation (7), a rigorous expression of the distribution function of the absorption coe$cients for a homogeneous medium, is, on the one hand, a consequence of mathematically correct calculations, and, on the other hand, a formal realization of the scheme of arranging the calculated absorption coe$cients in the increasing order. Formulas (16), (20) and (28) rigorously extend this procedure to include the case of a nonhomogeneous medium, overlapping spectra and integrals with the source function, thereby providing conditions for approximations and simpli"cations to be estimated. In particular, a heuristic approach [10] using the convolution integral to consider the overlapping spectra is obviously an approximation, which can be clearly seen from Eq. (22). Further, well-evaluated approximation (18) states that there is no acute need to create arti"cially `correlated k-distributionsa [13] for di!erent altitudes. Finally, approximation (29) following from the corresponding rigorous formula allows one to simplify considerably the calculations of the source function.
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Proceeding with our comments on mathematical peculiarities of Eqs. (3) and (4) we note that these formulas essentially imply that the replacement of the integration variables i(u)"s in integral (3) is incorrect (in fact, a similar idea is exploited in Refs. [3,4]). Furthermore, it is pertinent to raise the question as to the feasibility of calculating the coe$cients in the series of exponents directly from the experimental data without any need for the line-by-line procedure or absorption band models. Actually, the formulation of this kind of exponent series expansion method seems quite reasonable. The point is that the majority of climatic and geophysical problems do not require a detailed knowledge of the spectral behavior of the absorption coe$cient. While mathematical prerequisites for this action are basically available, an e$cient algorithm is yet to be found. Let us mention the major problems to be discussed and the basic results obtained in this paper. Section 2 presents the derivation of explicit formula (7) for the function g(s) known as the distribution function of the absorption coe$cient in the case of a homogeneous medium. Two ways of obtaining the actual series of exponents are described: the representation of the transmission function in terms of its Laplace transform f (s) and in terms of g(s). Arguments are given in favor of the latter variant. In Section 3, the derivation of the formula for the expansion of the transmission function in the case of nonhomogeneous media is given. It is these mathematical results, Eqs. (7) and (6b), which allow one to lift the requirement for the correlation of g(s) for di!erent atmospheric layers. The e$ciency of approximation (18) whose physical meaning is quite obvious, is substantiated. In Section 4 the problem of overlapping spectra is treated. The foregoing mathematical results clearly demonstrate that it is insu$cient to use only the parallels between the series of exponents and the probability theory as a means of solving the problem of overlapping spectra. A way of evaluating integrals with the source function is considered in Section 5.
2. Homogeneous medium The expression 1 P(x)" *u
S e\VGS du, SY
*u"u !u ,
(1)
where i(u) is the molecular absorption coe$cient at frequency u, de"nes the transmission function for an absorber amount x in a homogeneous gas medium. Let us write some relations generally regarded as a starting premise when the `k-distributiona is discussed. Function (1) is considered to be the Laplace transform of the corresponding function f (s):
P(x)"
ds f (s)e\VQ
(2a)
"
dg e\VQE,
(2b)
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A> dx eQVP(x), c'0, A\ Q A> dx 1 P(x)eQV, g(s)" f (s) ds" x 2pi A\ A more general formula 1 f (s)" 2pi
(3) s(g)"g\(s).
(4)
S $(i(u)) du" $(s(g)) dg" $(s) f (s) ds (5) SY is derived according to a conventional scenario of proving the Parseval theorem on the product of the Laplace transforms. Here $(s) is an arbitrary function of i(u). Series of exponents appear upon application of the relevant quadrature formulas in Eqs. (2a) and (2b): 1 *u
P(x)" a e\VQJ (6a) J J " b e\VQEJ (6b) J J with abscissas s , g and ordinates a , b . The quantities s , g and b are speci"ed by the J J J J J J J quadrature formula. The problem is to "nd a direct relationship between a and s(g ), on the one J J hand, and the absorption coe$cients, on the other. A correct way to solve the problem is best illustrated in the case of a homogeneous medium. Extensions to more complicated situations will also be given. It is quite clear that an answer to the question in hand should follow from the substitution of Eq. (1) into Eqs. (3) or (4) and transposition of integrations over x and u in the resultant double integrals. However, the latter operation is only possible for Eq. (4) and it is this fact which represents a subtle mathematical feature of the problem. Really, the requirement of a uniform convergence imposed to change the order of the integrals results in the following condition in the case of Eq. (3):
5
1 dx e VQ\GS " sin((s!i(u)) (= != ))(e p(s!i(u))
5 to be satis"ed for all u simultaneously at any e'0 and for arbitrary values of = , = . The case where = != PR must not be excluded from the consideration. Then the last expression is transformed into d(s!i(u)). This, however, testi"es that the integral diverges at particular values of u. At the same time, when the expression for g(s) in terms of P(x) is considered (Eq. (4)), a quantity appears in the estimate of the uniform convergence for su$ciently large = and = which is proportional to (= (s!i(u)))\ sin((s!i(u)) (= != )). Now even with the `criticala condition s"i(u) there exists an u-independent number (= != )/= which actually ensures a uniform convergence. Further elucidation is related to the fact that Eq. (2a) (and then Eq. (6a)) may appear to be derivable from Eq. (1) introducing formally a new integration variable s"i(u); then f (s)"1/"i (u)" (a similar idea is used in Refs. [3,4]). However, the well-known theorems of mathematical analysis [28] state that, in this case the substitution s"i(u) is incorrect. This operation is permissible if the quantity 1/i (u) is continuous. At the same time the points such that i (u)"0 are quite common to
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the absorption coe$cient curve that depicts the `palisadea of spectral lines with a great number of maxima and minima. A suggestion that the abscissas in numerical integration (2) should be chosen in a way de"ned by the condition i (u)O0 does not remedy the situation since, by de"nition, the value of the integral should not depend on the way *u is subdivided. The problems outlined do not occur if the substitution of Eq. (1) in Eq. (4) is performed to yield 1 g(s)" *u
du.
(7)
GSXQ SZ SY S
Mathematical details of the derivation of Eq. (7) are given in Appendix A. Fig. 1 is a graphic representation of Eq. (7). Clearly, the line shape does not present any further di$culties when Eq. (7) is used. The approach at hand does not necessarily assume the evaluation of the roots of the equation s"i(u). In e!ect, Eq. (7) merely implies ordering of the absorption coe$cients. In so doing, it should be borne in mind that, if N points are chosen from the interval *u at equal distances from each other and the maximum value of i(u) in this ordered sequence is s(1), the next term will be s(1!1/N), etc. In other words, Eq. (7) formalizes the ordering of i(u), i.e. a procedure suggested in a number of papers virtually from qualitative considerations. Rigorous mathematical formulation (7) enables one to extend this procedure to more complicated cases than that of a homogeneous medium. Comparison of Eqs. (6a) and (6b) seems purely pragmatic. In formal terms, these variants are equivalent. Yet there are strong arguments in favor of Eq. (6b), which can be regarded as a signi"cant mathematical accomplishment. One argument pertains to the case where function (7) is calculated immediately through the absorption coe$cient to give f (s)"g (s). The di!erentiation should be performed numerically on calculation of g(s), which is a well-known inverse problem with `swinginga of errors. The use of Eq. (6b) does not require this kind of di!erentiation. One further argument is related to the theory of the Dirichlet series. Some pertinent results are given in Supplement. The known exponent expansion plays an important part: 1 ¸(j) e\HJ V. e\HV" j!j ¸ (j ) J J J
Fig. 1. Function g(s) is the sum of intervals marked on abscissa axis.
(8)
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Here ¸(j) is an integral function: ¸(j)" (1!j/j ) (9) J J of the argument j, and the values j are simple roots of ¸(j). Substitution of Eq. (8) in Eq. (1) and J the use of Eq. (5) result in rigorous expressions for the expansion coe$cients in Eq. (6a):
¸(s) f (s) ds 1 S ¸(i(u)) 1 1 " du. (10) i(u)!j ¸ (j ) s!j ¸ (j ) *u SY J J J J To be sure, a desire to obtain the most rational (with a minimal number of terms) expansion in a series of exponents prevails. On the one hand, Eq. (10) implies that for this purpose ¸(j) should be among the polynomials orthogonal with a weight f (s). In this case the quantities a in Eq. (10) J become ordinates in the quadrature formula of the Guass type. On the other hand, to do so, expansion (8) must exist. However, this is the case where the description should start with the words: `If there exists such ¸(j) that 2a and the problem of satisfying these two conditions simultaneously is far from being solved automatically (see the pertinent analysis in Supplement). At the same time, the properties of s(g), such as monotony, continuity and limitedness that follow from Eq. (7) assure the e$ciency of any quadrature formula (of the type of the Gauss or Chebyshev formulas) for Eq. (2b). Another argument is also related to the theory of series of exponents. Relation (S.5) makes it possible (see Appendix A) to write a in Eq. (6a) directly in terms of i(u): J 1 du. (11) a " J *u SZ HJ\ XGSXHJ>
The use of the roots of function (9) is a powerful method at the level of `theorems of existencea. However, the actual construction of function (9) with the properties which are necessary for the existence of Eq. (8) is faced with certain (and very essential) di$culties (see Supplement). It is this problem which is eliminated due to the transition to Eq. (2b) or (6b), while retaining all the advantages of the explicit relation between the series of exponents and the absorption coe$cient. In e!ect, a comparison of Eqs. (11) and (7) indicates that s "j , J J
a " J
a "g(j )!g(j ), (12) J J> J\ and, as shown in Supplement, the in"nite (in the limit) set j "lls a limited interval everywhere J densely. Therefore Eq. (12) may be treated as an in"nitesimal increment of g, which is actually equivalent to the substitution of variable according to Eq. (4). Needless to say that, in formal terms expressions (6a) and(6b) are equivalent. Their comparison is dictated by pragmatic purposes. It should be emphasized that the quantities s , b and g are mere J J J numbers solely controlled by choosing the appropriate quadrature formula, whereas quantities (10) are undoubtedly dependent on the thermodynamic characteristics of the medium, which necessarily creates serious di$culties in the case of a nonhomogeneous medium and in the calculation of the source function. A desire to "nd parameters of series of exponents using empirical data on the transmission function seems quite logical in the context of the foregoing discussion. In principle, the solution appears to be feasible: substituting Eq. (9) in Eq. (10), coe$cients a are expressed in terms of the J
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moments of the form
*KP(x) S iK(u) du"(!1)K . (13) *xK SY V Again, the transition to Eq. (6b) appears to be useful, because we need only to have knowledge of the numbers k "s(g ). The properties of s(g) mentioned above and the procedure of constructing J J Eq. (9) described in Supplement ensure that the function K(k)" (1!k/k ) corresponds to the J J expansion of exp(!j x) according to Eq. (8). Then the function ¸ is eliminated by trick (S.9); the J resultant series is equal to the one from Eq. (6b), which gives rise to a system of equations for k : J 1 S K(i(u)) du"b K (k ) (14) J J i(u)!k *u SY J whose left-hand side is reduced to the quantities = (13). K Eqs. (13) and (14) are, in fact, the `existence theorema of a sort for the solution of the problem formulated earlier in the discussion of this subject. Such a theorem follows from the mathematical properties of the Dirichlet series and is undoubtedly of pragmatic importance. Indeed, spectroscopic peculiarities of i(u) in Eq. (1) whose description poses major problems are clearly smoothed out. Certain di$culties emerge when models of spectra for P(x) in Eqs. (3) and (4) are invoked (the way of principle importance according to Ref. [14]). Yet a workable algorithm remains to be found. To sum up the foregoing discussion focused on rigorous mathematical methods, the following statement can be formulated: the most rational way of expanding the absorption function into a series of exponents is a combination of Eqs. (7) and (6b) which appear to be closely related. 1 = " K *u
3. Nonhomogeneous medium In the case of a nonhomogeneous medium where the thermodynamic parameters and, consequently, the absorption coe$cient change from point to point, we have to replace Eq. (1) by the relation of the form
S e\OS_J du, q(u, l)" i(u; l ) dl , (15) SY J where in the general case q(u; l) involves a curvilinear integral over points l along the ray path. The argument l in Q and q (and in similar expressions below) indicates a function of the upper limit, i.e. of the "nal point of the ray path in the integral dl (2). J A representation of Q(l) by a series of exponents can be really obtained: the quantity l is assumed to be a parameter and q in Eq. (15) must be replaced by qx, where x is an arti"cial variable. Formally, the problem is reduced to Eq. (1) but with the function 1 Q(l)" *u
1 g (s, l)" *u
du (16) OS JXQ SZ SY S
used instead of g(s) (see Eq. (7)). Let us now turn to Eq. (6b), taking into account the fact that b and J g are mere numbers, i.e. they are independent of thermodynamic parameters. In addition, let us J
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assume that x"1 to get (17) Q(l)" b e\Q EJ J, J J where s (g) is the inverse function of g (s, l) (Eq. (16)). It is essential that Eqs. (16) and (17) are rigorous formulas and Eq. (16) implies the abovementioned ordering procedure for q(u, l) as a function of u for given dl(2) when s is calculated. J There are problems, of course, wherein Eq. (17) is a convenient computational tool. For example, for a ray path through the plane-parallel horizontally homogeneous atmosphere running at an angle h to the vertical from a prescribed altitude between z and z , we have q"X i(u, z) dz sec h,q (u)sec h, and Q" b exp(!s (g )sec h), where s (g ) is obtained from X J J J J Eq. (16) upon substitution of q for i. However, because of the two arguments in Eq. (16) the calculations may appear to be too cumbersome, whereas rigorous formula (17) gives a good chance for estimations and approximations. The variant \ QEJ JY JY , (18) Q" b e J J J where s(g, l) is the inverse function of Eq. (7) with i(u)Pi(u, l), is very popular. It can be veri"ed that Eq. (18) is a reliable approximation and there is no need for purely heuristic reasoning relative to the correlation of k-representation (see, for example, Refs. [11,13]). Really, the di!erence between Eqs. (15) and (18) is reduced to the following quantity (see Appendix B): 1 <" *u
S
SY
du
J
dg(X(u, l)!>(g, l)),
where
GS JY JY, QE JY JY >(g, l)"¹K e J (19) X(u, l)"¹K e J J J and ¹K is the operator of the Dyson ordering with respect to variable l. It is possible (see details in J Ref. [27]) to introduce in X the parameters averaged over l and identical for all lines, thereby returning formally to a homogeneous medium. While a search for an adequate approximation and expression for averaged thermodynamic characteristics is far from being simple, at this point `the existence theorema su$ces, speaking in the mathematical language. A similar procedure is possible for > (cf. Eqs. (15) and (17)), which is obvious having the formal equivalence of k and s in mind in the calculation of Q. The equality of k and s averaged over l is, to be sure, an assumption, but it appears quite natural. Now the integral dl (2) in Eq. (19) vanishes, and, consequently, so does the operator ¹K . Thereafter
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251
and these values correspond to a sequence of arguments g "0(g (g 2 (i.e. q(u )"s (g ), q(u )"s (g ), etc.). The corresponding values u (a"1, 2, 3,2) are, to be sure, ? placed in a fashion which is far from being ordered because of the `palisadea of spectral lines. The quantity s (g) appearing in Eq. (18) is equal to s(g, l ) dl "s (g) if (and only if) the ordering for u is ? the same for any value of l. Actually, it is this fact which is implied where the correlation of k-representation is concerned. Needless to say, it is not generally the case. Therefore, s is but only an approximation. The function s (g) is monotone by construction. In algorithms including an ordering procedure di!erent from Eq. (18) (see, e.g., Ref. [13]) there appear other functions s (g) which are also approximate but not monotone any longer. A natural question arises concerning the e$ciency of di!erent approximations. First it is to be noted that, for any variant related to the transposition of i values the condition m (g )"0 or m (g) dg"0 must be ful"lled for m "s !s . The value of m will be assumed T to be `su$ciently smalla. Then the estimate "Q!Q ""0(" dg m (g)exp(!s(g))") follows from Eq. (17) where Q is an approximate value of (15). If s (g) is a monotone function the well-known mean-value theorem, the ordering procedure and the property of m (g) mentioned above provide the estimate "Q!Q ""0("E m (g) dg")"0(" m (g) dg") with 0(g (l. The possibility to choose E a minimal integral of the two and the sign-changing structure of m(g) show that the monotony of s (g) contributes undoubtedly to the reliability of the approximation. As for the nonmonotone s (g), we have to restrict ourselves to the estimate
"Q!Q ""O
e\Q Em (g) dg
whose smallness is not quite evident. Another circumstance concerning the comparison of s and s is the number of abscissas in the quadrature formulas: the smoother the integrand the smaller the number of abscissas. The advantage of the monotone function is quite evident here. Both of the factors mentioned above are essential in terms of the pragmatic aspect relating to the estimate of the number of terms in series (6b), (17) and (18) and in similar series below. As an illustration of the problem discussed let us consider a typical example for the region 5000}5050 cm\ of the CO spectrum examined in Ref. [10] and for the strati"cation of the atmosphere adopted there. The value of (15) obtained by the numerical integration with i precalculated using a line-by-line scheme is denoted as Q ; Q and Q are estimated by Eqs. (17) and (18). It emerges that Q !Q "!0.00718, Q
Q !Q "!0.0035 Q
for "ve terms only(!) and for b and g given by quadrature Gauss formula (Q "0.6557, which J J implies the di!erence in the third signi"cant digit). In the calculation [13] by an algorithm with a nonmonotone s we have to write a series with more than 100 terms.
252
S.D. Tvorogov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 66 (2000) 243}262 Table 1 Calculation of transmittance along nonhomogeneous path with known coe$cients of series of exponents for individual layers using the 5-term Gauss formula l 1
2
3
4
5
0.2393
0.2844
0.2392
0.1185
a J 0.1185 Layer number
a exp[!( s(g , l )*l )m ] J H J H H P
1 2 3 4 5 6 7 8
0.1185 0.11845 0.1178 0.1157 0.1083 0.09843 0.0822 0.06071
0.2393 0.2389 0.2340 0.2193 0.1737 0.1262 0.07029 0.02642
P(l Pl ) H 0.2844 0.2834 0.2696 0.2311 0.1332 0.0633 0.0165 0.0017
0.2393 0.2346 0.1780 0.0780 0.001 0.00001 0 0
0.1185 0.09453 0.0050 0.00001 0.00001 0 0 0
1.0 0.970 0.804 0.415 0.288 0.169 0.078 0.034
The simplicity of handling the Dirichlet series with a small number of terms is illustrated by Table 1 pertaining to this example. The atmosphere is divided into seven homogeneous layers (it is clear that Eq. (18) takes the form b exp[! s(g , l )*l ] where l is the altitude and *l is the J J H J H H H H thickness of the layer). Table 1 includes all the necessary information for calculating radiation #uxes.
4. Overlapping spectra When spectra of di!erent gas species overlap i(u)" i (u), where i is the absorption K K K coe$cient of the species in question and the corresponding concentration is included in i . K A trivial generalization of Eq. (7) is as follows: 1 g(s)" *u
du
(20)
K K
G SXQ SZ SY S
with subsequent application of Eq. (6b) (x has the meaning of the ray path length). The transition to Eq. (17) or (18) is performed readily. It should be noted again that (20) together with (2b) and (6b) are rigorous formulas and the ordering procedure described is extended to include the case of overlapping spectra. It is natural to use this exact mathematical basis to "nd approximate procedures.
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Eq. (20) presents a possibility of discussing an interesting suggestion [10] that g(s) for (4) and(2a) be written in the form
Q
f (s )g (s!s ) ds "
1 g(s)" (*u)
Q
A> dx P (x)P (x)eQV, (21) x A\ where indices 1 and 2 refer, as previously, to components of the mixture (the third equality in Eq. (21) follows from mathematical problems of permutation of integration over u and s discussed in Section 2). However, Eq. (21) results in the relation of the form g(s)"
1 f (s)g (s!s ) ds " 2pi
du du . (22) G S>G SXQ S S Z SY S
The di!erence between Eqs. (20) and (22) is illustrated in Fig. 2. Eq. (22) implies integration over a `trianglea whereas in Eq. (20) the integration is performed over a `diagonala. To reduce the problem to Eq. (21), i.e. to the characteristics of the components of the mixture, some additional assumptions or approximations are called for. Eq. (21) makes use of the belief that f (s) can be treated as `the probability that the absorption coe$cient will be equal to sa (the relation f (s)'0 results from Eq. (4) and from Eq. (7) for g(s)). Further statement [10] that the spectra are noncorrelated is clear. Indeed, in view of the fact that the concentration of absorbing gases in atmospheric conditions is generally low the contribution of species `1a to the spectral line width of species `2a may well be neglected and vice versa. Then the rules of the probability theory are used to write the distribution function for the sum of the distribution functions for individual species [10]. Thus, the problem reduces to the question whether f (s) can be interpreted as this kind of probability. A negative answer follows from reductio ad adsurdum. Really, if the order of integrals can be changed upon substitution of Eq. (1) into Eq. (3), then f (s)"(1/*u)S du d(s!i(u)), and this expression, by de"nitions of the probability theory, is the SY distribution function with all the ensuing formal consequences. In this case, however, it is impossible to perform the permutation of integrations (see Section 2). It can be made possible using Eq. (4), which leads to Eq. (20).
Fig. 2. On construction of function g(s) for overlapping spectra.
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Another possibility is to consider the following function instead of Eq. (1): 1 P(x , x )" *u
S e\V G S\V G S du, SY
(23)
where gas concentrations are included in x and x . Applying the two-dimensional Laplace transform to Eq. (23) yields
P(x , x )"
ds ds f (s , s )e\Q V \Q V ,
*g(s , s ) , f (s , s )" *s *s
(24b)
g(s , s )" 1 *u
(24a)
du,
(24c)
G SXQ G SXQ SZ SY S
S $(i (u),i (u)) du" ds ds f (s , s )$(s , s ) SY
(24d)
instead of Eqs. (2), (4), (7) and (5). Unfortunately, it turns out that Eqs. (24a)}(24d) are irreducible to the expression of the type given by Eq. (2b) or by its consequence Eq. (6b) with all their practical advantages. And again attention should be paid to the di!erence between g(s) from Eq. (20) and g(s , s ) from Eq. (22). A pragmatic desire to write Eq. (23) in terms of s (g) and s (g), calculated for each gas separately is quite understandable. To this end, an approximation is often used (see, e.g., Refs. [7,20]) based on an analogy between (23) and (15): P(x , x )" b e\V Q EJ \V Q EJ J J
(25)
with an obvious generalization of Eq. (25) to a multicomponent mixture. However, the validity of approximation (25) cannot be substantiated in the way it was done for Eq. (18) for the reasons given in Appendix B. Incidentally, there exist some su$cient conditions where Eq. (25) can be met exactly but these are most likely to be highly exotic as compared to the actual properties of P(x , x ) from Eq. (23).
5. Source function First let us consider the following relation: 1 H(x)" *u
S u(u)e\VGS du, SY
(26)
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where the function u(u) is normalized by the condition (*u)\S u(u) du"1. Performing operaSY tions on Eq. (26) used for Eq. (1) to derive Eq. (6), we obtain
(27) dg e\QS EV" b e\QS EJ V, J J where s is the inverse function of g (s) with S S 1 g (s)" u(u) du. (28) S *u GSXQ SZ S SY
It is apparent that Eqs. (27) and (28) represent a mere extension of Eq. (7) to the case of Eq. (26) with all the foregoing comments as to a rigorous solution of the problem. The need for going to the dimensionless normalized function u is quite evident. It is dictated by a desire that the quantity g be kept dimensionless. Therefore, if an arbitrary function ; appears in S H, we can return to Eq. (26) introducing u";/((*u)\S ;(u) du). The Planck function, radiation SY from some source, arti"cial light spectrum and instrument function or a combination of these can be chosen as the function ;. It should be emphasized that, as in the previous case, rigorous formulas (27) and (28) extend the ordering operation to (26). In order to understand this it will su$ce to introduce a new variable p(u)"u #S u(u) du with the inverse function u"u(p); i(u)"i(u(p)),i(p), and, by de"niSY dz which is equivalent to Eq. (27). tion of u, p3[0,*u]. Eq. (28) takes the form (1/*u) GNXQ NZ S
Indeed, we only have to go from u to p scale. Once again, this procedure allows for the use of a series of exponents with a small number of terms. The example given in Table 2 refers to the 1380}1900 cm\ range of the water vapor spectrum and u in Eq. (26) is related to the Planck function (it is assumed that the temperature is 250 K, the total pressure is 1 atm, and the water vapor amount has the dimensionality g cm\). The ratios between quantity (27) using the ordering procedure just described and quantity (26) calculated according to the line-by-line scheme are given in Table 2. In the "rst column the results obtained by the 5-point Gauss formula are listed, the next three columns tabulate the values for the 6-, 7-, and 9-point Chebyshev formulas, and the last column gives line-by-line result (26) calculated with the Lorentzian line shape cut at 10 cm\ from the line center. As an example of the approximation based on Eq. (28) let us consider separation of variables which, in fact, reduces Eq. (28) to Eq. (7). Pragmatic advantages of this operation are evident. H(x)"
Table 2 Ratios of Planck-weighted transmission function with function calculated from series of exponents and those calculated according to the line-by-line scheme n w, g cm\
5
6
7
9
F
0.0001 0.001 0.01 0.1
1.003 0.9995 0.9982 1.0040
1.007 0.9901 0.9944 0.9697
1.005 0.9934 0.9952 0.9607
1.003 1.0000 0.9985 1.005
0.9488 0.7854 0.4558 0.1189
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Simple arguments taking into account Fig. 1 and the mean-value theorem lend support to the following approximation: g (s) y(s)g(s), S
S Q u(u) du, S Q
1 y(s)" s*u
u (s)"u #(1!s)*u,
u (s)"u !(1!s)*u.
(29)
Actually, y(1)"g(1)"1, which provides the necessary relation g (1)"1; if u(u)"const"1 then S y(s)"1, and we return to Eq. (7). Further, y(0)"u((u #u )/2), i.e. according to the scheme of Fig. 1 the `motiona in integral (29) starts from the center of the interval *u. In e!ect, we merely have to do with the choice of the characteristic point in the mean-value theorem for the integral in Eq. (28).
6. Summary The objective of the paper formulated above is to combine speci"c problems of atmospheric optics and the mathematical theory of the Dirichlet series. The integration of this kind o!ers a possibility to translate methods based on intuitive or heuristic considerations (the `physicala level of rigor) into the exact mathematical language. The formalization of the ordering procedure for the absorption coe$cients by means of relation (7) in the case of a homogeneous medium, its extension to more complicated problems of atmospheric optics (see the discussion of relations (17), (20),(27)), arguments in favor of practical advantages of relation (6b), the interpretation of the correlation concept of k-representations as a purely mathematical approximation and the insu$ciency of reading Eq. (3) as the distribution function density of the absorption coe$cients (together with its consequences for overlapping spectra) can be regarded as obvious. Importantly, the foregoing issues are not merely methodical in any way for they enable one to decrease drastically the number of terms in the series of exponents. In other words, they serve to reduce radiation codes used in climate models and in geophysical applications of atmospheric optics, which is the main purpose of the k-representation method. We believe that the potentialities of the mathematical program outlined are far from being exhausted. For example, an idea was mentioned that the series of exponents may be constructed from experimental data on the transmission function. Approximate formulas for the source function and overlapping spectra remain to be derived. Exact mathematical relations pertaining to the properties of the series of exponents can play a part in the construction of and search for numerical algorithms.
Supplement: On the Dirichlet series We present here some relations of a repeatedly cited theory of the Dirichlet series [24}26] and some discussion concerning functions (9) and their connection to orthonormal polynomials for the quadrature formulas.
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By de"nition, the Dirichlet series is a function F(z)" h e\HJ X J J of the complex variable z, and j are zeros of integral function (9). The coe$cients J 1
(z)F(z) dz h " J J 2pi ! and the closed contour C encircles all singular points of the functions
1 L\ X ¸(j)
(z)" eHX dj. J ¸ (j ) j!j J If F(z) is an integral function Eqs. (31), (32) and (9) lead to the relation of the form
(30)
(31)
(32)
1 @\ P j@\A\FA(0)(!1)A, h " (33) @ J ¸ (j ) J @ A where P are coe$cients of j@ in Eq. (9). The expansion of exponent (8) follows from Eqs. (33) and @ (30) if F(z)"e\HX. Let us further assume that, as in the case of the transmission function, we need only real z'0 in Eq. (30), and F(R)"0. Now it is worthwhile to construct ¸(x) (Eq. (9)) using j '0. The following L relation is a consequence of these assumptions:
J dz m! A> " (m!j )Kh , (34) eKXF(z) @ @ zK> 2pi A\ @ where m"0, 1, 2,2; j (m(j . J J> Eq. (34) emphasizes the intrinsic relationship between Eq. (30) and the Laplace transform (cf. with Eqs. (1)}(4), (6)). Let us write
A> F(z)eQX dz. A\ It actually follows from Eqs. (34) and (35) that F(z)"
e\XQN(s) ds,
N(s)"
(35)
1 *K> J (m!j )Kh "N(m) (36) @ J m! *mK> @ and Eq. (36) states that N from Eq. (6) solves in principle the problem of "niding h and g . Note J J that the derivation of Eqs. (11) and (12) is merely a good case in point. Expression (10) in the present notation has the form
N(s)¸(s) ds . (37) (s!j )¸ (j ) J J The relationships between Eqs. (9) and h (Eq. (37)) as the ordinates of the quadrature formulas of J the Gauss type have already been discussed in Section 2. Recall that this plays a part in the mathematical interpretation of Eq. (6). The question that now arises can be formulated in the h " J
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following way. Let ¸ be a polynomial of degree j orthogonal with the weight N(s) from Eq. (36). H The function M(k)"lim ¸ (k) is, of course, integral, and k are its roots. It is neccessary to "nd H H J conditions wherein M(k) corresponds to the expansion of exponent (8). Treating F(z) in Eq. (30) as an integral function (see Eq. (1)) for which the quadrature Gauss formula is written, we change j for k and h for Eq. (37) with the substitution of M for ¸ implicit J J J from the context. Furthermore, let us apply expansion (8) to exp(!k z) with ¸ properly chosen J and require that the Dirichlet series appear but with M in Eq. (9). The resulting condition is of the form ¸(s) ¸(k )M(s) 1 @ " . M (k ) (k !j )(s!k ) s!j J @ J @ @ @ Equality (38) in equivalent to the integral
(38)
¸(z) dz "0 M(z)(z!j )(s!z) ! J over a closed contour C with all zeros of the functions ¸ and M localized within it. It becomes clear that the functions ¸ and M have condensation points of zeros. The C can readily be moved to in"nity. Furthermore, the inequality m 5m must be ful"lled, where m 2 is an indicator of an + * increase in the corresponding function. To continue the discussion of the problem of choosing ¸(j) (Eq. (9)) let us restrict ourselves to the version z'0 and 0(j(1 (it will su$ce to make an identical replacement in Eq. (1): xPx/i
and iPi , where i is the maximum absorption coe$cient in the interval [u ,u ]).
Let ¸H(j) be a set of polynomials of degree j, whose roots jH "ll the interval [0,1] irrespective of j with the value of j falling within this interval (it is su$cient that the interval be limited, then it can be converted to the required interval with the help of a simple substitution variable). Let us consider the following integral: 1 %" 2pi
e\XQ ds ¸H(s)(j!s)
(39)
! over a closed contour surrounding the interval [0,1]. Obviously, J ¸ (j) , e\XH, , !e\XH ¸H(j)%" ¸HY(jH)(j!jH) J J J and expansion (8) will appear (with j "lim jH) if J H J
lim ¸H(j)%"0. (40) H Of course, integral (39) must be estimated asymptotically to be used in Eq. (40), taking into account the fact that for j chosen the quantity "¸H(j)" is limited even with jPR (for example, in the case of the Chebyshev polynomials ¸H(j)"cos( j arc cos j)). The asymptotic behavior of quantity (39) is de"ned by the relation exp[!(zs#ln ¸H(s))] exp[!(zs#j ln s)]
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Fig. 3. Integration contour in calculation of integral (S.10).
and subsequent standard actions [28] are illustrated in Fig. 3. In this way we obtain the estimate "%""0(exp(!j ln j)) which insures the ful"llment of Eq. (40). Further, Eq. (36) can be used by treating them as a set of equations in j and h and applying, J J say, the least-squares technique. A similar idea [29] consists in using the exponential orthogonal functions D (z)" K d exp(!j z), where d "d (2j 2) are chosen in such a way that K J J J J J J D (z)D (z) dz"d . A subsequent conventional expansion of F(z) in D (z) makes us return K KY KKY K formally to Eq. (30). Once again, it is recommended that the numbers j be found by means of the L least-squares technique, limiting the search to "nite m. It should be added that the way based on D is simply a variation on the theme `biorthogonal setsa since functions (32) are of the form which K yields (2pi)\ dz exp(!j z)u (z)"d . ! J J JJY Acknowledgements This work was supported by the Russian Foundation for Basic Research (grant No. 97-0565985).
Appendix A Changing the order of integrations dx and du leads to the emergence of the following integral [30] from Eqs. (1) and (4): 1 2pi
1, A> dx eVQ\GS" , x A\ 0,
s'i(u), s"i(u),
(A.1)
s(i(u).
We must calculate now an integral of the form
I"
S
S
'(u) du,
(A.2)
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where '(u) is a discontinous function:
1, u (u(u , , u"u (or u ). Let us rewrite Eq. (A.2) as the Stieltjes integral of the form '(u)"
S
S
I"(S)
'(u) dh(u)
(A.3)
(A.4)
S with h(u)"u. The integration of Eq. (A.4) by parts results in
S S '(u) dh(u)"'(u)h(u) " !(S) h(u) d'(u). (A.5) S S S The "rst term in Eq. (A.5) equals (u !u ). The function '(u) is equal to , as it approaches a discontinuous jump from the left, and it will be ! when approaching the discontinuity from the right. Therefore, the second term in Eq. (A.5) is (u !u ), see Ref. [31]. Thus, I"u !u , which leads to Eq. (7). The derivation of Eq. (11) is immediately related to Eq. (34). Writing (34) for l and l#1 (with m"0) and subtracting the expression obtained for l from that derived for l#1 we get I"(S)
A> dz (A.6) F(z)(eK X!eKYX) . z A\ The arrangement of points m and m is shown in Fig. 4. Now we replace F in Eq. (A.6) by P(x) (Eq. (1)) to take into account the feasibility of interchaning the integrations and to return to Eq. (A.1). The arbitrariness inherent to the de"nition of m in Eq. (34) and Fig. 4 allow us to choose m Pj and m Pj . A comparison of Eq. (30) and Eqs. (6) and (10) implies that we are J\ J> dealing with Eq. (6a). The "nal result appears to be Eq. (11), and upon the transformations described the orgin of Eq. (12) becomes quite clear. 1 h " J 2pi
Appendix B To estimate Z, i.e. the di!erence between Eqs. (15) and(18), let us represent the integral dl as J an integral sum (with an element of length *l) and expand all the exponents exp(2) into a series (a is the index of summation). The term without l in Z vanishes because l "1 for any quadrature JJ formula. The terms including i and s with the same argument l will also vanish if we make use of their independence from b and g , Eq. (15) and the fact that and dg are equivalent. Finally, the J J J
Fig. 4. Arrangement of points m and m .
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261
expressions where the degree of *l does not exceed the number of summations over a are eliminated, as is often the case in the theory of multiplicative integrals [32]. Following these simpli"cations Z becomes
Z"
(*l) S i(u; la )i(u; la ) du! b s(g ; la )s(g ; la )(*l) J J J *u SY ? ? J ? ?
#
(*l) S i(u; la )i(u; la )i(u; la ) du *u SY ? ? ?
! b s(g ; la )s(g ; la )s(g ; la )(*l) #2. J J J J J ? ? ?
(B.1)
It is impossible to use Eq. (5) once more because Eq. (5) is not ful"lled in the case of the sum of i(u). Let us denote the sums of the terms in Eq. (B.1) appearing after (*u)\S and b as XI (u; l) SY J J and >I (u; l), respectively, and introduce the quantities X and > from Eq. (19) which are the solutions of the following equations:
X(u; l)"1#
J
>(u; l)"1#
J
dl i(u; l )X(u; l), dl i(u; l )>(u; l).
(B.2)
The representation of X and > in (B.2) as the Neumann series makes us return to the integrals over l in XI and >I . The replacement b Pdg shows that Z coincides with < given just before Eq. (19) J J (the `excessivea terms have no e!ect on the estimate, as it was emphasized in writing Eq. (B.1). Of course, the foregoing consideration is unsuitable for the di!erence between Eqs. (25) and (23) because the expansion in terms of *l cannot be performed. Formula (5) is now replaced by Eq. (24d). However, if the previous expansion of exp(2) is applied and Eq. (24d) used, a su$cient condition for exact coincidence of Eqs. (25) and (27) can readily be derived: 1 *u
S
SY
du iK (u)iK (u), "
ds ds f (s , s )sK sK
dg(g\(g)K g\(g)K ).
(B.3)
The meaning of the subscripts is the same as in Eq. (26). To avoid confusion between the notation of arguments and functions s(g) is replaced by g\(g). Substituting the integration variables s for g (according to Eq. (4) with the subscript `1a) we can see that equality (B.3) is satis"ed if f (s , s )"f (s )d(s !g\(g (s ))).
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This relation and Eq. (24b) imply that g(s , s ) calculated according to Eq. (24c) must coincide with
s 'g\(g (s )), g(s , s )" , s "g\(g (s )) f (s ), (B.4) f (s ), s (g\(g (s )). It is obvious that the condition obtained from (B.4) by interchanging the indices `1a and `2a is also valid. 0,
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
Jamamoto G, Tanaka M, Asano S. J Atmos Sci 1970;27:282. Arking A, Grossman K. J Atmos Sci 1972;29:937. Lacis AA, Hansen JE. J Atmos Sci 1974;31:118. Liou KN, Sasamori T. J Atmos Sci 1975;32:2166. Chou MD, Arking A. J Atmos Sci 1980;37:855. Chou MD, Arking A. J Atmos Sci 1981;38:798. Isaacs RG, Wang W-C, Worsham RD, Goldenberg S. Appl Opt 1987;26:1272. Wang W-C, Shi G-Yu. JQSRT 1988;39:387. Xun Zhu. J Atmos Sci 1989;46:511. Goody R, West R, Chen L, Grisp D. JQSRT 1989;42:539. Phillips WJ. JQSRT 1990;43:13. Si-Chee Tsay, Stamnes K, Jaywecre K. JQSRT 1990;43:133. West R, Crisp D, Chen L. JQSRT 1990;43:191}9. Lacis AA, Oinas V. J Geophys Res 1991;96D:9027. Xun Zhu. JQSRT 1991;45:33. Xun Zhu. JQSRT 1992;47:159. Riviere Ph, Sou"ani A, Taine J. JQSRT 1992;48:187. Hollweg H-D. J Geophys Res 1993;98D:2747. Tvorogov SD. Atmos Oceanic Opt 1994;7:165. Firsov KM, Smirnov AB. Atmos Oceanic Opt 1995;8:659. Tvorogov SD, Nesmelova LI, Rodimova OB. Atmos Oceanic Opt 1996;9:239. Ellington RG, Ellis J, Fels S. J Geophys Res 1991;96D:8929. Fouquart Y, Bonnel B, Ramaswamy V. J Geophys Res 1991;96D:8955. Leont'ev AF. Series of exponents. Moscow: Nauka, 1976. Leont'ev AF. Sequence of polinomials of exponents. Moscow: Nauka, 1980. Leont'ev AF. Integral functions. Series of exponents. Moscow: Nauka, 1980. Zuev VE. Propagation of visible and infrared radiation in the atmosphere. New York: Wiley, 1974. De Bruijn NG. Asymptotic methods in analysis. Amsterdam: North-Holland Publishing Co., 1958. Dmitriev AA. Orthogonal exponential functions in hydrometeorology. Leningrad: Gidrometeoizdat, 1973. Lavrent'ev MA, Shabat BV. Methods of the theory of functions depending on complex variable. Moscow: Gos. Izdat. Fiz-Mat. Lit., 1968. [31] Fichtengolts GM. Course of di!erential and integral calculus, vol. 2. Moscow: Gos. Izdat. Fiz-Mat. Lit., 1969. [32] Gantmacher FP. Matrix theory. Moscow: Nauka, 1966.
k-distribution of transmission function and theory of Dirichlet series S.D. Tvorogov*, L.I. Nesmelova, O.B. Rodimova Institute of Atmospheric Optics, Siberian Branch of the Russian Academy of Sciences, Tomsk, 634055, Russia
Abstract A representation of the transmission function by a series of exponents is discussed in the context of the Dirichlet series theory. A rigorous expression for the distribution function of the absorption coe$cient is obtained for a homogeneous medium. In this case it is a formalization of the ordering procedure for absorption coe$cients. A rigorous extension of this expression to the case of a nonhomogeneous medium, overlapping spectra and integrals with the source function is also performed. The resulting relations can be used to derive approximate formulas. Thus, such a formula whose accuracy can be evaluated is written for a nonhomogeneous medium and in relation to this formula, the meaning of the correlation of k-distributions is discussed. It is shown that the number of terms in the series of exponents used for the transmission function can be greatly reduced by means of the foregoing results. As a general conclusion, it follows that transmission functions should be represented by series of exponents using the distribution function for absorption coe$cients rather than the distribution function density for absorption coe$cients. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction A representation of the transmission function by a series of exponents or the k-distribution, the term generally accepted in atmospheric optics, is widely covered in the literature [1}21]. It is also known as the Dirichlet series in the mathematical language. The method is treated as a tool intended for reducing calculations in radiation codes used in climate models and geophysical applications [7,22,23]. Conventional problems of atmospheric optics, such as calculations of characteristics of a medium integrated over the frequency spectrum in the cases of overlapping
* Corresponding author. 0022-4073/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 1 2 1 - 1
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bands [3] and nonhomogeneous media [3,10], and estimation of radiation #uxes (the source function), are all directly related to this subject. The problem of this kind of expansion appears to be straightforward. Indeed, we are dealing with integrals over the frequency of well-known functions. A purely computational problem is caused by a `palisadea of a great number of spectral lines, and replacement of this `capriciousa integrand with a monotone function allows one to approximate the corresponding integrals by a sum with a small number of terms. The method itself has been known for a few decades. The idea suggested as early as 1936 has gradually assumed features of a formal scheme and has given rise to a number of versions and speci"c algorithms. It should be noted, however, that these results were obtained at the `physicala level of rigor. At the same time special mathematical literature (mainly in Russian) has covered in detail [24}26] the Dirichlet series which actually involve series of exponents used in spectroscopy (some results are given in Supplement for convenience of reference). We believe that combining exact formal results with physical aspects will provide an adequate mathematical climate for the development of appropriate methods and algorithms. Strictly speaking, it is this combination and what it may give for the problems discussed which are the main objective of the paper. It goes without saying that such a formalization is by no means a basis for an improvement in each individual algorithm available. More likely, it can help develop new algorithms and give an insight into a niche occupied by a speci"c algorithm and its advantages. It should be noted that, since a fairly long list of expedients is already available and some formulas given below can be found in the literature, the discussion that follows will sometimes take the form of a review. At this point, however, we can underline some peculiar features illustrating the main goal of the paper. It is important that two mathematically equivalent forms (2a) and (2b) of the expression for the transmission function based on the Laplace transform, in terms of the distribution function of the absorption coe$cients g(s) and in terms of the distribution function density of the absorption coe$cients f (s), possess distinctly di!erent qualities when used to construct computational algorithms. When expressions (2a) and (2b) are considered as the basis for expansion (6a) and (6b) into a series of exponents the dependence on particular values of the absorption coe$cients is centered on the coe$cients of the exponents in the case of (6a) and on the exponents in the case of (6b). In either case rigorous formulas are derived for this dependence. Computationally, a comparison of the two ways of expansion points clearly to the advantages of Eqs. (2b) and (6b). Relation (7), a rigorous expression of the distribution function of the absorption coe$cients for a homogeneous medium, is, on the one hand, a consequence of mathematically correct calculations, and, on the other hand, a formal realization of the scheme of arranging the calculated absorption coe$cients in the increasing order. Formulas (16), (20) and (28) rigorously extend this procedure to include the case of a nonhomogeneous medium, overlapping spectra and integrals with the source function, thereby providing conditions for approximations and simpli"cations to be estimated. In particular, a heuristic approach [10] using the convolution integral to consider the overlapping spectra is obviously an approximation, which can be clearly seen from Eq. (22). Further, well-evaluated approximation (18) states that there is no acute need to create arti"cially `correlated k-distributionsa [13] for di!erent altitudes. Finally, approximation (29) following from the corresponding rigorous formula allows one to simplify considerably the calculations of the source function.
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245
Proceeding with our comments on mathematical peculiarities of Eqs. (3) and (4) we note that these formulas essentially imply that the replacement of the integration variables i(u)"s in integral (3) is incorrect (in fact, a similar idea is exploited in Refs. [3,4]). Furthermore, it is pertinent to raise the question as to the feasibility of calculating the coe$cients in the series of exponents directly from the experimental data without any need for the line-by-line procedure or absorption band models. Actually, the formulation of this kind of exponent series expansion method seems quite reasonable. The point is that the majority of climatic and geophysical problems do not require a detailed knowledge of the spectral behavior of the absorption coe$cient. While mathematical prerequisites for this action are basically available, an e$cient algorithm is yet to be found. Let us mention the major problems to be discussed and the basic results obtained in this paper. Section 2 presents the derivation of explicit formula (7) for the function g(s) known as the distribution function of the absorption coe$cient in the case of a homogeneous medium. Two ways of obtaining the actual series of exponents are described: the representation of the transmission function in terms of its Laplace transform f (s) and in terms of g(s). Arguments are given in favor of the latter variant. In Section 3, the derivation of the formula for the expansion of the transmission function in the case of nonhomogeneous media is given. It is these mathematical results, Eqs. (7) and (6b), which allow one to lift the requirement for the correlation of g(s) for di!erent atmospheric layers. The e$ciency of approximation (18) whose physical meaning is quite obvious, is substantiated. In Section 4 the problem of overlapping spectra is treated. The foregoing mathematical results clearly demonstrate that it is insu$cient to use only the parallels between the series of exponents and the probability theory as a means of solving the problem of overlapping spectra. A way of evaluating integrals with the source function is considered in Section 5.
2. Homogeneous medium The expression 1 P(x)" *u
S e\VGS du, SY
*u"u !u ,
(1)
where i(u) is the molecular absorption coe$cient at frequency u, de"nes the transmission function for an absorber amount x in a homogeneous gas medium. Let us write some relations generally regarded as a starting premise when the `k-distributiona is discussed. Function (1) is considered to be the Laplace transform of the corresponding function f (s):
P(x)"
ds f (s)e\VQ
(2a)
"
dg e\VQE,
(2b)
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A> dx eQVP(x), c'0, A\ Q A> dx 1 P(x)eQV, g(s)" f (s) ds" x 2pi A\ A more general formula 1 f (s)" 2pi
(3) s(g)"g\(s).
(4)
S $(i(u)) du" $(s(g)) dg" $(s) f (s) ds (5) SY is derived according to a conventional scenario of proving the Parseval theorem on the product of the Laplace transforms. Here $(s) is an arbitrary function of i(u). Series of exponents appear upon application of the relevant quadrature formulas in Eqs. (2a) and (2b): 1 *u
P(x)" a e\VQJ (6a) J J " b e\VQEJ (6b) J J with abscissas s , g and ordinates a , b . The quantities s , g and b are speci"ed by the J J J J J J J quadrature formula. The problem is to "nd a direct relationship between a and s(g ), on the one J J hand, and the absorption coe$cients, on the other. A correct way to solve the problem is best illustrated in the case of a homogeneous medium. Extensions to more complicated situations will also be given. It is quite clear that an answer to the question in hand should follow from the substitution of Eq. (1) into Eqs. (3) or (4) and transposition of integrations over x and u in the resultant double integrals. However, the latter operation is only possible for Eq. (4) and it is this fact which represents a subtle mathematical feature of the problem. Really, the requirement of a uniform convergence imposed to change the order of the integrals results in the following condition in the case of Eq. (3):
5
1 dx e VQ\GS " sin((s!i(u)) (= != ))(e p(s!i(u))
5 to be satis"ed for all u simultaneously at any e'0 and for arbitrary values of = , = . The case where = != PR must not be excluded from the consideration. Then the last expression is transformed into d(s!i(u)). This, however, testi"es that the integral diverges at particular values of u. At the same time, when the expression for g(s) in terms of P(x) is considered (Eq. (4)), a quantity appears in the estimate of the uniform convergence for su$ciently large = and = which is proportional to (= (s!i(u)))\ sin((s!i(u)) (= != )). Now even with the `criticala condition s"i(u) there exists an u-independent number (= != )/= which actually ensures a uniform convergence. Further elucidation is related to the fact that Eq. (2a) (and then Eq. (6a)) may appear to be derivable from Eq. (1) introducing formally a new integration variable s"i(u); then f (s)"1/"i (u)" (a similar idea is used in Refs. [3,4]). However, the well-known theorems of mathematical analysis [28] state that, in this case the substitution s"i(u) is incorrect. This operation is permissible if the quantity 1/i (u) is continuous. At the same time the points such that i (u)"0 are quite common to
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the absorption coe$cient curve that depicts the `palisadea of spectral lines with a great number of maxima and minima. A suggestion that the abscissas in numerical integration (2) should be chosen in a way de"ned by the condition i (u)O0 does not remedy the situation since, by de"nition, the value of the integral should not depend on the way *u is subdivided. The problems outlined do not occur if the substitution of Eq. (1) in Eq. (4) is performed to yield 1 g(s)" *u
du.
(7)
GSXQ SZ SY S
Mathematical details of the derivation of Eq. (7) are given in Appendix A. Fig. 1 is a graphic representation of Eq. (7). Clearly, the line shape does not present any further di$culties when Eq. (7) is used. The approach at hand does not necessarily assume the evaluation of the roots of the equation s"i(u). In e!ect, Eq. (7) merely implies ordering of the absorption coe$cients. In so doing, it should be borne in mind that, if N points are chosen from the interval *u at equal distances from each other and the maximum value of i(u) in this ordered sequence is s(1), the next term will be s(1!1/N), etc. In other words, Eq. (7) formalizes the ordering of i(u), i.e. a procedure suggested in a number of papers virtually from qualitative considerations. Rigorous mathematical formulation (7) enables one to extend this procedure to more complicated cases than that of a homogeneous medium. Comparison of Eqs. (6a) and (6b) seems purely pragmatic. In formal terms, these variants are equivalent. Yet there are strong arguments in favor of Eq. (6b), which can be regarded as a signi"cant mathematical accomplishment. One argument pertains to the case where function (7) is calculated immediately through the absorption coe$cient to give f (s)"g (s). The di!erentiation should be performed numerically on calculation of g(s), which is a well-known inverse problem with `swinginga of errors. The use of Eq. (6b) does not require this kind of di!erentiation. One further argument is related to the theory of the Dirichlet series. Some pertinent results are given in Supplement. The known exponent expansion plays an important part: 1 ¸(j) e\HJ V. e\HV" j!j ¸ (j ) J J J
Fig. 1. Function g(s) is the sum of intervals marked on abscissa axis.
(8)
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Here ¸(j) is an integral function: ¸(j)" (1!j/j ) (9) J J of the argument j, and the values j are simple roots of ¸(j). Substitution of Eq. (8) in Eq. (1) and J the use of Eq. (5) result in rigorous expressions for the expansion coe$cients in Eq. (6a):
¸(s) f (s) ds 1 S ¸(i(u)) 1 1 " du. (10) i(u)!j ¸ (j ) s!j ¸ (j ) *u SY J J J J To be sure, a desire to obtain the most rational (with a minimal number of terms) expansion in a series of exponents prevails. On the one hand, Eq. (10) implies that for this purpose ¸(j) should be among the polynomials orthogonal with a weight f (s). In this case the quantities a in Eq. (10) J become ordinates in the quadrature formula of the Guass type. On the other hand, to do so, expansion (8) must exist. However, this is the case where the description should start with the words: `If there exists such ¸(j) that 2a and the problem of satisfying these two conditions simultaneously is far from being solved automatically (see the pertinent analysis in Supplement). At the same time, the properties of s(g), such as monotony, continuity and limitedness that follow from Eq. (7) assure the e$ciency of any quadrature formula (of the type of the Gauss or Chebyshev formulas) for Eq. (2b). Another argument is also related to the theory of series of exponents. Relation (S.5) makes it possible (see Appendix A) to write a in Eq. (6a) directly in terms of i(u): J 1 du. (11) a " J *u SZ HJ\ XGSXHJ>
The use of the roots of function (9) is a powerful method at the level of `theorems of existencea. However, the actual construction of function (9) with the properties which are necessary for the existence of Eq. (8) is faced with certain (and very essential) di$culties (see Supplement). It is this problem which is eliminated due to the transition to Eq. (2b) or (6b), while retaining all the advantages of the explicit relation between the series of exponents and the absorption coe$cient. In e!ect, a comparison of Eqs. (11) and (7) indicates that s "j , J J
a " J
a "g(j )!g(j ), (12) J J> J\ and, as shown in Supplement, the in"nite (in the limit) set j "lls a limited interval everywhere J densely. Therefore Eq. (12) may be treated as an in"nitesimal increment of g, which is actually equivalent to the substitution of variable according to Eq. (4). Needless to say that, in formal terms expressions (6a) and(6b) are equivalent. Their comparison is dictated by pragmatic purposes. It should be emphasized that the quantities s , b and g are mere J J J numbers solely controlled by choosing the appropriate quadrature formula, whereas quantities (10) are undoubtedly dependent on the thermodynamic characteristics of the medium, which necessarily creates serious di$culties in the case of a nonhomogeneous medium and in the calculation of the source function. A desire to "nd parameters of series of exponents using empirical data on the transmission function seems quite logical in the context of the foregoing discussion. In principle, the solution appears to be feasible: substituting Eq. (9) in Eq. (10), coe$cients a are expressed in terms of the J
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moments of the form
*KP(x) S iK(u) du"(!1)K . (13) *xK SY V Again, the transition to Eq. (6b) appears to be useful, because we need only to have knowledge of the numbers k "s(g ). The properties of s(g) mentioned above and the procedure of constructing J J Eq. (9) described in Supplement ensure that the function K(k)" (1!k/k ) corresponds to the J J expansion of exp(!j x) according to Eq. (8). Then the function ¸ is eliminated by trick (S.9); the J resultant series is equal to the one from Eq. (6b), which gives rise to a system of equations for k : J 1 S K(i(u)) du"b K (k ) (14) J J i(u)!k *u SY J whose left-hand side is reduced to the quantities = (13). K Eqs. (13) and (14) are, in fact, the `existence theorema of a sort for the solution of the problem formulated earlier in the discussion of this subject. Such a theorem follows from the mathematical properties of the Dirichlet series and is undoubtedly of pragmatic importance. Indeed, spectroscopic peculiarities of i(u) in Eq. (1) whose description poses major problems are clearly smoothed out. Certain di$culties emerge when models of spectra for P(x) in Eqs. (3) and (4) are invoked (the way of principle importance according to Ref. [14]). Yet a workable algorithm remains to be found. To sum up the foregoing discussion focused on rigorous mathematical methods, the following statement can be formulated: the most rational way of expanding the absorption function into a series of exponents is a combination of Eqs. (7) and (6b) which appear to be closely related. 1 = " K *u
3. Nonhomogeneous medium In the case of a nonhomogeneous medium where the thermodynamic parameters and, consequently, the absorption coe$cient change from point to point, we have to replace Eq. (1) by the relation of the form
S e\OS_J du, q(u, l)" i(u; l ) dl , (15) SY J where in the general case q(u; l) involves a curvilinear integral over points l along the ray path. The argument l in Q and q (and in similar expressions below) indicates a function of the upper limit, i.e. of the "nal point of the ray path in the integral dl (2). J A representation of Q(l) by a series of exponents can be really obtained: the quantity l is assumed to be a parameter and q in Eq. (15) must be replaced by qx, where x is an arti"cial variable. Formally, the problem is reduced to Eq. (1) but with the function 1 Q(l)" *u
1 g (s, l)" *u
du (16) OS JXQ SZ SY S
used instead of g(s) (see Eq. (7)). Let us now turn to Eq. (6b), taking into account the fact that b and J g are mere numbers, i.e. they are independent of thermodynamic parameters. In addition, let us J
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assume that x"1 to get (17) Q(l)" b e\Q EJ J, J J where s (g) is the inverse function of g (s, l) (Eq. (16)). It is essential that Eqs. (16) and (17) are rigorous formulas and Eq. (16) implies the abovementioned ordering procedure for q(u, l) as a function of u for given dl(2) when s is calculated. J There are problems, of course, wherein Eq. (17) is a convenient computational tool. For example, for a ray path through the plane-parallel horizontally homogeneous atmosphere running at an angle h to the vertical from a prescribed altitude between z and z , we have q"X i(u, z) dz sec h,q (u)sec h, and Q" b exp(!s (g )sec h), where s (g ) is obtained from X J J J J Eq. (16) upon substitution of q for i. However, because of the two arguments in Eq. (16) the calculations may appear to be too cumbersome, whereas rigorous formula (17) gives a good chance for estimations and approximations. The variant \ QEJ JY JY , (18) Q" b e J J J where s(g, l) is the inverse function of Eq. (7) with i(u)Pi(u, l), is very popular. It can be veri"ed that Eq. (18) is a reliable approximation and there is no need for purely heuristic reasoning relative to the correlation of k-representation (see, for example, Refs. [11,13]). Really, the di!erence between Eqs. (15) and (18) is reduced to the following quantity (see Appendix B): 1 <" *u
S
SY
du
J
dg(X(u, l)!>(g, l)),
where
GS JY JY, QE JY JY >(g, l)"¹K e J (19) X(u, l)"¹K e J J J and ¹K is the operator of the Dyson ordering with respect to variable l. It is possible (see details in J Ref. [27]) to introduce in X the parameters averaged over l and identical for all lines, thereby returning formally to a homogeneous medium. While a search for an adequate approximation and expression for averaged thermodynamic characteristics is far from being simple, at this point `the existence theorema su$ces, speaking in the mathematical language. A similar procedure is possible for > (cf. Eqs. (15) and (17)), which is obvious having the formal equivalence of k and s in mind in the calculation of Q. The equality of k and s averaged over l is, to be sure, an assumption, but it appears quite natural. Now the integral dl (2) in Eq. (19) vanishes, and, consequently, so does the operator ¹K . Thereafter
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and these values correspond to a sequence of arguments g "0(g (g 2 (i.e. q(u )"s (g ), q(u )"s (g ), etc.). The corresponding values u (a"1, 2, 3,2) are, to be sure, ? placed in a fashion which is far from being ordered because of the `palisadea of spectral lines. The quantity s (g) appearing in Eq. (18) is equal to s(g, l ) dl "s (g) if (and only if) the ordering for u is ? the same for any value of l. Actually, it is this fact which is implied where the correlation of k-representation is concerned. Needless to say, it is not generally the case. Therefore, s is but only an approximation. The function s (g) is monotone by construction. In algorithms including an ordering procedure di!erent from Eq. (18) (see, e.g., Ref. [13]) there appear other functions s (g) which are also approximate but not monotone any longer. A natural question arises concerning the e$ciency of di!erent approximations. First it is to be noted that, for any variant related to the transposition of i values the condition m (g )"0 or m (g) dg"0 must be ful"lled for m "s !s . The value of m will be assumed T to be `su$ciently smalla. Then the estimate "Q!Q ""0(" dg m (g)exp(!s(g))") follows from Eq. (17) where Q is an approximate value of (15). If s (g) is a monotone function the well-known mean-value theorem, the ordering procedure and the property of m (g) mentioned above provide the estimate "Q!Q ""0("E m (g) dg")"0(" m (g) dg") with 0(g (l. The possibility to choose E a minimal integral of the two and the sign-changing structure of m(g) show that the monotony of s (g) contributes undoubtedly to the reliability of the approximation. As for the nonmonotone s (g), we have to restrict ourselves to the estimate
"Q!Q ""O
e\Q Em (g) dg
whose smallness is not quite evident. Another circumstance concerning the comparison of s and s is the number of abscissas in the quadrature formulas: the smoother the integrand the smaller the number of abscissas. The advantage of the monotone function is quite evident here. Both of the factors mentioned above are essential in terms of the pragmatic aspect relating to the estimate of the number of terms in series (6b), (17) and (18) and in similar series below. As an illustration of the problem discussed let us consider a typical example for the region 5000}5050 cm\ of the CO spectrum examined in Ref. [10] and for the strati"cation of the atmosphere adopted there. The value of (15) obtained by the numerical integration with i precalculated using a line-by-line scheme is denoted as Q ; Q and Q are estimated by Eqs. (17) and (18). It emerges that Q !Q "!0.00718, Q
Q !Q "!0.0035 Q
for "ve terms only(!) and for b and g given by quadrature Gauss formula (Q "0.6557, which J J implies the di!erence in the third signi"cant digit). In the calculation [13] by an algorithm with a nonmonotone s we have to write a series with more than 100 terms.
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2
3
4
5
0.2393
0.2844
0.2392
0.1185
a J 0.1185 Layer number
a exp[!( s(g , l )*l )m ] J H J H H P
1 2 3 4 5 6 7 8
0.1185 0.11845 0.1178 0.1157 0.1083 0.09843 0.0822 0.06071
0.2393 0.2389 0.2340 0.2193 0.1737 0.1262 0.07029 0.02642
P(l Pl ) H 0.2844 0.2834 0.2696 0.2311 0.1332 0.0633 0.0165 0.0017
0.2393 0.2346 0.1780 0.0780 0.001 0.00001 0 0
0.1185 0.09453 0.0050 0.00001 0.00001 0 0 0
1.0 0.970 0.804 0.415 0.288 0.169 0.078 0.034
The simplicity of handling the Dirichlet series with a small number of terms is illustrated by Table 1 pertaining to this example. The atmosphere is divided into seven homogeneous layers (it is clear that Eq. (18) takes the form b exp[! s(g , l )*l ] where l is the altitude and *l is the J J H J H H H H thickness of the layer). Table 1 includes all the necessary information for calculating radiation #uxes.
4. Overlapping spectra When spectra of di!erent gas species overlap i(u)" i (u), where i is the absorption K K K coe$cient of the species in question and the corresponding concentration is included in i . K A trivial generalization of Eq. (7) is as follows: 1 g(s)" *u
du
(20)
K K
G SXQ SZ SY S
with subsequent application of Eq. (6b) (x has the meaning of the ray path length). The transition to Eq. (17) or (18) is performed readily. It should be noted again that (20) together with (2b) and (6b) are rigorous formulas and the ordering procedure described is extended to include the case of overlapping spectra. It is natural to use this exact mathematical basis to "nd approximate procedures.
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Eq. (20) presents a possibility of discussing an interesting suggestion [10] that g(s) for (4) and(2a) be written in the form
Q
f (s )g (s!s ) ds "
1 g(s)" (*u)
Q
A> dx P (x)P (x)eQV, (21) x A\ where indices 1 and 2 refer, as previously, to components of the mixture (the third equality in Eq. (21) follows from mathematical problems of permutation of integration over u and s discussed in Section 2). However, Eq. (21) results in the relation of the form g(s)"
1 f (s)g (s!s ) ds " 2pi
du du . (22) G S>G SXQ S S Z SY S
The di!erence between Eqs. (20) and (22) is illustrated in Fig. 2. Eq. (22) implies integration over a `trianglea whereas in Eq. (20) the integration is performed over a `diagonala. To reduce the problem to Eq. (21), i.e. to the characteristics of the components of the mixture, some additional assumptions or approximations are called for. Eq. (21) makes use of the belief that f (s) can be treated as `the probability that the absorption coe$cient will be equal to sa (the relation f (s)'0 results from Eq. (4) and from Eq. (7) for g(s)). Further statement [10] that the spectra are noncorrelated is clear. Indeed, in view of the fact that the concentration of absorbing gases in atmospheric conditions is generally low the contribution of species `1a to the spectral line width of species `2a may well be neglected and vice versa. Then the rules of the probability theory are used to write the distribution function for the sum of the distribution functions for individual species [10]. Thus, the problem reduces to the question whether f (s) can be interpreted as this kind of probability. A negative answer follows from reductio ad adsurdum. Really, if the order of integrals can be changed upon substitution of Eq. (1) into Eq. (3), then f (s)"(1/*u)S du d(s!i(u)), and this expression, by de"nitions of the probability theory, is the SY distribution function with all the ensuing formal consequences. In this case, however, it is impossible to perform the permutation of integrations (see Section 2). It can be made possible using Eq. (4), which leads to Eq. (20).
Fig. 2. On construction of function g(s) for overlapping spectra.
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Another possibility is to consider the following function instead of Eq. (1): 1 P(x , x )" *u
S e\V G S\V G S du, SY
(23)
where gas concentrations are included in x and x . Applying the two-dimensional Laplace transform to Eq. (23) yields
P(x , x )"
ds ds f (s , s )e\Q V \Q V ,
*g(s , s ) , f (s , s )" *s *s
(24b)
g(s , s )" 1 *u
(24a)
du,
(24c)
G SXQ G SXQ SZ SY S
S $(i (u),i (u)) du" ds ds f (s , s )$(s , s ) SY
(24d)
instead of Eqs. (2), (4), (7) and (5). Unfortunately, it turns out that Eqs. (24a)}(24d) are irreducible to the expression of the type given by Eq. (2b) or by its consequence Eq. (6b) with all their practical advantages. And again attention should be paid to the di!erence between g(s) from Eq. (20) and g(s , s ) from Eq. (22). A pragmatic desire to write Eq. (23) in terms of s (g) and s (g), calculated for each gas separately is quite understandable. To this end, an approximation is often used (see, e.g., Refs. [7,20]) based on an analogy between (23) and (15): P(x , x )" b e\V Q EJ \V Q EJ J J
(25)
with an obvious generalization of Eq. (25) to a multicomponent mixture. However, the validity of approximation (25) cannot be substantiated in the way it was done for Eq. (18) for the reasons given in Appendix B. Incidentally, there exist some su$cient conditions where Eq. (25) can be met exactly but these are most likely to be highly exotic as compared to the actual properties of P(x , x ) from Eq. (23).
5. Source function First let us consider the following relation: 1 H(x)" *u
S u(u)e\VGS du, SY
(26)
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255
where the function u(u) is normalized by the condition (*u)\S u(u) du"1. Performing operaSY tions on Eq. (26) used for Eq. (1) to derive Eq. (6), we obtain
(27) dg e\QS EV" b e\QS EJ V, J J where s is the inverse function of g (s) with S S 1 g (s)" u(u) du. (28) S *u GSXQ SZ S SY
It is apparent that Eqs. (27) and (28) represent a mere extension of Eq. (7) to the case of Eq. (26) with all the foregoing comments as to a rigorous solution of the problem. The need for going to the dimensionless normalized function u is quite evident. It is dictated by a desire that the quantity g be kept dimensionless. Therefore, if an arbitrary function ; appears in S H, we can return to Eq. (26) introducing u";/((*u)\S ;(u) du). The Planck function, radiation SY from some source, arti"cial light spectrum and instrument function or a combination of these can be chosen as the function ;. It should be emphasized that, as in the previous case, rigorous formulas (27) and (28) extend the ordering operation to (26). In order to understand this it will su$ce to introduce a new variable p(u)"u #S u(u) du with the inverse function u"u(p); i(u)"i(u(p)),i(p), and, by de"niSY dz which is equivalent to Eq. (27). tion of u, p3[0,*u]. Eq. (28) takes the form (1/*u) GNXQ NZ S
Indeed, we only have to go from u to p scale. Once again, this procedure allows for the use of a series of exponents with a small number of terms. The example given in Table 2 refers to the 1380}1900 cm\ range of the water vapor spectrum and u in Eq. (26) is related to the Planck function (it is assumed that the temperature is 250 K, the total pressure is 1 atm, and the water vapor amount has the dimensionality g cm\). The ratios between quantity (27) using the ordering procedure just described and quantity (26) calculated according to the line-by-line scheme are given in Table 2. In the "rst column the results obtained by the 5-point Gauss formula are listed, the next three columns tabulate the values for the 6-, 7-, and 9-point Chebyshev formulas, and the last column gives line-by-line result (26) calculated with the Lorentzian line shape cut at 10 cm\ from the line center. As an example of the approximation based on Eq. (28) let us consider separation of variables which, in fact, reduces Eq. (28) to Eq. (7). Pragmatic advantages of this operation are evident. H(x)"
Table 2 Ratios of Planck-weighted transmission function with function calculated from series of exponents and those calculated according to the line-by-line scheme n w, g cm\
5
6
7
9
F
0.0001 0.001 0.01 0.1
1.003 0.9995 0.9982 1.0040
1.007 0.9901 0.9944 0.9697
1.005 0.9934 0.9952 0.9607
1.003 1.0000 0.9985 1.005
0.9488 0.7854 0.4558 0.1189
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Simple arguments taking into account Fig. 1 and the mean-value theorem lend support to the following approximation: g (s) y(s)g(s), S
S Q u(u) du, S Q
1 y(s)" s*u
u (s)"u #(1!s)*u,
u (s)"u !(1!s)*u.
(29)
Actually, y(1)"g(1)"1, which provides the necessary relation g (1)"1; if u(u)"const"1 then S y(s)"1, and we return to Eq. (7). Further, y(0)"u((u #u )/2), i.e. according to the scheme of Fig. 1 the `motiona in integral (29) starts from the center of the interval *u. In e!ect, we merely have to do with the choice of the characteristic point in the mean-value theorem for the integral in Eq. (28).
6. Summary The objective of the paper formulated above is to combine speci"c problems of atmospheric optics and the mathematical theory of the Dirichlet series. The integration of this kind o!ers a possibility to translate methods based on intuitive or heuristic considerations (the `physicala level of rigor) into the exact mathematical language. The formalization of the ordering procedure for the absorption coe$cients by means of relation (7) in the case of a homogeneous medium, its extension to more complicated problems of atmospheric optics (see the discussion of relations (17), (20),(27)), arguments in favor of practical advantages of relation (6b), the interpretation of the correlation concept of k-representations as a purely mathematical approximation and the insu$ciency of reading Eq. (3) as the distribution function density of the absorption coe$cients (together with its consequences for overlapping spectra) can be regarded as obvious. Importantly, the foregoing issues are not merely methodical in any way for they enable one to decrease drastically the number of terms in the series of exponents. In other words, they serve to reduce radiation codes used in climate models and in geophysical applications of atmospheric optics, which is the main purpose of the k-representation method. We believe that the potentialities of the mathematical program outlined are far from being exhausted. For example, an idea was mentioned that the series of exponents may be constructed from experimental data on the transmission function. Approximate formulas for the source function and overlapping spectra remain to be derived. Exact mathematical relations pertaining to the properties of the series of exponents can play a part in the construction of and search for numerical algorithms.
Supplement: On the Dirichlet series We present here some relations of a repeatedly cited theory of the Dirichlet series [24}26] and some discussion concerning functions (9) and their connection to orthonormal polynomials for the quadrature formulas.
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By de"nition, the Dirichlet series is a function F(z)" h e\HJ X J J of the complex variable z, and j are zeros of integral function (9). The coe$cients J 1
(z)F(z) dz h " J J 2pi ! and the closed contour C encircles all singular points of the functions
1 L\ X ¸(j)
(z)" eHX dj. J ¸ (j ) j!j J If F(z) is an integral function Eqs. (31), (32) and (9) lead to the relation of the form
(30)
(31)
(32)
1 @\ P j@\A\FA(0)(!1)A, h " (33) @ J ¸ (j ) J @ A where P are coe$cients of j@ in Eq. (9). The expansion of exponent (8) follows from Eqs. (33) and @ (30) if F(z)"e\HX. Let us further assume that, as in the case of the transmission function, we need only real z'0 in Eq. (30), and F(R)"0. Now it is worthwhile to construct ¸(x) (Eq. (9)) using j '0. The following L relation is a consequence of these assumptions:
J dz m! A> " (m!j )Kh , (34) eKXF(z) @ @ zK> 2pi A\ @ where m"0, 1, 2,2; j (m(j . J J> Eq. (34) emphasizes the intrinsic relationship between Eq. (30) and the Laplace transform (cf. with Eqs. (1)}(4), (6)). Let us write
A> F(z)eQX dz. A\ It actually follows from Eqs. (34) and (35) that F(z)"
e\XQN(s) ds,
N(s)"
(35)
1 *K> J (m!j )Kh "N(m) (36) @ J m! *mK> @ and Eq. (36) states that N from Eq. (6) solves in principle the problem of "niding h and g . Note J J that the derivation of Eqs. (11) and (12) is merely a good case in point. Expression (10) in the present notation has the form
N(s)¸(s) ds . (37) (s!j )¸ (j ) J J The relationships between Eqs. (9) and h (Eq. (37)) as the ordinates of the quadrature formulas of J the Gauss type have already been discussed in Section 2. Recall that this plays a part in the mathematical interpretation of Eq. (6). The question that now arises can be formulated in the h " J
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following way. Let ¸ be a polynomial of degree j orthogonal with the weight N(s) from Eq. (36). H The function M(k)"lim ¸ (k) is, of course, integral, and k are its roots. It is neccessary to "nd H H J conditions wherein M(k) corresponds to the expansion of exponent (8). Treating F(z) in Eq. (30) as an integral function (see Eq. (1)) for which the quadrature Gauss formula is written, we change j for k and h for Eq. (37) with the substitution of M for ¸ implicit J J J from the context. Furthermore, let us apply expansion (8) to exp(!k z) with ¸ properly chosen J and require that the Dirichlet series appear but with M in Eq. (9). The resulting condition is of the form ¸(s) ¸(k )M(s) 1 @ " . M (k ) (k !j )(s!k ) s!j J @ J @ @ @ Equality (38) in equivalent to the integral
(38)
¸(z) dz "0 M(z)(z!j )(s!z) ! J over a closed contour C with all zeros of the functions ¸ and M localized within it. It becomes clear that the functions ¸ and M have condensation points of zeros. The C can readily be moved to in"nity. Furthermore, the inequality m 5m must be ful"lled, where m 2 is an indicator of an + * increase in the corresponding function. To continue the discussion of the problem of choosing ¸(j) (Eq. (9)) let us restrict ourselves to the version z'0 and 0(j(1 (it will su$ce to make an identical replacement in Eq. (1): xPx/i
and iPi , where i is the maximum absorption coe$cient in the interval [u ,u ]).
Let ¸H(j) be a set of polynomials of degree j, whose roots jH "ll the interval [0,1] irrespective of j with the value of j falling within this interval (it is su$cient that the interval be limited, then it can be converted to the required interval with the help of a simple substitution variable). Let us consider the following integral: 1 %" 2pi
e\XQ ds ¸H(s)(j!s)
(39)
! over a closed contour surrounding the interval [0,1]. Obviously, J ¸ (j) , e\XH, , !e\XH ¸H(j)%" ¸HY(jH)(j!jH) J J J and expansion (8) will appear (with j "lim jH) if J H J
lim ¸H(j)%"0. (40) H Of course, integral (39) must be estimated asymptotically to be used in Eq. (40), taking into account the fact that for j chosen the quantity "¸H(j)" is limited even with jPR (for example, in the case of the Chebyshev polynomials ¸H(j)"cos( j arc cos j)). The asymptotic behavior of quantity (39) is de"ned by the relation exp[!(zs#ln ¸H(s))] exp[!(zs#j ln s)]
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259
Fig. 3. Integration contour in calculation of integral (S.10).
and subsequent standard actions [28] are illustrated in Fig. 3. In this way we obtain the estimate "%""0(exp(!j ln j)) which insures the ful"llment of Eq. (40). Further, Eq. (36) can be used by treating them as a set of equations in j and h and applying, J J say, the least-squares technique. A similar idea [29] consists in using the exponential orthogonal functions D (z)" K d exp(!j z), where d "d (2j 2) are chosen in such a way that K J J J J J J D (z)D (z) dz"d . A subsequent conventional expansion of F(z) in D (z) makes us return K KY KKY K formally to Eq. (30). Once again, it is recommended that the numbers j be found by means of the L least-squares technique, limiting the search to "nite m. It should be added that the way based on D is simply a variation on the theme `biorthogonal setsa since functions (32) are of the form which K yields (2pi)\ dz exp(!j z)u (z)"d . ! J J JJY Acknowledgements This work was supported by the Russian Foundation for Basic Research (grant No. 97-0565985).
Appendix A Changing the order of integrations dx and du leads to the emergence of the following integral [30] from Eqs. (1) and (4): 1 2pi
1, A> dx eVQ\GS" , x A\ 0,
s'i(u), s"i(u),
(A.1)
s(i(u).
We must calculate now an integral of the form
I"
S
S
'(u) du,
(A.2)
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where '(u) is a discontinous function:
1, u (u(u , , u"u (or u ). Let us rewrite Eq. (A.2) as the Stieltjes integral of the form '(u)"
S
S
I"(S)
'(u) dh(u)
(A.3)
(A.4)
S with h(u)"u. The integration of Eq. (A.4) by parts results in
S S '(u) dh(u)"'(u)h(u) " !(S) h(u) d'(u). (A.5) S S S The "rst term in Eq. (A.5) equals (u !u ). The function '(u) is equal to , as it approaches a discontinuous jump from the left, and it will be ! when approaching the discontinuity from the right. Therefore, the second term in Eq. (A.5) is (u !u ), see Ref. [31]. Thus, I"u !u , which leads to Eq. (7). The derivation of Eq. (11) is immediately related to Eq. (34). Writing (34) for l and l#1 (with m"0) and subtracting the expression obtained for l from that derived for l#1 we get I"(S)
A> dz (A.6) F(z)(eK X!eKYX) . z A\ The arrangement of points m and m is shown in Fig. 4. Now we replace F in Eq. (A.6) by P(x) (Eq. (1)) to take into account the feasibility of interchaning the integrations and to return to Eq. (A.1). The arbitrariness inherent to the de"nition of m in Eq. (34) and Fig. 4 allow us to choose m Pj and m Pj . A comparison of Eq. (30) and Eqs. (6) and (10) implies that we are J\ J> dealing with Eq. (6a). The "nal result appears to be Eq. (11), and upon the transformations described the orgin of Eq. (12) becomes quite clear. 1 h " J 2pi
Appendix B To estimate Z, i.e. the di!erence between Eqs. (15) and(18), let us represent the integral dl as J an integral sum (with an element of length *l) and expand all the exponents exp(2) into a series (a is the index of summation). The term without l in Z vanishes because l "1 for any quadrature JJ formula. The terms including i and s with the same argument l will also vanish if we make use of their independence from b and g , Eq. (15) and the fact that and dg are equivalent. Finally, the J J J
Fig. 4. Arrangement of points m and m .
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261
expressions where the degree of *l does not exceed the number of summations over a are eliminated, as is often the case in the theory of multiplicative integrals [32]. Following these simpli"cations Z becomes
Z"
(*l) S i(u; la )i(u; la ) du! b s(g ; la )s(g ; la )(*l) J J J *u SY ? ? J ? ?
#
(*l) S i(u; la )i(u; la )i(u; la ) du *u SY ? ? ?
! b s(g ; la )s(g ; la )s(g ; la )(*l) #2. J J J J J ? ? ?
(B.1)
It is impossible to use Eq. (5) once more because Eq. (5) is not ful"lled in the case of the sum of i(u). Let us denote the sums of the terms in Eq. (B.1) appearing after (*u)\S and b as XI (u; l) SY J J and >I (u; l), respectively, and introduce the quantities X and > from Eq. (19) which are the solutions of the following equations:
X(u; l)"1#
J
>(u; l)"1#
J
dl i(u; l )X(u; l), dl i(u; l )>(u; l).
(B.2)
The representation of X and > in (B.2) as the Neumann series makes us return to the integrals over l in XI and >I . The replacement b Pdg shows that Z coincides with < given just before Eq. (19) J J (the `excessivea terms have no e!ect on the estimate, as it was emphasized in writing Eq. (B.1). Of course, the foregoing consideration is unsuitable for the di!erence between Eqs. (25) and (23) because the expansion in terms of *l cannot be performed. Formula (5) is now replaced by Eq. (24d). However, if the previous expansion of exp(2) is applied and Eq. (24d) used, a su$cient condition for exact coincidence of Eqs. (25) and (27) can readily be derived: 1 *u
S
SY
du iK (u)iK (u), "
ds ds f (s , s )sK sK
dg(g\(g)K g\(g)K ).
(B.3)
The meaning of the subscripts is the same as in Eq. (26). To avoid confusion between the notation of arguments and functions s(g) is replaced by g\(g). Substituting the integration variables s for g (according to Eq. (4) with the subscript `1a) we can see that equality (B.3) is satis"ed if f (s , s )"f (s )d(s !g\(g (s ))).
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This relation and Eq. (24b) imply that g(s , s ) calculated according to Eq. (24c) must coincide with
s 'g\(g (s )), g(s , s )" , s "g\(g (s )) f (s ), (B.4) f (s ), s (g\(g (s )). It is obvious that the condition obtained from (B.4) by interchanging the indices `1a and `2a is also valid. 0,
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