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Effects of turbulence on average refraction angles in occultations by planetary atmospheres
ICARUS34, 396--405 (1978)
Effects of Turbulence on Average Refraction Angles in Occultations by Planetary Atmospheres V O N R. E S H L E M A N
AND B J A R N E
S. H A U G S T A D ~
Center for Radar Astronomy, Stanford University, Stanford, California 9~305 Received June 23, 1977; revised October 3, 1977 Four separable effects of atmospheric turbulence on average refraction angles in occultation experiments are derived from a simplified analysis, and related to more general formulations by B. S. Haugstad. The major contributors are shown to be due to gradients in height of the strength of the turbulence, and the sense of the resulting changes in refraction angles is explained in terms of Fermat's principle. Because the results of analyses of such gradient effects by W. B. Hubbard and J. R, Jokipii are expressed in other ways, a special effort is made to compare all of the predictions on a common basis. We conclude that there are fundamental differences, and use arguments based on energy conservation and Fermat's principle to help characterize the discrepancies. INTRODUCTION
results are then c o m p a r e d with those obtained f r o m o t h e r analyses. Since several critical aspects of the various results a p p e a r to be incompatible, we a t t e m p t to develop a c o m m o n basis for c o m p a r i s o n as an aid b o t h for discussing the n a t u r e of these differences, and for seeking a b e t t e r u n d e r s t a n d i n g of w h a t m i g h t constitute the correct answers. All of the work discussed here is limited to the weak scattering a p p r o x i m a t i o n , a l t h o u g h this is not always m a d e clear in the original papers. All of the c o m p a r i s o n is m a d e on the basis of results derived f r o m geometrical optics, a l t h o u g h the connection with wave-optical effects is given in A p p e n d i x A.
T h e D o p p l e r frequencies a n d intensities of radio signals observed during spacecraft oceultations b y planets can be used to derive altitude profiles of relative t e m p e r a ture a n d pressure in their atmospheres. Optical intensities observed during stellar occultations provide similar information, usually over fewer scale heights centered at greater altitudes. I n t e n s i t y and D o p p l e r f r e q u e n c y are related since the f o r m e r depends u p o n differential angles of refraction (assuming no loss due to scattering or absorption), while the latter is a measure of the total refraction. ]¢ecent theoretical studies of the effects of a t m o s p h e r i c t u r b u lence on refraction angles have yielded diverse results. We present here a simplified t r e a t m e n t of the average change or bias in angles of refraction due to the presence of t u r b u lence in the occulting atmosphere. T h e
ANGULAR BIAS DUE TO TURBULENCE
1 On leave from the Norwegian Defence Research Establishment, Kjeller, Norway.
This section is based on the analytical f o r m u l a t i o n b y H a u g s t a d (1977a, referred to here as H-77a). He finds t h a t there are several sources of the net turbulenceinduced angular bias. T h e y can be classified
;396 0019-1035/78/0342-0396502.00/0 Copyright ~) 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.
TURBULENCE EFFECTS IN OCCULTATIONS into two groups, one containing those t e r m s t h a t require the average refractivity to v a r y with altitude (or in a n y direction normal to the propagation path), and the other including those t h a t require the strength of the turbulent fluctuations in refractivity to v a r y in directions normal to the path. For obvious reasons, we call the former mean-gradient and the latter fluctuation-gradient terms. This division follows naturally from a representation based on a derivative of the ratio of the two variables t h a t constitute the second-order optical p a t h length. Our model is a spherically s y m m e t r i c isothermal average a t m o sphere with superimposed turbulence which is isotropic and locally homogeneous, and whose strength decreases exponentially with altitude. Consider a dielectric slab representation of the turbulent atmosphere, a normally incident plane wave, and the receiver at a distance D behind the slab, where D is much greater t h a n the slab thickness. As discussed in H-77a, and also in E s h l e m a n and H a u g s t a d (1977, referred to as EH-77), homogeneous turbulence added to an initially homogeneous dielectric slab causes a second-order change in the average 2 optical p a t h length of ($2) = --D(Sa2)/2 -- D(5132)/2,
(1)
where (Sa 2) and (~/~2) are the mean-square fluctuations in the vertical and horizontal angles of arrival at the receiver. (A physical explanation for the negative signs is offered below.) If we next add an exponential mean gradient to the dielectric to represent the average atmospheric density in an isothermal, spherically s y m m e t r i c a t m o sphere, the m e a n of the second-order 2Average optical path lengths and refraction angles in these treatments are simple ensemble averages (i.e., not weighted by signal intensities), since if one formed an ensemble average of the atmospherically induced Doppler frequencies, as measured, it would be proportional to the latter.
397
optical p a t h length becomes
($2) = -D(~a2)/2(M q- 1) -- D@~2}/2(N q- 1),
(2)
where M ~ a a D / H , 5 : ~ - % D / R , aq is the angle of refraction caused b y the mean gradient in the absence of turbulence, H is the constant atmospheric scale height, and R is the p l a n e t a r y radius. I n effect, the addition of the mean vertical gradient suppresses the vertical angular fluctuations at the receiver, while increasing the horizontal fluctuations. I t m a y be of interest to note t h a t the relative signal intensity in the absence of turbulence is (M q- 1 ) - I ( N q- 1) i, where (M -~- 1) -1 represents defocusing in the vertical plane due to the atmosphere, and (N q - 1 ) -1 is the focusing in the horizontal plane due to limb curvature. Correspondingly, the principal Fresnel zone axes are a p p r o x i m a t e l y [ X D / ( M + 1)] 1/~ and [XD/(N + 1)] 1/2 in the vertical and horizontal directions, respectively, where X is the radiation wavelength in free space. Note t h a t for a given occultation distance, M and N are proportional to O~q,which in turn is approxim a t e l y proportional to the refractivity, density, and pressure in the atmosphere. I t is assumed here t h a t (N + 1) is positive, although b y replacing this factor with I N q- 11 the equations can be applied to both near- and far-limb oeeultations, except when the observer is extremely close to the focal evolute where N = - 1 , usually within a small fraction of a meter. In the presence of both turbulence and a vertical mean gradient, the average vertical angle of refraction, s0, is aq plus d(S~)/dz, where z is the coordinate at the receiver which is normal to the incoming ray a s y m p t o t e and in the direction of increasing c~q. This added t e r m represents the extra average wave-front tilt due to the turbulence. As shown in H-77a,
d(S2)/dz
=
--(M
+ 1)-ld(S2)/'dh,
(3)
398
ESHLEMAN AND HAUGSTA1)
where h is the altitude in the slab, so that OL 0
--
O~(l
D
- 2(M + I ) -
L Mgl
(~<~2}~M/m,. (;l + 1)2
-I-
X+
1
(~a~),~m/m~ -] j.
(4)
The four terms in the bracket of (4) make obvious the characterization of the effects of turbulence as being separable fluctuation-gradient and mean-gradient phenomena. For isotropic turbulence ((~a2) = (~f12})having an exponential vertical variation of rms intensity with scale height Ht, d(Sa2),/dh = i 2 ( ~ o / 2 ) J H t . Since dM/dh = - - M / H and dN/dh = - - N / H for an isothermal atmosphere of scale height H, and for 7 --= 2H/Ht, we have
O/0 -- ~(t ( OL(t2 N~
M
F
3`
2(M+I) L •
:l
I
3'
M+I
N+I
N
+ (M + 1) 2 + ( N +
1 1) 2 " (5)
Equation (5) is the full geometricaloptics representation for the turbulenceinduced change in the average angle of refraction, normalized as indicated, for the atmospheric model being considered here. It is based on the H-77a analysis, which goes further to indicate the changes needed to make the results conform with wave optics. These wave-optical results are also derived in a more general treatment b y Haugstad (1977b), and we show their relationship to (5) in Appendix A. F r o m this connection, we conclude that (5) predicts both the net fluctuation-gradient and individual mean-gradient effects to the right approximate relative magnitude and proper sign, while indicating their dependence on factors such as the geometry of the problem, the characteristics of the
average atmosphere, and the strength of the turbulence. What is not available from the geometrical-optics terms in (5) are the wavelength dependence of the angular bias, the specific effects of different turbulence spectra, and the absolute magnitudes of the effects, which are overestimated by a large factor in (5). These wave-optical characteristics are given in Appendix A, but we use (5) for comparison with the other treatments because they can be shown to be limited to geometrical optics. The four terms of (5) are plotted in Fig. 1 assuming H = Ht (~y = 2). The two terms containing the effects of limb focusing (those containing N) are given for values of RJH = - M / N of 103 and infinity. The latter value corresponds to the omission of the focussing effect of the curved planetary limb. The ordinate of Fig. 1 is the normalized angular offset or bias given on the left-hand side of (5). The abscissa is given in terms of M, and also of 0q = (M -4- l) -1, the relative signal intensity neglecting limb curvature. Note from (4) and (5) that the results plotted in Fig. 1 are separable effects of turbulence due to vertical or horizontal angular fluctuations caused by the turbulence, combined with those due to vertical gradients in either the strength of the turbulence (fluctuation-gradient) or the average atmospheric background (mean gradient). The two-letter identification on each curve results from F and M representing fluctuation-gradient and mean-gradient effects, respectively, and V and H representing vertical and horizontal angular fluctuations due to the turbulence. The MV curve shows a positive change (increase) in the refraction angle due to turbulence, while all the others represent negative changes. To the extent that the fluctuation gradient terms overwhelm the others, we see that turbulence causes a negative change in the angle so that the total refraction is less than it would be in the absence of turbulence. We a t t e m p t a physical explanation of this characteristic in EH-77. In
TURBULENCE EFFECTS IN OCCULTATIONS
399
~ q = ( M + I ) -r I. 20. 999
0.99 J-
0.91 T
0.5 [
0.091. ]
IO -z /
ti,I G) LL h
1.0
o i'Y
0.8
EXP. SCAT., 5 0 : - F V - F H + M V - M " EXP. SCAT., 2D:-FV+MV CONST. SCAT., 5D: +MV-MH CONST. SCAT., 2D:+MV /
/"
/ / ~/~
F" , R , ( --~---- co )
I0 -3 I
I
I I
I I [ 1
z I.L
0 i,i r~ Z) I--
0.6
Z
0.4 UA
>
0.2 _J tlA n.-"
0 10-3
iO-Z
iO-I
i00 M
i0 f
102
103
FIG. 1. Four effects of atmospheric turbulence on the average bias in angles of refraction in occultation experiments. The exact form of the ordinate is given on the left hand side of (5), whose terms are plotted as a function of M, which is approximately proportional to atmospheric pressure, and ¢~, which is the relative occultation signal intensity neglecting the curvature of the planetary limb. The double-letter curve identifiers follow from FH representing the effect of a combined fluctuation-gradient and horizontal angular fluctuations, M V representing meangradient and vertical fluctuations, etc., where the four terms of (5) are in order the FV, FH, MV, and M H curves. The branch in the FH and MH curves is for different ratios of the planetary radius and atmospheric scale height. The indicated combinations of the curves apply to three or two dimensional analyses of exponential or constant turbulence added to an isothermal at~nosphere. its m o s t f u n d a m e n t a l form, the explanation can be u n d e r s t o o d in terms of the a d d i t i o n of turbulence to a h o m o g e n e o u s dielectric region. W i t h o u t turbulence, p r o p a g a t i o n is in a s t r a i g h t line, t h u s minimizing the optical p a t h length, which is the integral of the refractive index along the p r o p a gation path. W i t h turbulence, the optical p a t h l e n g t h m u s t be an e x t r e m u m , according to F e r m a t ' s principle, b u t this m u s t be a m i n i m u m in this case where weak s c a t t e r i n g is i n v o l v e d a n d o n l y one p a t h exists at a n y one time. If the p a t h s were s t r a i g h t for different realizations of the turbulence or at different times in one, the ensemble or time average of the optical p a t h lengths would be the same as before since on the average, the same a m o u n t of dielectric material would be encountered.
B u t it is obvious t h a t the p r o p a g a t i o n p a t h s are n o t s t r a i g h t in the t u r b u l e n t medium. Some m i g h t s a y t h a t this creates a paradox, since the rays follow sinuous a n d hence longer geometrical p a t h s in order to minimize their optical p a t h lengths, while if t h e y had followed a straight path, their average optical p a t h length would have been the same as existed before turbulence was added. T h e only w a y o u t is for the average sinuous p a t h to be shorter, optically, t h a n the original s t r a i g h t path. As detailed in H-77a a n d E H - 7 7 , this is e x a c t l y w h a t happens. T h e increase in a v e r a g e optical p a t h length due to r a y sinuosity is c o u n t e r e d a n d reversed b y an a v e r a g e s h o r t e n i n g which is just twice as great, due to the fact t h a t the rays now go. on the
400
ESHLEMAN AND ItAUGSTA1) TABI,E I
DIFFERENT PREDICTIONSOF THE I~]FFECTOF TURBULENCE ON OCCULTATIONREFRACTION ANGLESIL Reference (equations)
Equivalent definition for fgure erdinate in original notation
Result in terms of M (definition of M in original notation)
Result in terms of ¢~q
HJ-75 (3)
(5~) -- v±E~I''~eo2 viso ~(8~)]
+ M / ( M + 1) (~1I = flRAeo)
1 -- (~q
HJ-77a
(h) -- h. [ 1 '~
+ (M + ½)/(M + 1)2
¢,,(1 -- ½¢%)
(26)
h,,
k~]
(M=~-~=~hq)
(weighted) HJ-77a (unweighted)
As above, but intensityweighting changed by us to rio weighting for comparison with other results
½ -- M ( M + 1) (M + 1)2
¢~(1 + ½¢q) -- 1
EH-77 (10 and 11)
m, - an [ '~aq 2 c~ ~ 2 ) )
-2M (M=o~qD/H, assumed small compared to unity)
-2(¢q-' -- 1)
H-77a (6.7 and 6.8)
-X(f~} [ %2 .~ v,~q ~ , ~ }
- M ( M + 2)I(M + 1)2 (M = Mq = aqn/u~)
- ( 1 - ¢Oq2)
The quantity under consideration is defined in colunm 2 from the original papers of column 1, and is expressed in comn,on terms in columns 3 and 4 for ph)tting in Fig. 2. This quantity (the figure ordinate) is the normalized increase in the average refraction angle due to the turbulence, divided by the normalized, mean-square, fluctuating angle that would be observed if it were not also affected by the average atmosphere. This angular spread is directly related to the mean-square fluctuations in refractivity at the lowest part of the ray path. The table entries are discussed in detail in the text. The HJ-77a (weighted) row should not be compared directly with the others. average, t h r o u g h regions of lower t h a n average refractivity. Thus atmospheric t u r b u l e n c e causes a n a d v a n c e of t h e a v e r a g e phase f r o n t s 2 F o r the e x p o n e n t i a l s c a t t e r i n g m o d e l considered here, the increasing r e f r a c t i v i t y f l u c t u a t i o n s w i t h d e p t h t h e r e b y cause a n u p w a r d tilt of the wave f r o n t a n d hence a r e d u c t i o n in the d o w n w a r d angle of r e f r a c t i o n caused b y the average a t m o s p h e r e , a t least for small v a l u e s of M. T h e more complete a n a l y s i s of H - 7 7 a shows t h a t the a n g u l a r offset r e m a i n s n e g a t i v e for all values of M for this model. Because the four c o m b i n a t i o n s of effects s As in our discussion of phase paths and refraction angles, an average phase front is a simple, unweighted average position of individual rippled surfaces of the same constant phase for different realizations of the turbulence.
g i v e n b y the d o u b l e - l e t t e r curve identifiers i n Fig. 1 are separable, the curves in Fig. 1 c a n be used to d e t e r m i n e the results for either a three- or t w o - d i m e n s i o n a l analysis, a n d for either e x p o n e n t i a l s c a t t e r i n g (3' = 2) or c o n s t a n t s c a t t e r i n g (3' = 0) in a n isothermal atmosphere. (By two-dimensional we m e a n t h a t there are o n l y vertical a n g u l a r f l u c t u a t i o n s ((852) = 0), a n d t h e p r o b l e m is r e s t r i c t e d to the a c t u a l a v e r a g e p l a n e of p r o p a g a t i o n . ) F o r example, the i n d i c a t e d c o m b i n a t i o n of all four curves gives the a n s w e r for a t h r e e - d i m e n s i o n a l a n a l y s i s of a n i s o t h e r m a l a t m o s p h e r e w i t h e x p o n e n t i a l s c a t t e r i n g . A t the o t h e r extreme, a t w o - d i m e n s i o n a l a n a l y s i s of a n i s o t h e r m a l a t m o s p h e r e in which the r m s s t r e n g t h of the t u r b u l e n c e is c o n s t a n t w i t h a l t i t u d e yields o n l y the M V curve.
401
TURBULENCE EFFECTS IN 0CCULTATIONS q~q=(M+I) -' L999
0,99
0.91
0.5
0.091
H- 77o
./
NEGATIVE-
7
IO-2
10-3
.f ,//7
/ ,/ /
H J - 7 7 o (unweighted) POSITIVE BRANCH ~,.,
u
..
/
.//
/
:
. . . . .
/~,/'~ //
/
NEGATIVE BRANCH
.J-7
fi~/
POSITIVE
-,, , / / / x.
//
/ /
"\// /// /
POSITIVE
/A/ I
o.,
/"
HJ-77o(weighted)
,
o.
,o.,
",o.
o,
M FIG. 2. Relative turbulenee-induced changes in average refraction angles in occultation experiments, as predicted in four publications for the same exponential model, and as summarized in Table I. The coordinates are the same as for Fig. I, but the effectsof limb curvature are omitted here. The H-77a and EH-77 curves correspond in the region of applieability of the latter, while
the remaining results exhibit fundamental differences which are discussed in the text.
There is a predicted singularity in the turbulence-induced angular bias at the abscissa M -- R/H, where the geometricaloptics signal intensity in the absence of turbulence is also singular. We plan to consider this interesting focal region in a later paper. COMPARISONS WITH OTHER RESULTS Average effects of turbulence in occultation experiments were introduced and have been considered subsequently by H u b b a r d and Jokipii (1975, 1977a, 1977b, referred to here as H J-75, HJ-77a, and HJ-77b, respectively). We a t t e m p t here a comparison of the angular results of H-77a, as given in (5), with corresponding predictions in H J-75, HJ-77a, and also in EH-77. We restrict this comparison to the particular exponential scattering model where H = Ht, so that 1' = 2 in (5). EWe discuss the HJ-77a, b result for their constant scattering model elsewhere (Haugstad and Eshleman, 1977).-] Comparison is made difficult b y the different nomenclature, model dimension-
ality, analytical methods, and averaging techniques used in the various treatments. However, it is clear that in each case an a t t e m p t is being made to predict what effects would actually occur for an isothermal average atmosphere with superimposed turbulence whose rms strength changes with height. Table I and Fig. 2 show the results of our comparison. There is a notable lack of agreement. We explain first just what is displayed and then discuss the differences. The first column in Table I gives the source and the second the original notation for the normalized angular bias as defined on the left-hand side of (5). Columns three and four present the different predictions for plotting, in terms of M and of Cq = (M 4- 1) -t. For the H-77a or equivalent (5) results, we have assumed N = 0 since limb curvature is not considered in the other treatments. Also, only the sum of the two fluctuation-gradient terms is plotted since the remaining mean-gradient effect is so small b y comparison. Note t h a t the only agreement in Fig. 2 is between the
402
ESHLEMAN AND HAUGSTAD
H-77a results and that from EH-77 for M << 1, the only region of applicability of the latter. As required from the F e r m a t argument, both start out negative for small M. The H J-75 result for Ht = H is similar in magnitude to that of H-77a, but of opposite sign. Several aspects of the H J-75 analysis are discussed in EH-77, where for example it is shown that to use this result at all in the comparison, one of the two definitions for the bending angle must be discarded. The principal complication in our conlparisons comes from consideration of the HJ-77a result. This is because the average angular bias in HJ-77a is formed using the intensity of the signal as a weighting factor, while the other analyses involve a simple unweighted ensemble average. (As discussed in Appendix B, we see no application for the weighted result.) In addition, HJ-77~ is a two-dimensional analysis while the H-77a and EH-77 are three dimensional. Thus the HJ-77a weighted results in Table I and Fig. 2 should not be compared directly with the others. However, we do plot this weighted answer as published in order to see the effects of removing the weighting and expanding the result dimensionally. The detailed nature of our interpretation of how to make these changes is explained in Appendix B. The resulting unweighted three-dimensional prediction is given in line three of Table I and is
plotted for comparison with the other results in Fig. 2. There are several areas of basic disagreement among the various results. At large values of M, the H J-75 and HJ-77a (unweighted) results agree with each other in magnitude but not in sign, while at small M the opposite is true. Of all of the investigations, only the HJ-77a analysis predicts a finite value of the ordinate as M approaches zero, while all saturate at a magnitude of unity for large M where, however, both signs are predicted. We also note in Appendix B that the H J-75 angular result along with the HJ-77a intensity and angular results for this model atmosphere, are not compatible on the basis of geometrical-optics energy conservation. Perhaps the most fundamental problem disclosed by the comparison is the different signs of the various predictions for the same effect, particularly at small values of M. This is the region discussed in the last section, where it is concluded that the angular bias must be negative. Thus we submit that in addition to the above problems, both the H J-75 and HJ-77a results fail to conform with the requirements of Fermat's principle. APPENDIX A: WAVE-OPTICALCOMPARISON The average angular bias has been derived from wave optics by Haugstad (1977b). For an exponential atmosphere with exponential scattering, his Eqs. (18) to (22) can be used to yield
0o
sin
(2P)duzduy-4-~/Inff~(~)Q(2P)du~du~ cos
,
:o
cos `) (P)du~duy
(A-l)
TURBULENCE EFFECTS IN OCCULTATIONS where
D ( Uz2
?~y2 I
"
(A-2)
,v+i ;Iu~fl
Nuy 2
Q =
-I(M+I)
~
.
(A-a)
(N+I)"
In these expressions q)(u) is the normalized turbulence power spectrum, uz, uu are respectively the vertical and horizontal components of the turbulence wavenumber u, and k is the radiation wavenumber. The double integral in the denominator of (A-l) multiplied by 27rl(vt2) is the definition of (5a2), while if u, 2 in this integral were replaced b y u~2 it would be (~52), but these angular effects are equal in our model where the turbulence is assumed to be isotropic. For the above multipliers, 1 is the equivalent tangential thickness of the atmosphere, approximately (RI-[) 1/2 when H and Ht are comparable, and (vt2) is the mean-square value of the fluctuations of the atmospheric refractivity at the lowest point on the ray path. Since P ~ k -1 -~ 0 for geometrical optics, (A-l) is readily seen to reduce to (5) under this condition. It is also a straightforward m a t t e r to obtain from (A-l) the dependence of the average angular offset on the radiation wavelength, as introduced in EH-77 and discussed more fully in Haugstad and Eshleman (1977). For turbulence power spectra that v a r y as the (--p) exponent of the wavenumber of the turbulence, the angular offset is proportional to X"/~-~ (X-1/6 for Kolmogorov turbulence) for 2 < p < 4 a n d l 0 < at < L0, where 10 and L0 are the inner and outer scales of the turbulence, respectively, and at is the radius of the principle Fresnel zone. APPENDIX B: INTENSITYWEIGHTING In HJ-77a,b, the average angular bias is formed using the concurrent signal intensity as a weighting factor, while in the other analyses the simple unweighted ensemble average is derived. We submit that the
403
unweighted result is the quantity that is needed in practice, but indicate below the relationship between the two. Angles are measured only in radio occultation experiments, through their effects on Doppler frequencies. Such experiments involve a point source of coherent radiation which, due to the weak scattering approximation which applies to all of the analyses discussed here, propagates through the atmosphere in going from source to observer along only one path at a time. (In ionospheric occultations, there m a y be multiple rays, but these appear to be deterministic in terms of ionospheric layers.) In practice, the measured Doppler frequencies are used without regard to intensities in deriving profiles of atmospheric temperature and pressure. (Signal intensities are used either alone to obtain independent profile information, or in conjunction with the Doppler-derived profiles to determine the distribution of absorbing regions in the atmosphere.) When turbulence is present, it follows that one would want to know to what extent the average of the measured unweighted Doppler frequencies or refraction angles (if indeed it is possible to form a meaningful average over a sufficiently small height interval in one or a few oecultations) m a y be affected by the atmospheric irregularities. Correspondingly, it would be important to be able to predict the expected magnitude and sign of the average biases in the derived atmospheric profiles due to the turbulence. In both eases, what is needed is a theory giving the nonweighted ensemble average of the turbulence-induced changes in the refraction angles. However, it is of interest to see in what way the different averaging procedure used in HJ-77a affects the comparisons in Table I and Fig. 2. We use the method of HJ-77b to find the intensity-weighted average angle of refraction, in order to show its relationship to the simple average used for the other results. F r o m HJ-77b Eqs. (3)
404
ESHLEMAN AND HAUGSTAD
and (5), (B-l)
E = ex+i~, kai = I m ( E * O E / O z } / ( E * E } ,
(B-2)
where E is the complex field at the zposition of the receiver, having log-amplitude x and phase q', k is the radiation wavenumber, and a1 is the intensityweighted average angle of refraction. Expanding x and ,I, to second order, x = xq -t- xl -t- x2 and ,I, = ,I,q -t- ,Iq + ,I% so t h a t I m (E*OE/Oz} --- e2x~([(1 q- 2Xl -b 2x2 q- 2x~ ~) (g-3)
to second order. B u t O~n/Oz is just /Caq, O~l/OZ is kay, etc., where aq, al, as are the zero-, first-, and second-order bending angles. Using also the simple average angle of refraction ~0 -= ( ~ + ~ + ~2) = ~ + ( ~ ) ,
I m (E*OE/Oz) + 2(a~x~)].
(B-4)
( E E * ) = e2x.[-1 + 2((×~ 2) + (x2))],
(B-5)
But since
we find t h a t (B-6)
through second order in the turbulence. I t follows from (B-6) t h a t if we have a normalized angular bias like the left side of (5) but expressed in terms of ai instead of a0, and wish to change it to one in terms of the simple average a0, then we can use 13/0 - - O L q ( ~q2
O/I __ OQt
+
2(M + 1)
O/q2
-r
L(~ ~)
The t e r m in square brackets in (B-7) can be evaluated b y the methods used in
.
q- 1
(B-S)
N-k
B y using the above results and (5), we find that, ai -- aq
aq '2
a,,
;][
2(211 + 1) X
X (O~ffq/OZ -'[- O~I~l/oZ -'[- O~2/OZ)])
ar = a0 ~- 2(a~×~)
H-77a to find in geometrical optics t h a t it equals
L ( M + 1) 2
-[-
(N + 1) i
(B-9)
for a n y value of % and in fact for a n y variation of the strength of the turbulence with altitude. I t is interesting in this regard t h a t the constant-scattering result given in HJ-77a [-Eq. (17)] is more nearly applicable to the exponential-scattering model t h a n is the H J - 7 7 a [-Eq. (26)], which was derived specifically for the latter model. I t also follows from (B-9) t h a t there is no difference for this intensity-weighted and normalized angular offset between a two-dimensional analysis and a threedimensional one where it is assumed t h a t there is no limb curvature, since the t e r m containing N vanishes in both cases, although for different reasons. The point of these last remarks is t h a t the HJ-77a [-Eq. (26)] intensity-weighted, two-dimensional, geometric-optics result can be transformed to the equivalent unweighted three-dimensional prediction using (B-7) and (B-S). Thus the HJ-77a unweighted column three e n t r y is obtained from the weighted e n t r y in Table I using these equations with N = 0 and ~ = 2. A corresponding translation has also been made in column four to express the unweighted result in terms of Cq. The relationship between weighted and unweighted angular offsets can also be used in an additional comparison of results. Two-dimensional, geometrical-optics expressions for signal intensities and refrac-
TURBULENCE EFFECTS IN OCCULTATIONS tion angles m u s t be related for c o n s e r v a t i o n of e n e r g y as (¢} - Cq = ¢(~
Cq [d(ao -- aq)/dh-]dx,
(B-10) where (~) is the ensemble average signal intensity. Using the m e t h o d s of t r a n s l a t i o n described above, we find t h a t the expressions for i n t e n s i t y a n d b e n d i n g angle effects given in H J - 7 7 a ['Eqs. (25) a n d (26)1 are n o t compatible on these grounds. W e s u b m i t t h a t these two effects can be o b t a i n e d using (5), (B-9), a n d (B-10) for t w o dimensions, so t h a t t h e y should be equal to (¢)
-
~bq =
(1/24)
C2¢q(1
-
~q)2
X (5 -~ 10¢q -]- 9¢q2), (ce~ -- C~q)D/H -- ½C2(1 -- Cq)a,
(B-ll) (B-12)
respectively, instead of the values given in H J - 7 7 a . (In these expressions, the H J - 7 7 a f o r m u l a t i o n is used on the righth a n d sides, where C ~ = - ( ~ } / a q 2 in the present n o t a t i o n . ) I t also follows f r o m these considerations t h a t the H J-75 a n g u l a r result is n o t consistent with the H J - 7 7 a i n t e n s i t y prediction.
405
ACKNOWLEDGMENTS We thank R. W. Lee, G. L. Tyler, J. B. Keller, and A. T. Young for helpful discussion. This work was supported by the Royal Norwegian Council for Scientific and Industrial Research and by the U.S. National Aeronautics and Space Administration. REFERENCES ESHLEMAN, V. R., AND HAUGSTAD, B. S. (1977).
Lowest-order average effect of turbulence on atmospheric profiles derived from radio occultation. Astrophys. J. 214, 928-933. (EH-77.) HAUGSTAD, B. S. (1977a). Turbulence in planetary occultations: I. A theoretical formulation. Submitted to Icarus. (H-77a.) HAUGSTAD, B. S. (1977b). Effects of an inhomogeneous background on radiation propagating through turbulent planetary atmospheres. Submitted to Radio Sci. HAUGSTAD,B. S., AND ESHLEMAN, V. R. (1977). On the wavelength dependence of the effects of turbulence on average refraction angles in occultations by planetary atmospheres. Submitted to Astrophys. J. HUBBARD, W. B., AND JOKIPII, J. R. (1975). Effects
of turbulence on radio-occultation scale heights. Astrophys. J. 199, L193-L196. (HJ-75.) HUBBARD, W. B., AND JoKIPII, J. R. (1977a).
Stellar occultations by turbulent planetary atmospheres: A heuristic scattering model. Icarus 30, 531-536. (HJ-77a.) HUBBARD, W. B., AND JOKIPII, J. R. (1977b). Turbulent scattering in an exponential atmosphere: A wave-optical solution. Astrophys. J. 214, 924-927. (HJ-77b.)
Effects of Turbulence on Average Refraction Angles in Occultations by Planetary Atmospheres V O N R. E S H L E M A N
AND B J A R N E
S. H A U G S T A D ~
Center for Radar Astronomy, Stanford University, Stanford, California 9~305 Received June 23, 1977; revised October 3, 1977 Four separable effects of atmospheric turbulence on average refraction angles in occultation experiments are derived from a simplified analysis, and related to more general formulations by B. S. Haugstad. The major contributors are shown to be due to gradients in height of the strength of the turbulence, and the sense of the resulting changes in refraction angles is explained in terms of Fermat's principle. Because the results of analyses of such gradient effects by W. B. Hubbard and J. R, Jokipii are expressed in other ways, a special effort is made to compare all of the predictions on a common basis. We conclude that there are fundamental differences, and use arguments based on energy conservation and Fermat's principle to help characterize the discrepancies. INTRODUCTION
results are then c o m p a r e d with those obtained f r o m o t h e r analyses. Since several critical aspects of the various results a p p e a r to be incompatible, we a t t e m p t to develop a c o m m o n basis for c o m p a r i s o n as an aid b o t h for discussing the n a t u r e of these differences, and for seeking a b e t t e r u n d e r s t a n d i n g of w h a t m i g h t constitute the correct answers. All of the work discussed here is limited to the weak scattering a p p r o x i m a t i o n , a l t h o u g h this is not always m a d e clear in the original papers. All of the c o m p a r i s o n is m a d e on the basis of results derived f r o m geometrical optics, a l t h o u g h the connection with wave-optical effects is given in A p p e n d i x A.
T h e D o p p l e r frequencies a n d intensities of radio signals observed during spacecraft oceultations b y planets can be used to derive altitude profiles of relative t e m p e r a ture a n d pressure in their atmospheres. Optical intensities observed during stellar occultations provide similar information, usually over fewer scale heights centered at greater altitudes. I n t e n s i t y and D o p p l e r f r e q u e n c y are related since the f o r m e r depends u p o n differential angles of refraction (assuming no loss due to scattering or absorption), while the latter is a measure of the total refraction. ]¢ecent theoretical studies of the effects of a t m o s p h e r i c t u r b u lence on refraction angles have yielded diverse results. We present here a simplified t r e a t m e n t of the average change or bias in angles of refraction due to the presence of t u r b u lence in the occulting atmosphere. T h e
ANGULAR BIAS DUE TO TURBULENCE
1 On leave from the Norwegian Defence Research Establishment, Kjeller, Norway.
This section is based on the analytical f o r m u l a t i o n b y H a u g s t a d (1977a, referred to here as H-77a). He finds t h a t there are several sources of the net turbulenceinduced angular bias. T h e y can be classified
;396 0019-1035/78/0342-0396502.00/0 Copyright ~) 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.
TURBULENCE EFFECTS IN OCCULTATIONS into two groups, one containing those t e r m s t h a t require the average refractivity to v a r y with altitude (or in a n y direction normal to the propagation path), and the other including those t h a t require the strength of the turbulent fluctuations in refractivity to v a r y in directions normal to the path. For obvious reasons, we call the former mean-gradient and the latter fluctuation-gradient terms. This division follows naturally from a representation based on a derivative of the ratio of the two variables t h a t constitute the second-order optical p a t h length. Our model is a spherically s y m m e t r i c isothermal average a t m o sphere with superimposed turbulence which is isotropic and locally homogeneous, and whose strength decreases exponentially with altitude. Consider a dielectric slab representation of the turbulent atmosphere, a normally incident plane wave, and the receiver at a distance D behind the slab, where D is much greater t h a n the slab thickness. As discussed in H-77a, and also in E s h l e m a n and H a u g s t a d (1977, referred to as EH-77), homogeneous turbulence added to an initially homogeneous dielectric slab causes a second-order change in the average 2 optical p a t h length of ($2) = --D(Sa2)/2 -- D(5132)/2,
(1)
where (Sa 2) and (~/~2) are the mean-square fluctuations in the vertical and horizontal angles of arrival at the receiver. (A physical explanation for the negative signs is offered below.) If we next add an exponential mean gradient to the dielectric to represent the average atmospheric density in an isothermal, spherically s y m m e t r i c a t m o sphere, the m e a n of the second-order 2Average optical path lengths and refraction angles in these treatments are simple ensemble averages (i.e., not weighted by signal intensities), since if one formed an ensemble average of the atmospherically induced Doppler frequencies, as measured, it would be proportional to the latter.
397
optical p a t h length becomes
($2) = -D(~a2)/2(M q- 1) -- D@~2}/2(N q- 1),
(2)
where M ~ a a D / H , 5 : ~ - % D / R , aq is the angle of refraction caused b y the mean gradient in the absence of turbulence, H is the constant atmospheric scale height, and R is the p l a n e t a r y radius. I n effect, the addition of the mean vertical gradient suppresses the vertical angular fluctuations at the receiver, while increasing the horizontal fluctuations. I t m a y be of interest to note t h a t the relative signal intensity in the absence of turbulence is (M q- 1 ) - I ( N q- 1) i, where (M -~- 1) -1 represents defocusing in the vertical plane due to the atmosphere, and (N q - 1 ) -1 is the focusing in the horizontal plane due to limb curvature. Correspondingly, the principal Fresnel zone axes are a p p r o x i m a t e l y [ X D / ( M + 1)] 1/~ and [XD/(N + 1)] 1/2 in the vertical and horizontal directions, respectively, where X is the radiation wavelength in free space. Note t h a t for a given occultation distance, M and N are proportional to O~q,which in turn is approxim a t e l y proportional to the refractivity, density, and pressure in the atmosphere. I t is assumed here t h a t (N + 1) is positive, although b y replacing this factor with I N q- 11 the equations can be applied to both near- and far-limb oeeultations, except when the observer is extremely close to the focal evolute where N = - 1 , usually within a small fraction of a meter. In the presence of both turbulence and a vertical mean gradient, the average vertical angle of refraction, s0, is aq plus d(S~)/dz, where z is the coordinate at the receiver which is normal to the incoming ray a s y m p t o t e and in the direction of increasing c~q. This added t e r m represents the extra average wave-front tilt due to the turbulence. As shown in H-77a,
d(S2)/dz
=
--(M
+ 1)-ld(S2)/'dh,
(3)
398
ESHLEMAN AND HAUGSTA1)
where h is the altitude in the slab, so that OL 0
--
O~(l
D
- 2(M + I ) -
L Mgl
(~<~2}~M/m,. (;l + 1)2
-I-
X+
1
(~a~),~m/m~ -] j.
(4)
The four terms in the bracket of (4) make obvious the characterization of the effects of turbulence as being separable fluctuation-gradient and mean-gradient phenomena. For isotropic turbulence ((~a2) = (~f12})having an exponential vertical variation of rms intensity with scale height Ht, d(Sa2),/dh = i 2 ( ~ o / 2 ) J H t . Since dM/dh = - - M / H and dN/dh = - - N / H for an isothermal atmosphere of scale height H, and for 7 --= 2H/Ht, we have
O/0 -- ~(t ( OL(t2 N~
M
F
3`
2(M+I) L •
:l
I
3'
M+I
N+I
N
+ (M + 1) 2 + ( N +
1 1) 2 " (5)
Equation (5) is the full geometricaloptics representation for the turbulenceinduced change in the average angle of refraction, normalized as indicated, for the atmospheric model being considered here. It is based on the H-77a analysis, which goes further to indicate the changes needed to make the results conform with wave optics. These wave-optical results are also derived in a more general treatment b y Haugstad (1977b), and we show their relationship to (5) in Appendix A. F r o m this connection, we conclude that (5) predicts both the net fluctuation-gradient and individual mean-gradient effects to the right approximate relative magnitude and proper sign, while indicating their dependence on factors such as the geometry of the problem, the characteristics of the
average atmosphere, and the strength of the turbulence. What is not available from the geometrical-optics terms in (5) are the wavelength dependence of the angular bias, the specific effects of different turbulence spectra, and the absolute magnitudes of the effects, which are overestimated by a large factor in (5). These wave-optical characteristics are given in Appendix A, but we use (5) for comparison with the other treatments because they can be shown to be limited to geometrical optics. The four terms of (5) are plotted in Fig. 1 assuming H = Ht (~y = 2). The two terms containing the effects of limb focusing (those containing N) are given for values of RJH = - M / N of 103 and infinity. The latter value corresponds to the omission of the focussing effect of the curved planetary limb. The ordinate of Fig. 1 is the normalized angular offset or bias given on the left-hand side of (5). The abscissa is given in terms of M, and also of 0q = (M -4- l) -1, the relative signal intensity neglecting limb curvature. Note from (4) and (5) that the results plotted in Fig. 1 are separable effects of turbulence due to vertical or horizontal angular fluctuations caused by the turbulence, combined with those due to vertical gradients in either the strength of the turbulence (fluctuation-gradient) or the average atmospheric background (mean gradient). The two-letter identification on each curve results from F and M representing fluctuation-gradient and mean-gradient effects, respectively, and V and H representing vertical and horizontal angular fluctuations due to the turbulence. The MV curve shows a positive change (increase) in the refraction angle due to turbulence, while all the others represent negative changes. To the extent that the fluctuation gradient terms overwhelm the others, we see that turbulence causes a negative change in the angle so that the total refraction is less than it would be in the absence of turbulence. We a t t e m p t a physical explanation of this characteristic in EH-77. In
TURBULENCE EFFECTS IN OCCULTATIONS
399
~ q = ( M + I ) -r I. 20. 999
0.99 J-
0.91 T
0.5 [
0.091. ]
IO -z /
ti,I G) LL h
1.0
o i'Y
0.8
EXP. SCAT., 5 0 : - F V - F H + M V - M " EXP. SCAT., 2D:-FV+MV CONST. SCAT., 5D: +MV-MH CONST. SCAT., 2D:+MV /
/"
/ / ~/~
F" , R , ( --~---- co )
I0 -3 I
I
I I
I I [ 1
z I.L
0 i,i r~ Z) I--
0.6
Z
0.4 UA
>
0.2 _J tlA n.-"
0 10-3
iO-Z
iO-I
i00 M
i0 f
102
103
FIG. 1. Four effects of atmospheric turbulence on the average bias in angles of refraction in occultation experiments. The exact form of the ordinate is given on the left hand side of (5), whose terms are plotted as a function of M, which is approximately proportional to atmospheric pressure, and ¢~, which is the relative occultation signal intensity neglecting the curvature of the planetary limb. The double-letter curve identifiers follow from FH representing the effect of a combined fluctuation-gradient and horizontal angular fluctuations, M V representing meangradient and vertical fluctuations, etc., where the four terms of (5) are in order the FV, FH, MV, and M H curves. The branch in the FH and MH curves is for different ratios of the planetary radius and atmospheric scale height. The indicated combinations of the curves apply to three or two dimensional analyses of exponential or constant turbulence added to an isothermal at~nosphere. its m o s t f u n d a m e n t a l form, the explanation can be u n d e r s t o o d in terms of the a d d i t i o n of turbulence to a h o m o g e n e o u s dielectric region. W i t h o u t turbulence, p r o p a g a t i o n is in a s t r a i g h t line, t h u s minimizing the optical p a t h length, which is the integral of the refractive index along the p r o p a gation path. W i t h turbulence, the optical p a t h l e n g t h m u s t be an e x t r e m u m , according to F e r m a t ' s principle, b u t this m u s t be a m i n i m u m in this case where weak s c a t t e r i n g is i n v o l v e d a n d o n l y one p a t h exists at a n y one time. If the p a t h s were s t r a i g h t for different realizations of the turbulence or at different times in one, the ensemble or time average of the optical p a t h lengths would be the same as before since on the average, the same a m o u n t of dielectric material would be encountered.
B u t it is obvious t h a t the p r o p a g a t i o n p a t h s are n o t s t r a i g h t in the t u r b u l e n t medium. Some m i g h t s a y t h a t this creates a paradox, since the rays follow sinuous a n d hence longer geometrical p a t h s in order to minimize their optical p a t h lengths, while if t h e y had followed a straight path, their average optical p a t h length would have been the same as existed before turbulence was added. T h e only w a y o u t is for the average sinuous p a t h to be shorter, optically, t h a n the original s t r a i g h t path. As detailed in H-77a a n d E H - 7 7 , this is e x a c t l y w h a t happens. T h e increase in a v e r a g e optical p a t h length due to r a y sinuosity is c o u n t e r e d a n d reversed b y an a v e r a g e s h o r t e n i n g which is just twice as great, due to the fact t h a t the rays now go. on the
400
ESHLEMAN AND ItAUGSTA1) TABI,E I
DIFFERENT PREDICTIONSOF THE I~]FFECTOF TURBULENCE ON OCCULTATIONREFRACTION ANGLESIL Reference (equations)
Equivalent definition for fgure erdinate in original notation
Result in terms of M (definition of M in original notation)
Result in terms of ¢~q
HJ-75 (3)
(5~) -- v±E~I''~eo2 viso ~(8~)]
+ M / ( M + 1) (~1I = flRAeo)
1 -- (~q
HJ-77a
(h) -- h. [ 1 '~
+ (M + ½)/(M + 1)2
¢,,(1 -- ½¢%)
(26)
h,,
k~]
(M=~-~=~hq)
(weighted) HJ-77a (unweighted)
As above, but intensityweighting changed by us to rio weighting for comparison with other results
½ -- M ( M + 1) (M + 1)2
¢~(1 + ½¢q) -- 1
EH-77 (10 and 11)
m, - an [ '~aq 2 c~ ~ 2 ) )
-2M (M=o~qD/H, assumed small compared to unity)
-2(¢q-' -- 1)
H-77a (6.7 and 6.8)
-X(f~} [ %2 .~ v,~q ~ , ~ }
- M ( M + 2)I(M + 1)2 (M = Mq = aqn/u~)
- ( 1 - ¢Oq2)
The quantity under consideration is defined in colunm 2 from the original papers of column 1, and is expressed in comn,on terms in columns 3 and 4 for ph)tting in Fig. 2. This quantity (the figure ordinate) is the normalized increase in the average refraction angle due to the turbulence, divided by the normalized, mean-square, fluctuating angle that would be observed if it were not also affected by the average atmosphere. This angular spread is directly related to the mean-square fluctuations in refractivity at the lowest part of the ray path. The table entries are discussed in detail in the text. The HJ-77a (weighted) row should not be compared directly with the others. average, t h r o u g h regions of lower t h a n average refractivity. Thus atmospheric t u r b u l e n c e causes a n a d v a n c e of t h e a v e r a g e phase f r o n t s 2 F o r the e x p o n e n t i a l s c a t t e r i n g m o d e l considered here, the increasing r e f r a c t i v i t y f l u c t u a t i o n s w i t h d e p t h t h e r e b y cause a n u p w a r d tilt of the wave f r o n t a n d hence a r e d u c t i o n in the d o w n w a r d angle of r e f r a c t i o n caused b y the average a t m o s p h e r e , a t least for small v a l u e s of M. T h e more complete a n a l y s i s of H - 7 7 a shows t h a t the a n g u l a r offset r e m a i n s n e g a t i v e for all values of M for this model. Because the four c o m b i n a t i o n s of effects s As in our discussion of phase paths and refraction angles, an average phase front is a simple, unweighted average position of individual rippled surfaces of the same constant phase for different realizations of the turbulence.
g i v e n b y the d o u b l e - l e t t e r curve identifiers i n Fig. 1 are separable, the curves in Fig. 1 c a n be used to d e t e r m i n e the results for either a three- or t w o - d i m e n s i o n a l analysis, a n d for either e x p o n e n t i a l s c a t t e r i n g (3' = 2) or c o n s t a n t s c a t t e r i n g (3' = 0) in a n isothermal atmosphere. (By two-dimensional we m e a n t h a t there are o n l y vertical a n g u l a r f l u c t u a t i o n s ((852) = 0), a n d t h e p r o b l e m is r e s t r i c t e d to the a c t u a l a v e r a g e p l a n e of p r o p a g a t i o n . ) F o r example, the i n d i c a t e d c o m b i n a t i o n of all four curves gives the a n s w e r for a t h r e e - d i m e n s i o n a l a n a l y s i s of a n i s o t h e r m a l a t m o s p h e r e w i t h e x p o n e n t i a l s c a t t e r i n g . A t the o t h e r extreme, a t w o - d i m e n s i o n a l a n a l y s i s of a n i s o t h e r m a l a t m o s p h e r e in which the r m s s t r e n g t h of the t u r b u l e n c e is c o n s t a n t w i t h a l t i t u d e yields o n l y the M V curve.
401
TURBULENCE EFFECTS IN 0CCULTATIONS q~q=(M+I) -' L999
0,99
0.91
0.5
0.091
H- 77o
./
NEGATIVE-
7
IO-2
10-3
.f ,//7
/ ,/ /
H J - 7 7 o (unweighted) POSITIVE BRANCH ~,.,
u
..
/
.//
/
:
. . . . .
/~,/'~ //
/
NEGATIVE BRANCH
.J-7
fi~/
POSITIVE
-,, , / / / x.
//
/ /
"\// /// /
POSITIVE
/A/ I
o.,
/"
HJ-77o(weighted)
,
o.
,o.,
",o.
o,
M FIG. 2. Relative turbulenee-induced changes in average refraction angles in occultation experiments, as predicted in four publications for the same exponential model, and as summarized in Table I. The coordinates are the same as for Fig. I, but the effectsof limb curvature are omitted here. The H-77a and EH-77 curves correspond in the region of applieability of the latter, while
the remaining results exhibit fundamental differences which are discussed in the text.
There is a predicted singularity in the turbulence-induced angular bias at the abscissa M -- R/H, where the geometricaloptics signal intensity in the absence of turbulence is also singular. We plan to consider this interesting focal region in a later paper. COMPARISONS WITH OTHER RESULTS Average effects of turbulence in occultation experiments were introduced and have been considered subsequently by H u b b a r d and Jokipii (1975, 1977a, 1977b, referred to here as H J-75, HJ-77a, and HJ-77b, respectively). We a t t e m p t here a comparison of the angular results of H-77a, as given in (5), with corresponding predictions in H J-75, HJ-77a, and also in EH-77. We restrict this comparison to the particular exponential scattering model where H = Ht, so that 1' = 2 in (5). EWe discuss the HJ-77a, b result for their constant scattering model elsewhere (Haugstad and Eshleman, 1977).-] Comparison is made difficult b y the different nomenclature, model dimension-
ality, analytical methods, and averaging techniques used in the various treatments. However, it is clear that in each case an a t t e m p t is being made to predict what effects would actually occur for an isothermal average atmosphere with superimposed turbulence whose rms strength changes with height. Table I and Fig. 2 show the results of our comparison. There is a notable lack of agreement. We explain first just what is displayed and then discuss the differences. The first column in Table I gives the source and the second the original notation for the normalized angular bias as defined on the left-hand side of (5). Columns three and four present the different predictions for plotting, in terms of M and of Cq = (M 4- 1) -t. For the H-77a or equivalent (5) results, we have assumed N = 0 since limb curvature is not considered in the other treatments. Also, only the sum of the two fluctuation-gradient terms is plotted since the remaining mean-gradient effect is so small b y comparison. Note t h a t the only agreement in Fig. 2 is between the
402
ESHLEMAN AND HAUGSTAD
H-77a results and that from EH-77 for M << 1, the only region of applicability of the latter. As required from the F e r m a t argument, both start out negative for small M. The H J-75 result for Ht = H is similar in magnitude to that of H-77a, but of opposite sign. Several aspects of the H J-75 analysis are discussed in EH-77, where for example it is shown that to use this result at all in the comparison, one of the two definitions for the bending angle must be discarded. The principal complication in our conlparisons comes from consideration of the HJ-77a result. This is because the average angular bias in HJ-77a is formed using the intensity of the signal as a weighting factor, while the other analyses involve a simple unweighted ensemble average. (As discussed in Appendix B, we see no application for the weighted result.) In addition, HJ-77~ is a two-dimensional analysis while the H-77a and EH-77 are three dimensional. Thus the HJ-77a weighted results in Table I and Fig. 2 should not be compared directly with the others. However, we do plot this weighted answer as published in order to see the effects of removing the weighting and expanding the result dimensionally. The detailed nature of our interpretation of how to make these changes is explained in Appendix B. The resulting unweighted three-dimensional prediction is given in line three of Table I and is
plotted for comparison with the other results in Fig. 2. There are several areas of basic disagreement among the various results. At large values of M, the H J-75 and HJ-77a (unweighted) results agree with each other in magnitude but not in sign, while at small M the opposite is true. Of all of the investigations, only the HJ-77a analysis predicts a finite value of the ordinate as M approaches zero, while all saturate at a magnitude of unity for large M where, however, both signs are predicted. We also note in Appendix B that the H J-75 angular result along with the HJ-77a intensity and angular results for this model atmosphere, are not compatible on the basis of geometrical-optics energy conservation. Perhaps the most fundamental problem disclosed by the comparison is the different signs of the various predictions for the same effect, particularly at small values of M. This is the region discussed in the last section, where it is concluded that the angular bias must be negative. Thus we submit that in addition to the above problems, both the H J-75 and HJ-77a results fail to conform with the requirements of Fermat's principle. APPENDIX A: WAVE-OPTICALCOMPARISON The average angular bias has been derived from wave optics by Haugstad (1977b). For an exponential atmosphere with exponential scattering, his Eqs. (18) to (22) can be used to yield
0o
sin
(2P)duzduy-4-~/Inff~(~)Q(2P)du~du~ cos
,
:o
cos `) (P)du~duy
(A-l)
TURBULENCE EFFECTS IN OCCULTATIONS where
D ( Uz2
?~y2 I
"
(A-2)
,v+i ;Iu~fl
Nuy 2
Q =
-I(M+I)
~
.
(A-a)
(N+I)"
In these expressions q)(u) is the normalized turbulence power spectrum, uz, uu are respectively the vertical and horizontal components of the turbulence wavenumber u, and k is the radiation wavenumber. The double integral in the denominator of (A-l) multiplied by 27rl(vt2) is the definition of (5a2), while if u, 2 in this integral were replaced b y u~2 it would be (~52), but these angular effects are equal in our model where the turbulence is assumed to be isotropic. For the above multipliers, 1 is the equivalent tangential thickness of the atmosphere, approximately (RI-[) 1/2 when H and Ht are comparable, and (vt2) is the mean-square value of the fluctuations of the atmospheric refractivity at the lowest point on the ray path. Since P ~ k -1 -~ 0 for geometrical optics, (A-l) is readily seen to reduce to (5) under this condition. It is also a straightforward m a t t e r to obtain from (A-l) the dependence of the average angular offset on the radiation wavelength, as introduced in EH-77 and discussed more fully in Haugstad and Eshleman (1977). For turbulence power spectra that v a r y as the (--p) exponent of the wavenumber of the turbulence, the angular offset is proportional to X"/~-~ (X-1/6 for Kolmogorov turbulence) for 2 < p < 4 a n d l 0 < at < L0, where 10 and L0 are the inner and outer scales of the turbulence, respectively, and at is the radius of the principle Fresnel zone. APPENDIX B: INTENSITYWEIGHTING In HJ-77a,b, the average angular bias is formed using the concurrent signal intensity as a weighting factor, while in the other analyses the simple unweighted ensemble average is derived. We submit that the
403
unweighted result is the quantity that is needed in practice, but indicate below the relationship between the two. Angles are measured only in radio occultation experiments, through their effects on Doppler frequencies. Such experiments involve a point source of coherent radiation which, due to the weak scattering approximation which applies to all of the analyses discussed here, propagates through the atmosphere in going from source to observer along only one path at a time. (In ionospheric occultations, there m a y be multiple rays, but these appear to be deterministic in terms of ionospheric layers.) In practice, the measured Doppler frequencies are used without regard to intensities in deriving profiles of atmospheric temperature and pressure. (Signal intensities are used either alone to obtain independent profile information, or in conjunction with the Doppler-derived profiles to determine the distribution of absorbing regions in the atmosphere.) When turbulence is present, it follows that one would want to know to what extent the average of the measured unweighted Doppler frequencies or refraction angles (if indeed it is possible to form a meaningful average over a sufficiently small height interval in one or a few oecultations) m a y be affected by the atmospheric irregularities. Correspondingly, it would be important to be able to predict the expected magnitude and sign of the average biases in the derived atmospheric profiles due to the turbulence. In both eases, what is needed is a theory giving the nonweighted ensemble average of the turbulence-induced changes in the refraction angles. However, it is of interest to see in what way the different averaging procedure used in HJ-77a affects the comparisons in Table I and Fig. 2. We use the method of HJ-77b to find the intensity-weighted average angle of refraction, in order to show its relationship to the simple average used for the other results. F r o m HJ-77b Eqs. (3)
404
ESHLEMAN AND HAUGSTAD
and (5), (B-l)
E = ex+i~, kai = I m ( E * O E / O z } / ( E * E } ,
(B-2)
where E is the complex field at the zposition of the receiver, having log-amplitude x and phase q', k is the radiation wavenumber, and a1 is the intensityweighted average angle of refraction. Expanding x and ,I, to second order, x = xq -t- xl -t- x2 and ,I, = ,I,q -t- ,Iq + ,I% so t h a t I m (E*OE/Oz} --- e2x~([(1 q- 2Xl -b 2x2 q- 2x~ ~) (g-3)
to second order. B u t O~n/Oz is just /Caq, O~l/OZ is kay, etc., where aq, al, as are the zero-, first-, and second-order bending angles. Using also the simple average angle of refraction ~0 -= ( ~ + ~ + ~2) = ~ + ( ~ ) ,
I m (E*OE/Oz) + 2(a~x~)].
(B-4)
( E E * ) = e2x.[-1 + 2((×~ 2) + (x2))],
(B-5)
But since
we find t h a t (B-6)
through second order in the turbulence. I t follows from (B-6) t h a t if we have a normalized angular bias like the left side of (5) but expressed in terms of ai instead of a0, and wish to change it to one in terms of the simple average a0, then we can use 13/0 - - O L q ( ~q2
O/I __ OQt
+
2(M + 1)
O/q2
-r
L(~ ~)
The t e r m in square brackets in (B-7) can be evaluated b y the methods used in
.
q- 1
(B-S)
N-k
B y using the above results and (5), we find that, ai -- aq
aq '2
a,,
;][
2(211 + 1) X
X (O~ffq/OZ -'[- O~I~l/oZ -'[- O~2/OZ)])
ar = a0 ~- 2(a~×~)
H-77a to find in geometrical optics t h a t it equals
L ( M + 1) 2
-[-
(N + 1) i
(B-9)
for a n y value of % and in fact for a n y variation of the strength of the turbulence with altitude. I t is interesting in this regard t h a t the constant-scattering result given in HJ-77a [-Eq. (17)] is more nearly applicable to the exponential-scattering model t h a n is the H J - 7 7 a [-Eq. (26)], which was derived specifically for the latter model. I t also follows from (B-9) t h a t there is no difference for this intensity-weighted and normalized angular offset between a two-dimensional analysis and a threedimensional one where it is assumed t h a t there is no limb curvature, since the t e r m containing N vanishes in both cases, although for different reasons. The point of these last remarks is t h a t the HJ-77a [-Eq. (26)] intensity-weighted, two-dimensional, geometric-optics result can be transformed to the equivalent unweighted three-dimensional prediction using (B-7) and (B-S). Thus the HJ-77a unweighted column three e n t r y is obtained from the weighted e n t r y in Table I using these equations with N = 0 and ~ = 2. A corresponding translation has also been made in column four to express the unweighted result in terms of Cq. The relationship between weighted and unweighted angular offsets can also be used in an additional comparison of results. Two-dimensional, geometrical-optics expressions for signal intensities and refrac-
TURBULENCE EFFECTS IN OCCULTATIONS tion angles m u s t be related for c o n s e r v a t i o n of e n e r g y as (¢} - Cq = ¢(~
Cq [d(ao -- aq)/dh-]dx,
(B-10) where (~) is the ensemble average signal intensity. Using the m e t h o d s of t r a n s l a t i o n described above, we find t h a t the expressions for i n t e n s i t y a n d b e n d i n g angle effects given in H J - 7 7 a ['Eqs. (25) a n d (26)1 are n o t compatible on these grounds. W e s u b m i t t h a t these two effects can be o b t a i n e d using (5), (B-9), a n d (B-10) for t w o dimensions, so t h a t t h e y should be equal to (¢)
-
~bq =
(1/24)
C2¢q(1
-
~q)2
X (5 -~ 10¢q -]- 9¢q2), (ce~ -- C~q)D/H -- ½C2(1 -- Cq)a,
(B-ll) (B-12)
respectively, instead of the values given in H J - 7 7 a . (In these expressions, the H J - 7 7 a f o r m u l a t i o n is used on the righth a n d sides, where C ~ = - ( ~ } / a q 2 in the present n o t a t i o n . ) I t also follows f r o m these considerations t h a t the H J-75 a n g u l a r result is n o t consistent with the H J - 7 7 a i n t e n s i t y prediction.
405
ACKNOWLEDGMENTS We thank R. W. Lee, G. L. Tyler, J. B. Keller, and A. T. Young for helpful discussion. This work was supported by the Royal Norwegian Council for Scientific and Industrial Research and by the U.S. National Aeronautics and Space Administration. REFERENCES ESHLEMAN, V. R., AND HAUGSTAD, B. S. (1977).
Lowest-order average effect of turbulence on atmospheric profiles derived from radio occultation. Astrophys. J. 214, 928-933. (EH-77.) HAUGSTAD, B. S. (1977a). Turbulence in planetary occultations: I. A theoretical formulation. Submitted to Icarus. (H-77a.) HAUGSTAD, B. S. (1977b). Effects of an inhomogeneous background on radiation propagating through turbulent planetary atmospheres. Submitted to Radio Sci. HAUGSTAD,B. S., AND ESHLEMAN, V. R. (1977). On the wavelength dependence of the effects of turbulence on average refraction angles in occultations by planetary atmospheres. Submitted to Astrophys. J. HUBBARD, W. B., AND JOKIPII, J. R. (1975). Effects
of turbulence on radio-occultation scale heights. Astrophys. J. 199, L193-L196. (HJ-75.) HUBBARD, W. B., AND JoKIPII, J. R. (1977a).
Stellar occultations by turbulent planetary atmospheres: A heuristic scattering model. Icarus 30, 531-536. (HJ-77a.) HUBBARD, W. B., AND JOKIPII, J. R. (1977b). Turbulent scattering in an exponential atmosphere: A wave-optical solution. Astrophys. J. 214, 924-927. (HJ-77b.)