Resonance scattering by atmospheric sodium—VIII

Journalof AtmosphericzmdTfrrestrialPhysics,

1960. Vol. 1;. pp. 295 to 301. Petgm~~n

Press Ltd.

Printed in NorthernIreland

Resonance scattering by atmospheric sodium-VIII

TEE problem of deducing the vertical distribution of at~mospherie sodiwn front twilight nleas~lrenl~nts of t,he D-lines is one of long sta~lc~~~l~. It ~--as discussed in some detail in Paper VII of this series (RUNDLIE et ab., 1959) where a matrix method of solving the inte,Tal equat,ion was developed and test,ed. The results were useful but not whofly satisfactory. The present paper points out that the integral equation is analogous to the one which is found in smoothing problems in such fields as radio astronomy (BRACEWELL and ROBERTS, 1952) or spectral scanning with inadequate resolution (HARDY and POUXQ, 1919). A recent paper by DIXON and AITKEN (1965) has shown that the matrix method is apparently not suited to such problems, and our experience discussed in Paper VII seems to confirm this. The new viewpoint is to regard the t.ransmissio~l derivative p as sweeping over the sodium dist,ribution I&to generate the bri~ghtnessderivative r! according to equation I; thus B is regarded as a smeared version of E and the problem is to remove the smearing by a suitabIe restoring process. The references just mentioned are only a few of a large literature on that subject, and known methods may immediately be np$ied to our problem, In particular, we will use the elegant graphical method of BRACEWELL (1055) which is very easy to apply. BIZACEWELL and ROBERTS (1954) have shown that the correct’4 result is not always t,he same RS the original because the integral equation may not have a unique soltztion. We have enc~)untered some difficulties of this sort, and have been able to a1leviat.e them by sul~traeting from the observed 2? a standard curve which is the B for an . . assumed tlrstrzbution; restoration t,hen need be carried out only ou the difference. The result is, of course, biased towards the assumed one, but this map be preferable to the other difficulties. * The research reported in this paper was supported by the Geophysics Air Force Cambridge Research Center, Air Research and Developmenb AF 19(604)--1831. 295

Research DirecLorato of thlb Command, untlrr 01 mtr~ct,

D. M. HUNTEN When the sodium abu~ldance is large, resonance absorption may have an appreciabIe effect on the observed B; this effect was discussed in Paper VII, but the only result quoted was that it could be negIected for the lower abundances found in practice. A simyIe and compact method of presenting the results has now been devised, and they are given in graphical and tabular form. With their aid the observations may be corrected to the shape which they would have in the absence of resonance absorption. 2. CHORD The equation paper this is

TREATMENT OF THE INTEGRAL EQUATION

to be treated is (2) of Paper VII;

R&)

=

!Qz fin .zl -25

omitting

z&%(z) dz

a factor

as in that

(1)

The dot represents differentiation with respect to zl, the geometrical shadow height. As already mentioned, this equation can be regarded as describing the effect of scanning a function n(z) with an instrumental profile !i’ to produce an observed function B,; BRACEWELL’S equation (1) (1955) is essentially the same. It is convenient to have fp dz = 1; here this is assured since T( -25) = 0 and II{ co) = 1. In most smootlling problems the first moment of g vanishes; here this is not so unless the origin is shifted up by an amount which we call 4, where

s m

g=

II

-

xP(x) dx

25

and x = x - zl. We will also need to know the second moment 02, 2 and o2 were evaluated numerically for the “spring” and “autumn” transmission functions used in Paper VII (the variation is caused by the seasonal changes in ozone). The results were: spring: 2 = 27-O km, 20 = 194 km autumn :

36-3 km,

21.2 km

Treatment of the observations according to the BRACEWELL method is carried out as in Fig. 1, which shows the results of a test calculation. Plot Z?,(Q) from observation against (z., + 5); x1 follows from the solar depression. The result is to be considered as a broadened version of the sodium distribution n(x). The sharpening procedure does not appreciably change the position of the peak; this can therefore be seen immediately. A chord is drawn across the curve spanning a height-range 2a = 2~; the correction to the curve at the middle of the chord is equal to the distance from -the chord to the curve. &nce the shift cz and the span 3a vary with the ozone abundance, they must be chosen to suit the season. The variations are small and the exact choice makes little difference to the final result; thus, the method is happily insensitive to such variations and to other errors in the calculated 5?. There is a slight asymmetry in i7 which is not taken into account; the effect of this can be seen in Fig 1 and is fortunately slight. The model distribution shown in Fig. 1 was used to test the method. The models previously used contained discontinuities in slope which seem to upset 296

therefore the smooth curve n(z) = sech [(z - 55)/7*6] was used the results; (lengths in kilometres). This function goes over into esponentials at heights well above and below the peak, and shouId be similar to the atmospheric clistrihution of sodium. B,(zJ was calculated by electronic compter using the same programme as in Paper VII; it is shown also in Fig. 1 plotted against (zl --t 37 km) and t,he resemblance to a broadened version of ,E(:) may be seen. The curve corrected 1)~ the chord method agrees fairly well with the original; it. falls somewhat short at the peak. but the relat#ive error is not large. The undershoots about 45 km alcove

Fig. I. Test of the chord method. From the assumed distribution n(z) the corresponding brightness derivakive k(z,) xvas calculated. Tbc chord method then gave the restored distribution sh~mn by the CIWS~S.

alnrl below the peak are more annoying. They do not seem to be due to omission of higher-order terms of the correction series, since inclusion of the fourth-order term as suggested by BRACEWELL (1955) makes them worse. It seems rather t1la.t the slightly oscillating solution is mathematically just as good as the ori,$nal distribution; this is possible because equations like (1) may not have a unique solution (BRACEWELL and ROBERTS, l!kS). These authors give several examples of similar situa,tions. The difficulty seems to be caused by the presence of a fairly sharp peak in the distribut’ion ; it may be alleviated by sut)bracting an assumed solution so that t,hr remainder is better behaved. A suitable solution to subbract is t’he one in Fig. 1: let this be called no(z) and the corresponding brightness derivative R,(z). ‘l’h~n the actual dist~ribut~ion n = n,@ + An; substitute this in (1): R,(z,) M?(z,)

= .f!Pn, d.2 + f!F’4rz, (iz =LTI B,(z,) GE J$ -

B, = fP&iE.

+ JP&

$2. (2)

Equation (2) is of the same form as (1) and may be solved for An in the same way tha,t (1) is solved for n. a, may be moved up or down as well as changed in

D. M. HUNTEN amplitude to get the best agreement with the observed B,; then n, must be treated in the same way. When An has been found by the chord method it is added to this modified n, to give the observed distribution. There is a tendency for this to resemble n,, but this may be less serious than the oscillations produced by direct application of the chord method. Perhaps a combination of (1) for the region near the peak and (2) for the tails would be best. If the subtraction method is used on the curve of Fig. 1, An will be zero and the final result perfect. To obtain some idea of the behaviour of the method, a somewhat distorted version of the distribution was drawn and the corresponding B, computed, using Fig. 1 for n,; the result was very satisfactory and the oscillations were almost completely eliminated. Comparison of the direct and difference methods on experimentally measured 2, curves shows that they give almost indistinguishable results; undershoots are often present but they are often much larger than the ones found in Fig. 1. Apparently the defects in the observations or the assumed value of 2a are much larger than the defects in the chord method; thus, there is little use in including the refinement embodied in (2). The undershoots often seem to be due to a tendency to draw a curve through a noisy trace which falls too quickly to the axis; great care must be taken to draw a curve which is suitable for the sharpening procedure when the signal/noise ratio is low. 3. CORRECTIONFOR RESONANCE ABSORPTION The work described in this section is a continuation of Section 3 of Paper VII. Some light is lost by resonance absorption in the sodium layer on the sunlit and twilit sides of the earth. 9(z1) is the factor by which the intensity calculated by simple means must be multiplied to allow for this effect; it is a number usually between O-5 and 1 for the sodium abundances encountered. It was shown that the effect of 9’ on the shape of the derived distribution could be taken into account by the following expression: Z?, = Z? -

@B/Y

in which B and B are the observed brightness corrected derivative to be used in (1).

(3) and its derivative,

and B, is the

9 is a function of N, the total abundance of sodium, but plots of L? against z1 for various values of N appear to have the same shape, differing only by a factor. This was tested by plotting several of them together after normalizing each to 1 at .zl = 45 km. Even for an abundance as large as 17 x log atoms/cm2, these curves almost superposed; the one for 7 x log atoms/cm2 was taken as representative for further work. The correction term in (3) contains also B as well as Y which is taken constant: it is accurate enough to use a standard shape for B(q), such as the one from the model and the product

YB;

described

in Section

2.

Fig. 2 shows the shapes of $’

the latter is taken as equal to 1.0 at z1 = 43 km.

It is seen

that a simple straight line gives a satisfactory representation of gB; this, along with the several other approximations in this development, is justified by the fact that we are dealing with a small correct,ion. This straight line goes to zero 298

Hesonance scattering by atmospheric sodium-VIII

at=, = 56 km and has the value given in Table 1 at 43 km; the method of deriving these numbers is described in the Appendix. The abundance N to be used in Table 1 is found from the brightness at a depression of 6” 30’ according to the t,heory of Paper IV (CHAMBERLAIN et cd., 1958) or the equivalent by Donahue and collaborators.

21,

km

Fig. 2. Shapes of Y(z,) and of YB. The two points define the straight line which is used as an approximation to the latter.

Though it has not been mentioned so far, B and _8, are actually negative and we have been plotting them as if they were positive. Thus the correction term in (3) must also be plotted positive. The graphical procedure is then the following: Table 1. Values of &B/Y at 11 = 43 km as a function of sodium abundance N

9+3/Y

N (log atoms/cm2

(n/km)

I

0.42 1.60 3.33 5.48 8.00 10,s 13.7

x

j

8 9 10 12 14 17 “0

.+B/Y

16.8 19.9 33.0 39.3

plot B against (zl + Z); plot *B/9’ as a zero at (56 + 2) km and the value from Table 1 at (43 + 2) km, drawing a straight line through these points. Add the two curves to give B, which is to be treated as already described in Section 2. The scale of the plot must be known to do this, since the values in Table 1 are given in rayleighs/km. This is easy if an observed B is differentiated numerically t’o give Ii; if the conical-scanning method of Paper VII is used to measure AB directly, the scale must be calibrated in some way.

D. M. HUNTEN 4. DISCUSSION The methods just described have been used on the observations of nearly a year made with the birefringent photometer (Paper VII) by R. E. Bullock; the The chord-correction method is easier results will be reported in another paper. to apply than the matrix method and also more satisfactory. In a few cases where both methods have been used on the same observations, they have agreed well, though the matrix method gives only a few points instead of a continuous curve. As was mentioned in Section 2, the subtraction technique does not give much improvement because errors of measurement are usually greater than errors of treatment. It is interesting to notice that a by-product of the chord treatment is a natural definition of a “screening height”: the quantity 2 which is between 26 and 27 km. This agrees well with the 25 km found by DUFAY (1947) by taking the point at which T = 0.5; because 1’ is nearly symmetrical this height is approximately equal to b. Other values of screening height may be found according to the convention used; for example, HUNTEN (1956) obtained -2 km by weighting the transmission function by an exponential to represent the sodium distribution. This paper has presented a simple method of deriving the vertical distribution of sodium from the derivative of a measured curve of twilight brightness vs. shadow height. This curve is regarded as a smeared version of the distribution and the latter is found by means of the chord construction of BRACEWELL. Tests on computed results were satisfactory and the method has been used on a large number of observations. A modification was also tested; here a computed derivative curve is subtracted before the chord method is applied to the remainder; the result is then added to the assumed initial distribution. This gives a smoother result, but its use on observations we have been able to make does not seem justified. Resonance absorption may have some effect on the shape of the observed derivative; a simple method of correcting this has been developed. Once the abundance is known, a straight line can be drawn on the plot and added to the observations. A series of reliable measurements of vertical distribution may throw some light on the problems of the random and seasonal variations of abundance. Acknozcledgements-Much work in making observations? reducing them, and processing them by the new method has been done by R. E. BULLOCK, B. G. PRATT and J. K. WALKER. APPENDIX Normalization

of the resonance-absorption

correction

We want the value of the term $B/Y

in (3) at x1 = 43 km;

numbers given in Table 1. The curve representative

these are the

of the shape of &, shown in

Fig. 2, is called s(.z,); thus &‘(z,) = &‘(45)8(2,). Similarly, the brightness is written B,b(z,) where B, is the brightness at the plateau. Then L+B ___ = Y(45) 9 300

$‘.

ib.

B(Q)

Hesonanee scattering by &mospheric sodimn-\‘I11

Now, at the plateau becomes

the transmission

T is equal to I;

(1) writ,ten in full t,hetl

B, ==,%Yj-n dz = x.YLV where N is t)he tot’al abundance and k has the value 888 rayleighs/lOs atoms per cm2 according to Paper TV (CHAMBERLAIN et nl., 1958). Substituting for B,,/Ygives

From l?ig. 2, Bb is found Taobe 1.0 at 43 km;

it is convenient

to ta,bulate the numbers

for this shadow height since they are just equal to kN&‘(45). a function of N from the comput~ations of Paper VII.

&(45)

is known as

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