JOURNAL C F RESEARCH of the National Bureau of Standards- D. Radio Propagation Vol. 670, No.5, September-October 1963

Reflection of VLF Radio Waves From an Inhomogeneous Ionosphere. 1 Part II. Perturbed Exponential Model

James R. Wait and Lillie C. Walters

C ontribution From Central Radio Propagation Laboratory, National Bureau of Standards, Boulder, Colo.

(R eceived April 10, 1963)

The obli que refl ection of radio waves from a continuously stratified ionized medium is cons id ered . In this p ap er t he medium is assumed to be isotrop ic. T he h eight profile of the effect ive cond uctivity is a Gaussian curve superimposed on th e (und istu rbed) exponentia l form. The refl ection coefficient is shown to be influenced by the vertical location of the Gauss ian perturbat ion . In some cases the magnitude of the reflection coefficient is increased whi le, in other situat ions, it is decrea cd. In nearly all cases, insofa r as p hase is concerned, t he pre ence of t he perturbation corresponds to a lowering of the reflection h eight .

1. Introduction

In a previous communication from the present authors [W ait and W alters , 1963], obliqu e reflection of (VLF) radio waves from a continuously stratified ionized medium was considered. The profile of the effective conductivity was taken to be exponen tial in form. Actually, this is a fairly good representation of the actual D layer of the ionosphere under day­time conditions. Henceforth, t,hat paper will be referr ed to simply as (J) .

It is the pm'pose of the present paper to consider profiles which are no longer exponential in form. Since t he objPctive is to gain insight into the mecha­nism of r efl ection from per turbed layers, a number of idealizations are ffittde . First, it is assumed, under quiescent condi tions, that the ionosph eric conductiv ity varies exponen tially with height. Then th e idealized perturbation is assumed to have a Gaussian form . Again , f oj' sake of simplicity, the earth's magnetic field is neglected as in (1 ) . This is well justified wh en considering effects which result from ionization in the lowest ionosphere.

2. Description of the Profile

The notation follows that used in (1) as closely as possible. Thus the undisturbed profil e, as a func­tion of h eight z, is defined by the conductivity parameter 1/1-(z) where

1 1 L(z) =z exp ({3 z) , (1 )

and L is a constant, (3 is a gradient parameter and z is the h eight above the r eference level z= O. Under th e iso tropic assumptio n, i t is known Lhat [WaiL, 1962]

L w(v+iw) 2 ' Wo

(2)

1 'rho work in this paper was suppor ted by the Advanced Research Projects Agency, W ashington 25, D.C ., u nder AHPA Order No . 183-62.

519

in Lerms of Lbe a ngular frequency w, collis ion fre­quency v, and plasma frequen cy woo At VLF, v> >w, and th erefore

w w~ L~- wh ere W,=-' (3)

Wr v

to within a very good approximation .

In general , it is seen that L~z) is proportional to

~~~) where N (z) and v(z) are the elect ron density

a nd collis ion frequency r egard ed as a fun ction of height. The constant {3, in the exponent, is a mea,s­ure of the sharpness of the gmdient. For example, (3 = 1 km- 1 means that th e ratio of w,(z) or N (z)/v(z) increases by 2.71 for each km of ve rtical height. F rom the recent work of Darringto n et al. [1962], Kane [1962], and Belrose [1963], it appears that {3 for an undistul'bed ionosph ere m ay b e in the range from 0.2 to 0.8. If t he level from about 60 km to 70 km is considered, it appeal' t hat {3 = 0.3 typifies many of t hese daytime D-Iayer profiles. A detailed s tudy of th e influence of cha nging {3 is to be found in (1). For this paper, (3 is chosen Lo be 0.3.

H aving specified our undisturbed profil e, we now wish to introduce th e perturbation. It is assumed that the collision frequency profile is unchanged whereas the ionization is Lo be increased by an amount Il N (z) wh ere

IlN(z) = IlNo exp [ _(2 DF)] (4)

and t1No, F and Dare co nstanLs. Clearly, the maximum value of t1N(z) is t1No which is located at z= F. Furthermore, th e thiclmess of this layer is 2D which is the vertical dis tance between the levels where t1 N( z) drops to IlNO/L

In order to estimate co rrectly the influence of this Ga,ussian shaped layer , it is necessary to assume

something about the collision frequency profile. A careful study of the recent li terature indicates that an exponential variation of v(z) with height z is not unreasonable. The form chosen here is

v( z) = Vo exp ( -~ z). (5)

where 13= 0.3 lml- 1• Therefore, the resulting con­ductivity perturbation has the form

f>N(z)=tlNo exp (f!. z) exp [_(Z- F)2J. (6) v( z) Vo 2 D

_I§ 7

a:: 6 w I-w ::;: 5 <! a:: ct 4

>-l-S> ·3 i= u 6 2 z 8

). , 15km (OR f' 20 kciS) 0= 2km A' 2 p. O,3km"

2 6

FIGURE 1. The undisturbed and disturbed conductivity profiles used in this paper.

(0) \ = 15 km

0.8 A = 2 C = 0..0.5

c _ - OOS o = 2 km

f3 = 0.,3 km- I

0.7

0.1

g:; 0,6 0/

I-Z ~

0..15 u 0.15 LL 0.5 LL W 0 u

0.2 0.2 z 0.4 Q --l-

I , I

, , u I W I \

..J I \

LL 0.3 I , w I \

a:: ____ <2:3 I -~' 0.3 /

/ 't". '--~--.... _-_ ......

I , \

0,2 / / . c = 0..4 / / \c = 0..4 - -~-----~ 'f-' -.

" I '/ 0, I

10

The complete profile, under these idealized con­ditions, is given by

where the right-hand side is propor tional to the effective conductivity of the m.edium as a function of height above (or below) the reference level at z= O. The coefficient A defines the strength of the perturbation. In fact,

A=f>No, No

where No is the electron density of the undisturbed profile at the reference level z= O. In this paper, as in (I ), L (0 )=7.5/'A where A is the wavelength in kilometers.

It is admitted that other ways to define a pertur­bation in the profile may be preferable. Here the electron density anomaly, for a given value of A, does not change with its vertical location F. Con­sequently, we may anticipate that the influence of this type of perturbation will be diminished at sufficiently low heights because of the increasing

, (b)

, \ , ,

I \ 160 , , I \s I ,

\~ 1

120 10 I UJ

\ = 15 km I \ I

A = 2 \ \ \

80 0= 2 km \ I

\. f3 = 0.3 km- I \ \

0> \ \ <lJ \

"0 40 \ \

\ \ \ a:: \

\ \ LL \ 0 0 \ \ W \ (f) \

« \

\ \ I \ 0.. \

-40 \ '-...-. \ \ ,

____ .9~ __ , , -80

0.,20. 0.20.

0..15 0.15 -1 20

0..1 (, = o..o.'j 0.,1

10 F 1 km

FIGURES 2a and b . The reflection coefficient as a function of the vertical location, F, of the Gaussian perturbation, fOl' various angles of incidence.

520

collision frequency. However, we shall see the problem is not quite this simple as other factors come into play.

A sketch of the profiles used is given in figure 1. The undisturbed profile is the ex.'Ponential form, while the disturbed profiles have the superimposed Gaussian "bump." The location of the "bump" for five typical profiles is specified by the appropriate value of F.

3. Results of the Calculations

The method used to calculate the reflection co­efficient R has been described in detail in (I). The quantities considered are the amplitude IRI and the phase of R for a vertically polarized plane wave incident at an angle whose cosine is C. The reflec­tion coefficient is evaluated in the free space region corresponding to Z----7 - co . However, it is important to remember that the phase is 1'ejel'l'ed to the level z=O.

The plan of the calculations is to vary the value of one parameter while keeping the others constant. To obtain a complete understanding of the various phenomena, an enormous number of calculations is needed . In order to keep the problem within reason-

0.6 r--.--,--,---,---r-,----r--,.----,--.--r-----,

0.5

a:: I-~ 0.4 U LL LL W 8 03

z o f= ~ 0.2 -.l LL W 0::

0. 1

~ 50

A; 15 km C ; 0 .2 A ; 2 (3 ; 0 .3 km-I

-10 0 10

able bounds and to reduce the expense of the compu­tation, only a limited number of cases was considered. These results are shown in graphical form in figures 2 to 6. In all cases 13= 0.3 km- l .

In figure 20, the amplitude of the reflection co­efficient is plotted as a function of F for A= 15 km (j= 20 kc/s), A = 2, D = 2 km, and C values varying from 0.05 to 0.4. Small values of Chel'e correspond to angles near grazing. For lono' distance propaga­tion of VLF radio waves, value of C near 0.1 are most important. For this case, it is interesting to note, when F is near or above zero, that IRI takes the same value as for the undisturbed profile. As the "bump" or perturbed layer is lowered, the reflec­tion coefficient first increases then decreases. Even­tually, as the "bump" is brought down to very low heights, IHI returns to its undisturbed value. The other curves for highly oblique incidence have a similar behavior. Thus, the "bump" may either im­prove or degrade the reflection. Presumably, at the lower heights the Gaussian layer is acting as an absorber whereas, at greater heights, it enhances the reflection. At the steeper angles of incidence, the situation becomes more complicated. It is probable that this results from interference between mul tiple reflected rays between the upper side of the "bump"

180

IQO 0

.I>

100 A ; l5 km c ; 0.2 A ; 2

0' B ; 0.3 km-I

~ 60

0::

LL o 20 w (j) « I 0...

- 20 '

-60

-100

-1 40_50 -40 -30 -20 -10 10 F, km

FI GURI,S 3a and b . The refl ection coefficient, as a f 1tnction of F, f or va"ious widths of the Gaussian pertubation when C = 0.2 (i .e., angle of incidence is 78°).

521

0.7

LL LL W 8 0.5

z o E 04 w --.J LL W a::

03

A = 15 km C = 0.1 f3 = 03 km- I

A = 2

(0)

~ -80 -0

a: ~ -100 w if)

.<1:

iE cl20 (b)

10 F. km

FIGURES 4a and b. The Tefleetion eoeifieient, as a f 1!nction of F, fOT various widths of the Gaussian perttlrbation when C = 0.1 (i.e., angle of incidence is 84°) .

0.6 ii f-

~ 0.5 ~ LL LL W 8 04

z 0

~ 03 w -.J LL w et::

0.2

0.1

Q60

(0)

-50

C = 0..1 A = 2 D = 2 km {3 = 0..3 km- I

10

- 20

- 40

C = 0.1 -60 A = 2 (b)

D = 2 km

en f3 = 0.3 km- I

OJ -80 '0

a:: LL 0-100 w (f) <{ I lL -120

-140

-160 -60 -50 -40 -30 -20 -10 10

F, km

FIGURES Sa and b. The Teflection coefficient, as a function of F , for vaTious wavelengths (from 10 to SO km).

and the eXPQnentially varying layer. Such an inter­ference phenomenQn becomes more pronounced at steeper incidence because the vertical compQnent Qf the wavelength is becoming comparable with typical values of F.

The phase o.f R is shown in figure 2b for the same conditions as in figure 2a. Again, it is apparent that, when the Gaussian "bump" is such that F is near 0 or above, the phase of R attains its undisturbed value. When the angle of incidence is highly oblique the phase undergoes an increase (i.e., decrease of lag)

522

as the "bump" comes down to lQwer heights. Suffi­ciently far below the reference level, the phase o.f R returns to. its undisturbed value. It is well to. note that as 0 becomes small (i.e., appro.aching grazing incidence), the phase of R is approaching - 180°.

FQr highly o.blique incidence the influence of the "bump" is to lower the effective height of reflection fQr the who.le range o.f F. Ho.wever, a very interest­ing phenQmenQn o.ccurs at steeper incidence. As can be seen in figure 2b, when 0 = 0.2 the phase undergoes a rather rapid change as F varies fro.m about - 22

0.6

(0)

0.7

c;: A ~ 0.5

f-z ~ ~ LL LL W 0 U

Z 0 A ~ 15 km i= u C ~ 0. 1 w

0.4 --' D ~ 2 km

LL W j3 ~ 0.3 km-I n:

0.3

01 50 10

-60

-80

0-

'" '0

0: -100 LL 0

UJ (f)

<t it -110

-14~50

A~4

(b) A ~ 15 km C ~ 0.1

D ~ 2 km f3 ~ 0.3 km- I

10 F, km

FJGUHJeS Ga and b. The reflection coe.fJicient, as [L jllnction oj 11, ./'01' VCLI' ioIL8 amplitudes A oj the pel'tlll'bat ion.

km to - 27 km. As 0 is increased fur ther there is an /l,pparent ciisco ntinui ty when the phase cba.nges by 360°. Such a change of 271' i'<tdi,Ul s is quite perm is­sible sin ce the ord illitte is arbitmry to within [my integral number of 271' radians. T lms, the pbase CUlYes for 0 = 0.3 and 0.4 co ulcllttwe been dmwn

I in the ntnge below - 160°. The CUI.TeS in figure 2b, even if they show nothi ng

) else, demonstmte that phase s hifts in reflection p henomena may h,tve sO lll e unusual cycle am­biguities.

The influence of the widLit of Lhe Gnussi,LI1 pertUl'­bation 01' "hump" is shown in figUl'e 3tL ror tbe amplitude IHI and in figure :3b for the p hase of R. Here A = 2, A= 15 km, and 0 = 0.2. The ampliLude curves show that, whell D is increased, th e overall influence of the layer becomes somewhat greftter.

, There is some tendency for the thinner byers (i .e., smaller D ) to be more efl'ect ive at greater heights. The correspon ding phase curves show that the thicker layers always produce a larger pbase change. Furthermore, as D exceeds 2 km, il, point is reached whcre the "360° jump" takes place.

The curves in figures 4a and. b are for the Salne conditions as figures 3a and b except that now 0 = 0.1, corresponding to nearer grazing incidence. The amplitude curves have a very similar shape. The phase curves are also similar except that the "360° jump" is no longer present. .

The wavelength dependence or the reflectIOn coefficient is shown in figures 5a and b. For these 0 = 0.1, A = 2, and D = '2 . The wavelengths chosen (10, 15, 20, 25. 30lem) correspond to frequencies of 30, 20, 15, 12 , and 10 kc/s. Qualitatively, the curves have a very similar shape. There is some tendency for the shorter wavelengths to be accom­pfwied by more pronounced changes. In all eases the "bump" acts as an absorber at low heights while it enhances thc reflection at greater heights.

Finftlly, in figures 6a and b , t.he influence of A , the relative magnitude of the anomalous rlectron density, is showll. As expected , the individual CLll'ves are similar in sllape wit.h t llC l ftrger values of A cOl'l'esponding to an increased change over the llndisLurbed values. J t is importftnt to note that tbe phase anomaly is almost directly proporLional to A.

4 . Final Remarks

The results gi ven here consti tute a small portion or extensive cOll1put,ttio ns dealing with reflection or waves from inhomogeneous media. In subse­quent parts to t.his series other types or profiles will be considered. Al 0, the app lications to the mode theory o[ VLF propftgation are to be desc]'ibed in some detail.

The authors wish to thank A. G. Jean and D. D. Crombi e for t heir helpful suggestions during t ho course of this work.

5. References Barrington, R ., B. Landmark, O. Hol t, a nd E . Thrane

(June 1962), Experimental studies of the ionospheric D-regiolJ, Repor t No. 44, Norwegian Defence R esearch Establishment, Kjeller, Norway.

Belrose, J .S. (1 9G3, Low-frequency propagation (to be submittrc\ to Proe. lEE, Londo n).

Kan e J . A. (1962), Re-evaluation of ionosphrric electron densit ies and collision frrqurnc irs drrived from rocket measurements, Chap. 29 in Radio Wave Absorption in the I onosphere (Pergamon Press , Oxford) .

'Vait, J. R . (1962) , Elect romagnet ic ",aves in strat ifi ed media (Pergamon Prrss, Oxford).

Wait, J . R., and L. C. Walters (May- June 1963), Reflection of VLF radio wa ves from an inhomogeneous ionosphere. Part 1. Exponentially varyi ng iso tropic model, J . Research NBS 67D (Radio Prop.) No.3, :361- 367.

(Paper 67D5- 283)

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