Dispersion by Atmospheric Water Vapor at Frequencies Less Than 1 THz

The refractivity of atmospheric water vapor at frequencies less than 1 THz is expressed as the sum of (1) the empirical radio refractivity, (2) a simple approximation to the line-by-line summation of dispersion contributions by resonances above 1 THz, and (3) a line-by-line summation over resonances below 1 THz. This simple approximation, which is valid in the temperature range -40 degrees C to 40 degrees C, reduces labor and computer time required to calculate dispersion. A similar approximation for absorption predicted by three different line shapes is also given. Measured and calculated radio refractivity data are compared, revealing discrepancies in temperature dependence. The three different dispersion line shapes are then compared and shown to be in good agreement with both. >


I. INTRODUCTION HE REFRACTIVE index of water vapor is of interest in
T determining the phase delay between electromagnetic waves of differing frequency propagated through the atmosphere [l], [2].The dry-air constituents give an additive refractivity, and the microwave spectrum of oxygen provides some microwave dispersion [ 11, but these contributions are not discussed here.Because of foreign-gas pressure broadening, the dry-air constituents affect the water-vapor refractivity at frequencies within a few line widths of the center frequency of water-vapor resonances.The refractive index of pure water vapor has been measured in the frequency range 10-50 cm ~ ' (300-1500 GHz) and compared with calculations based on the Gross line shape by Kemp et al. [3].The refractive index of water vapor for millimeter-wave frequencies has also been calculated by Birnbaum [4], Zhevalun and Naumov [ 5 ] , and others [6], [ 11.The refractivity N(v) at electromagnetic frequency v is defined in terms of the refractive index n(v) by N ( v ) = [ n ( v ) -I ] x 106. (1) In principle, the refractivity of water vapor can be obtained by adding the radio-frequency extrapolation of the watervapor optical-frequency refractivity to a line-by-line summation over all infrared water-vapor resonances [6], although this calculation produces some discrepancies [7].Recently, measurements of millimeter-wave dispersion show good agreement with such line-by-line summations [2].For frequencies less than about 1 THz it is both easier and more accurate to Manuscript received December 17, 1986;revised February 27, 1987.express the water-vapor refractivity as the sum of three terms; these terms are 1) the empirical radio refractivity, 2) a simple approximation to the dispersive refractivity due to all resonances above 1 THz, and 3) the line-by-line sum of dispersive refractivity from resonances below 1 THz.By dispersive refractivity we mean the refractivity minus its value at zero frequency.These three terms are the subject of the next three sections.
The choice of 1 THz as the boundary between contributions 2) and 3) above is rather arbitrary; this choice is based on the fact that Liebe's millimeter-wave propagation model [ 11 extends to 1 THz.The motivation for obtaining a simple approximation to the dispersive refractivity from all resonances above 1 THz is to reduce the labor and computer time required to calculate dispersion.Indeed, the model described here has already been incorporated, in part, into Liebe's millimeter-wave propagation model [ 11.
It is well-known that line-by-line summations do not give the empirical window absorption [l].However, it is sometimes useful to produce such summations to demonstrate their inadequacy against measurements, as was recently done by Manabe et al. [2].To reduce the labor and computer time involved in such absorption calculations, we present in the Appendix a simple approximation to the absorption from lineby-line summation over resonances above 1 THz.This result in the Appendix applies to the Van Vleck-Weisskopf, full Lorentz, and Gross line shapes.

THE RADIO REFRACTIVITY
At frequencies below a few gigahertz the refractivity of water vapor is nondispersive for practical purposes.In this paper, the refractivity at such low frequencies is denoted by its value in the zero-frequency limit v --+ 0, namely, N(O), and is referred to as the radio refractivity.The notation N(0) is a convenience in the following sections of this paper.The radio refractivity of water vapor is given by where Tis absolute temperature, e is partial pressure of water vapor, and K2 and K3 are experimentally determined coefficients.In terms of the humidity Q in g.m -3 , the water-vapor partial pressure in millibars is given by e=[4.615144x10-3mbK-1 (g m-3)-llQT. (3) In the temperature range from -20 to 20°C the radio where Si is the temperature-dependent line-strength parameter, and Q is the absolute humidity.The product S j Q is the line-integrated absorption.The strength parameter S; obtains its dependence on absolute temperature T from the difference of two Boltzmann factors divided by the partition function (cf., [7]).Birnbaum [4] shows that (4) is obtained from the Kramers-Kronig relation for any sufficiently sharp absorption resonance, independent of details of the line shape.In particular, (4) is obtained from the Van Vleck-Weisskopf, full Lorentz, or Gross line shapes.At frequencies below 1 THz, the contribution to dispersive refractivity by lines above 1 THz is given by Using (4) we obtain where the summation extends only for lines i such that Y, > 1 THz.In (5a) we have subtracted the line-by-line summation for radio refractivity such that the more reliable empirical expression (2) can be added to D(v) to obtain N(v).We expand (5b) in Taylor series in powers of (v/v,)~.The dispersive refractivity is approximated by truncating the Taylor series; it is given by Using only terms up t o j = 4 produces an error, compared with the line-by-line summation over all lines above 33 cm-' (989.3GHz) using (5), of ten percent at 900 GHz, decreasing rapidly with decreasing frequency, passing below one percent at 690 GHz, 0.1 percent at 530 GHz, and 0.01 percent at 400 GHz .
The C,(T) are calculated from line-by-line summation over all lines above 1 THz in the 1982 AEGL line-parameters compilation [ll].More precisely, lines above 33 cm -I (989.3GHz) were included in the line-by-line summation.This calculation of the C,(T) is performed over the temperature range from -40°C to 40°C in steps of 5°C.The temperature dependences of the C,(T) were then fitted to formulas of the form where the coefficients ai, bj, and cj are found by the leastsquares method.The form of ( 8) was chosen because it conveniently and accurately fits the temperature dependence.
The worst percentage error of this fit for any of the Cj(T) at any of the calculated temperatures was only 0.034 percent.
The final expression for the dispersive refractivity contributed by resonances above 1 THz is conveniently given by ( 9) where Q is absolute humidity in g.m-3, T is absolute temperature, and the coefficients A,, a,, and B, are given in Table 11.The frequency in GHz is 29.97925 times its value in cm-I.Thus (v/33 cm-I) in ( 9) can be replaced by (d989.3GHz).The values of the coefficients A, in Table I1 show that the terms in (9) having higher powers of (d33 cm-I) are progressively smaller for v < 33 cm -I .
The considerable difference between the A,, a,, and B, f o r j = 1 as opposed t o j = 2, 3, 4 is due to the peculiarities of computer-generated best fits and the freedom to trade the temperature dependence between the coefficients; there is no computational problem in either obtaining or using these figure, using just j = 0 in ( 6) and ( 7) and summing over lines vi > 33 crngives slightly less temperature variation than shown by the short-dashed curve.Equation ( 9) includes a prediction for the temperature dependence of the dispersion in the temperature range -40°C to 40°C.Since the calculated temperature dependence of the radio refractivity is not in agreement with the empirical dependence, it may be that the temperature dependence in ( 9) is inaccurate.Therefore, precise measurements of water-vapor dispersion over a wide range of temperatures would be of interest.

IV. DISPERSIVE REFRACTIVITY FROM RESONANCES BELOW 1 THz
Having already discussed the contributions of the radio refractivity and of the dispersive refractivity from resonances beyond 1 THz, it remains to discuss the dispersive refractivity from resonances below 1 THz.This latter contribution is to be included by line-by-line summation.The three candidate line shapes most often used for absorption and dispersion are the Van Vleck-Weisskopf, full Lorentz, and Gross line shapes, abreviated by VVW, FL, and G, respectively.The refractivity contributed by a line i of strength SI, center frequency vi, and half-width y I is given by -.

SiQ
The significance of the parameters SI, vi, and yi in terms of the three absorption line shapes is given in [ 12, table 11.Note that (10)-( 12) all reduce to (4) if Ivv,I y,.
coefficients.The full six decimal places provided by the possible without affecting accuracy.
well as that of the radio refractivity in (2), are shown in Fig. 1 The value N,(o) is the contribution of the line i to the radio in its entirety, Nl(0) must be subtracted from N,(v) to computer are given in 1'9 Some is refractivity.Since the empirical radio refractivity is included The temperature dependences of the four terms in (9), as determine the dispersive refractivity given by for fixed absolute humidity (fixing water-vapor partial pres- sure would change the displayed temperature dependence by a factor of 273/T).The temperature dependences Shown are For the three line shapes the dispersive refractivity is given by normalized by their values at 0°C; as such, these dependences shows that the variation with temperature increases for D,(v)vvw =- are independent of frequency and absolute humidity.Fig. 1 increasing j, but the temperature dependence of two terms having consecutive values of j (i.e., j and j + 1) become more similar for larger j .In this regard, note that the radio refractivity predicted using line-by-line summation is a j = 0 term.That is, using justj = 0 in ( 6) and ( 7) and summing over all lines gives the calculated radio refractivity given by the short-dashed curve in Fig. 1.Although not shown in this (134 The VVW and F1 dispersive-refractivity shapes are given with the common denominator in (13b) and (14b) to emphasize the similarity between the three line shapes in (13)-( 15).Indeed, for v + 0 and Y 4 03, as well as for the frequency of the zero crossing near v = v,, the VVW, FL, and G shapes (13)-( 15) give the same results to within terms of order ( y , / ~, ) ~ relative to unity.They differ most near U = U, f y, where the difference is in terms of order y I / v , relative to unity.The same statements hold true for the N , ( v ) in (lo)-( 12).A similar study of differences in the VVW, FL, and G absorption line shapes is given in [ 121.
The three line shapes for absorption as well as for both refractivity and dispersive refractivity differ most for greater y,/u,.Hence the lowest frequency water-vapor line at 22 GHz is the natural choice for testing line shapes.For air at 1 atm pressure the ratio y,/u, is roughly 0.1 for the 22-GHz line.The distinction between the line shapes is much less important for the significant water-vapor lines at frequencies 183 GHz and above.It is shown in [12, table 31 that only the VVW absorption line shape fits data for the near-line-center absorption of the 22-GHz line; the FL and G shapes do not.These results presented in [12, table 31 were calculated without including the water-vapor-pressure-linear shift of the linecenter frequency of the 22-GHz line; however, the conclusions remain unchanged if this shift is included.Consequently, the VVW dispersive-refractivity shape is recommended over the FL and G shapes, although the distinction is considerable only within several line widths of 22 GHz.

V. COMPARISON WITH OTHER DISPERSION MODELS AND EXPERIMENT
Birnbaum [4] constructed a model for refractivity that is similar to that presented here.Birnbaum's model extends to 19 cm (569.6 GHz).Birnbaum adds the radio refractivity, the line-by-line sum of resonances below 19 cm-I (569.6 GHz), and an approximation to the dispersion from all resonances above 19 cm I .Since his model extends only to 19 cm-I, he requires only one term proportional to Y * , for his approximation to dispersive-refractivity by resonances above 19 cm -I ; this term IS analogous to t h e j = 1 term in (9).He assumes that this term has the same temperature dependence as the radio refractivity, whereas t h e j = 1 term in (9) does not, and the corresponding term from summing all resonances above 19 cm -I probably does not.

Chamberlain et al. [13] express the refractivity of water
vapor as the sum of the empirical radio refractivity and a lineby-line summation over the rotational transitions using the dispersive refractivity formula (5).Similar to Birnbaum [4], for v < 10 cm-' (300 GHz) they express the refractivity as the empirical radio refractivity plus a term proportional to v 2 with no further line-by-line summation.Unlike Birnbaum, they assume that the u z term is independent of temperature, which is very doubtful since the j = 1 term in (9) has temperature dependence.
Liebe [I] includes dispersion by water vapor as one of the aspects of his millimeter-wave propagation model for frequencies below 1 THz.Liebe's model likewise adds radio refractivity, line-by-line summation over resonances below 1 THz, and an approximation [ l , eq. (14b)l to dispersive refractivity by resonances above 1 THz.His equation [ 1 , eq. (14b)l is adapted from (9) but represents only the j = 1 term in (9).

Zrazhevskiy et al.
[I41 calculate the contribution to the dielectric constant of water vapor by the rotational transitions.This dielectric constant eror is related to the corresponding contribution to refractivity by N = 0.5(~,, -1) x lo6.

Zrazhevskiy and Malinkin
[ 151 obtain simplified expressions for (eror -1) at frequencies U < 30 cm (900 GHz) and temperatures between 173-333 K.They calculate a leastsquares fit to the frequency and temperature dependence of (erot -1) caused by lines having transition frequencies above 37 cm -I (1 109 GHz), and they add to this the line-by-line summation over lines below 37 cm -' using the equivalent of (12).There is some difficulty in interpreting [15]; one must assume that their quantity l ) y = l was intended everywhere that (erorol ) v = l appears and where (erot -1)"= appears in [15, eq. ( 7)]; otherwise the numerical value they give for (erot0 -1)"= I is low by roughly ten percent, and their model in [15] would predict radio refractivity that is roughly ten percent greater than calculated in [14].Also, their value of temperature for "normal conditions" is not given in [15] and is assumed to be the same as in [14].
In the model by Zrazhevskiy and Malinkin [15] the radio refractivity is obtained in the limit v -+ 0 as calculated using line-by-line summation, whereas in Section I1 we argue that the empirical radio refractivity formula should be used.We can alter the model in [15] to accommodate adding the empirical radio refractivity, a line-by-line summation over lines below either 33 or 37 cm-I using the dispersiverefractivity formulas in (13)-( 15), plus a simple formula for the dispersive refractivity by resonances above either 33 or 37 cm-'.This simple formula is obtained by subtracting from [15, eq.(7)] its value at U = 0 and adding the dispersive refractivity by the strong line at 36.6 cm -I (1097 GHz), which was the only line included in [15] between 33 and 37 cm-'.This formula for dispersive refractivity contributed by resonances above 33 cm-I, which should in principal yield the The first term in ( 16) appears peculiar; it arises from the method used in [ 151 for fitting the temperature dependence; it makes only a small contribution for typical atmospheric conditions.The second term in ( 16) dominates the fit in [15] to dispersion by resonances above 37 cm -I .The last term in ( 16) is the dispersive refractivity for the line at 36.6 cm-' according to the spectroscopic parameters in [15].This last term is included to make (16) the analog of ( 9), such that lineby-line summation over lines below 33 cm-' using the dispersive-refractivity shapes in (1 3)-( 15) must be added to either ( 16) or (9).However, the last term in ( 16) could be deleted if this line-by-line summation is extended to all lines below 37 cm -I.
Figs. 2-4 illustrate the different dispersive-refractivity models; the radio refractivity is not added.Fig. 2 shows the full behavior of dispersive refractivity for frequencies 0 to 1 THz; Fig. 3 emphasizes the region below 400 GHz, and Fig. 4 emphasizes the central portion of Fig. 2 by chopping off the peaks of the very strong dispersion resonances.The mediumlength dashed curve is (9), and the long-dashed curve is Liebe's equation [l, (14b)l which is approximately the j = 1 term in (9).Clearly, the terms in (9) having j > 1 are required for frequencies above 600 GHz.The solid curve is (9) plus a line-by-line summation of all water-vapor resonances below 1 THz (more precisely 33 cm-' = 989.3GHz) using the 1982 AFGL line-parameters compilation.The short-dashed curve is Birnbaum's model with the line-by-line summation likewise obtained from all water-vapor resonances below 19 cm-' (569.6 GHz) in the 1982 AFGL compilation.Birnbaum's model gives slightly less dispersive refractivity than our model; the agreement improves with increasing temperature but worsens with decreasing temperature.The crosses in Figs. 3 and 4 are from ( 16).The calculations for these figures used a temperature of O'C, total atmospheric pressure of 0.9 atm, and 90-percent relative humidity so e = 5.497 mb and Q = 4.361 gem -3 .The atmospheric pressure enters only in the line width of the resonances.The oxygen microwave resonances also produce dispersive refractivity below 1 THz, particularly near 60 GHz; this is, of course, not shown on the figures.
As seen in Figs. 3 and 4, ( 16) gives smaller values than does (9) for frequencies below about 300 GHz and larger values at frequencies above about 300 GHz for the temperature range - 40°C to 40°C.At frequencies above 650 GHz ( 16) gives values which are eight to 14 percent greater than values from (9).We noted in Section I11 that the error in (9) increases rapidly with increasing frequency relative to our line-by-line summation over lines above 33 cm-' (989.3GHz) in the AFGL compilation, such that (9) gives ten-percent underestimate at 900 GHz.Since (16) gives greater values than (9) at the higher frequencies, it follows that for frequencies above about 825 GHz (16) is closer to our line-by-line summation than is (9).Of course, retaining more than four terms in the Taylor series in (9) would improve its accuracy at the highest frequencies.The differences between (16) and (9) are partly because of the different methods used to fit to the frequency and temperature dependencies and partly because of the differing line-parameters compilations.I11 shows good agreement between measured and calculated refractivity.The calculated values were obtained by adding the empirical radio refractivity, a line-by-line summation over all lines below 33 cm-I in the AFGL compilation using the dispersive-refractivity formula (15), and either (9) or (16) to account for dispersive refractivity by lines above 33 cm-'.

VI. SUMMARY
The refractivity of water vapor at frequencies less than 1 THz is given by adding (2) and (9) to a line-by-line summation over resonances below 1 THz.The latter summation could use any of the dispersive-refractivity formulas (13)-( 1 3 , but the VVW shape is preferred [12].The model applies to the temperature range -40°C to 40°C.The model is in good agreement with other such models as well as experiment.Because of the deviations of the temperature dependence between measured and calculated radio refractivity, it would be worthwhile to compare measured and predicted temperature dependences of dispersive refractivity.The figures show that the dispersive refractivity above 500 GHz is strongly influenced by three strong resonances as well as by resonances above 1 THz.The dispersive refractivity contributed by resonances above 1 THz departs significantly from a v 2 behavior at frequencies above 600 GHz.although in the following the readers may insert their own choice for g(Pd, e, T).
The absorption coefficient a(v) is then given by where the summation over lines i is for vi > 1 THz.The parameter r is -1 /2 for the VVW line shape but is zero for the  FL and G shapes; thus (19) applies for all three line shapes.Also, (19) is identical for the FL and G shapes (this similarity is emphasized in [ 121).Approaching the limit Y 4 0, the VVW shape predicts half as much absorption as the FL and G shapes.
Retaining terms up to j = 5 in (19) produces an error relative to line-by-line summation using complete line-shape formulas of ten percent at 824 GHz, decreasing to one percent at 648 GHz, 0.1 percent at 516 GHz, and 0.01 percent at 41 1 GHz.The coefficients Fj(T) in (20) are calculated by line-byline summation over all lines vi > 33 cm -I (989.3GHz) in the 1982 AFGL compilation for temperatures of -40°C to 40°C in steps of 5°C and fitted to the functional form in (8).The worst percentage error of this fit at any of the calculated temperatures was only 0.17 percent.Finally, the absorption coefficient is conveniently given in dB-km-' by The coefficients in (21) are given in Table IV.
The sensitivity coefficients are the derivatives of the real and imaginary parts of the refractive index with respect to the logarithm of both absolute temperature and humidity [6].These coefficients describe the fluctuations of refraction and absorption caused by fluctuations of temperature and humidity.The sensitivity coefficient for refraction may be calculated by performing the corresponding differentiation on (l), (9), and the remaining summation over resonances below 1 THz.We compared this method against summations over all of the 1982 AFGL water-vapor compilation with excellent results.That is, the temperature and humidity derivatives of the approximation (9) (and of (21) as well) are within a few percent of these derivatives of the line-by-line summation that yields (9) (and (21)).

ACKNOWLEDGMENT
The author thanks Brenda Spaur for her help with the computations and Mildred Birchfield for her excellent typing of the paper.
This work was supported in part by the U .S .Army Research Office under Contract The author is with the U.S. Department of Commerce, NOAAIERL, 325 Broadway, Boulder, CO 80303.IEEE Log Number 8718775.MlPR 122-85.

Fig. 1 .
Fig. 1 .The temperature dependences, normalized by their values at O"C, of (2) and four terms in (9) are shown for fixed absolute humidity.Radio refractivity (2) using Boudouris' empirical coefficients inTable I gives solid curve, whereas calculated coefficients (1982 AFGL in Table I) give short-

Manabe
et al. [2] measured the dispersion between 82 and 246 GHz.They show that the data are in good agreement with refractivity calculated from line-by-line summation over the AFGL compilation.Liebe [ 11 shows that his model gives good agreement with the data by Manabe e? al.At these frequencies and to the necessary accuracy, these two models and the one presented here are equivalent.Thus the data by Manabe et al. likewise support our model of dispersive refractivity.Bradley and Gebbie [16] measure the refractivity of moist air, pure water vapor, and a mixture of nitrogen and water vapor at frequencies of 29.7 cm-I (890.4GHz) and 32.2 cm-I (965.3GHz).Table

APPENDIX
We seek an approximation for the absorption below 1 THz caused by water-vapor resonances above 1 THz as predicted by the VVW, F1, and G line shapes.Of course, this calculated absorption does not adequately predict the empirical absorption in the absorption windows [17], [l], [2].We begin with the absorption line shapes given in [12] and neglect terms in the denominators containing the line half-width since I vvi( >> yi.Next we expand in Taylor series in powers of (v/vi)'.The pressure, temperature, and humidity dependence of the line half-width yi is factored by expressing yi as where Pd is the partial pressure of dry air and yoi is the line half-width as given in the AFGL compilation at the reference values of pressure Po and temperature To.The values of PO and To for the AFGL compilation are Po = 1 atm and TO = 296 K .A possible choice for g(Pd, e, T ) is [18]

TABLE I1 THE COEFFICIENTS IN (9)
a The units of A, are g -' .m 3 , whereas a, and B, are dimensionless.

Table I gives solid curve, whereas calculated coefficients (1982 AFGL in Table I) give short- dashed curve.
j = 1-4 terms in (9) are given by X, medium-dashed, +,

TABLE 111 COMPARISON
OF MEASURED AND CALCULATED REFRACTIVITY AT 299 K FOR BOTH MOIST AIR (UPPER "WO ENTRIES) AND PURE WATER VAPOR (LOWER TWO ENTRIES)d v Pressure e a Errors

TABLE IV THE
COEFFICIENTS IN (21)