A method of full wave analysis with improved stability

PIdaet. Space Sci. VoL 30, No. 4, pp. 427-430, 1982

0032-0633/82/0404274)4503.00/0

Printed in Great Britain.

Pergamon Press Ltd.

SHORT PAPER

A METHOD OF FULL WAVE ANALYSIS WITH IMPROVED STABILITY

T.

Nygr~n

Department of Physics,

University of Oulu

SF-90570 Oulu 57, Finland (Received 9 February 1982)

ABSTRACT - A new version of a previously published method of f u l l wave analysis is presented. The purpose of the procedure is to avoid the numerical i n s t a b i l i t y which

in some cases is encountered

old method the r e f l e c t i o n r e c u r s i v e l y calculated

when the o r i g i n a l version is used.

In the

and inverse transmission c o e f f i c i e n t matrices

s t a r t i n g from the top

b e t t e r s t a b i l i t y is obtained,

if

of the l a y e r .

It

were

is found

that

the r e f l e c t i o n c o e f f i c i e n t s are calculated as

before, but the transmission c o e f f i c i e n t s are computed s t a r t i n g from the bottom of the layer and proceeding upwards in the d i r e c t i o n of the i n c i d e n t wave.

INTRODUCTION One of the main problems regions

in f u l l wave analysis

is numerical i n s t a b i l i t y in

where one or both of the two c h a r a c t e r i s t i c waves are evanescent.

d i f f i c u l t y arises,

because numerical

inaccuracies of one

The

c h a r a c t e r i s t i c wave

represent small f a u l t y amplitudes of the other mode. In an evanescent region the e r r o r grows r a p i d l y during the c a l c u l a t i o n and can e a s i l y swamp the s o l u t i o n . The s t a b i l i t y problem has been solved in several d i f f e r e n t ways (e.g. P i t t e way, 1 9 6 5 ; Inoue and Horowitz, 1966; Altman and Cory, 1969a, b; In

an e a r l i e r paper

method was introduced

(Nygr~n, 1981; h e r e a f t e r referred to applicable f o r

horizontally stratified 427

Wang, 1971).

as Paper I)

a

new

ionosphere and

428

Short Paper

v e r t i c a l propagation. Stable wave solutions were obtained by d i v i d i n g the ionosphere into a great number of thin elementary l a y e r s , of the wave f i e l d at every layer i n t e r f a c e ,

applying the

continuity

and calculating the r e f l e c t i o n and

inverse transmission c o e f f i c i e n t matrices r e c u r s i v e l y

s t a r t i n g from the top of

the l a y e r . The procedure in Paper I was tested using the results of Chessel (1971a, b). Since then the method has been employed in several papers dealing with coupling phenomena in sporadic E-layers (Nygr~n et a l . ,

1981; Jalonen et a l . ,

1981; Ja-

lonen, 1981). In a l l these tests and applications e i t h e r quite thin layers were studied fact

or only

the r e f l e c t i o n c o e f f i c i e n t s

were of i n t e r e s t .

that the c a l c u l a t i o n of the transmission c o e f f i c i e n t s

thick layers was not noticed

in the very beginning.

Therefore the

may be unstable in

Further investigations of

the matter f o r t u n a t e l y revealed that s t a b i l i t y can be gained by a small m o d i f i cation.

This new and more stable version of the f u l l wave method

is presented

in the following chapter.

THEORY The same notation as in Paper I is used. V e r t i c a l propagation

is assumed and

the h o r i z o n t a l l y s t r a t i f i e d ionosphere is divided into n thin s t r a t a . The height dependence of the r e f l e c t i o n c o e f f i c i e n t matrix R(i

is calculated as described

in Paper I . Evidently, i t

can be w r i t t e n T( i ) oo

T(i)oe ].[ u(O)(zo )Ox (1)

(i)(zi) Uey

u(O)(zo ) ey

T~ )

This equation defines the transmission c o e f f i c i e n t matrix of

the ionosphere

and equations

to the top of the i t h elementary l a y e r .

(7) and ( l l )

in Paper I ,

~(i) from the bottom Using this formula

G(i) and the downward propagating waves

can be eliminated from eq. (4) in Paper I , and the r e s u l t is

~(i).?(i).G(O)(zo) = ~(i+l).G(i+l)(zi)

,

(2

where = I l+r°°+P°re°

Po(l+ree)+roe

(3

t

Po(l+roo)+reo

l+ree+Poroe

[

l+R°°+P°Re°

Po(l+Ree)+Roe

and =

(4

L Po(l+Roo)+Reo

l+Ree+PoRoe

Short Paper

The f i e l d

~(i+l)

can t h e n be s o l v e d

0ilz fox sphere

to

is

components

taking

of

numerically

into

are

account

Using these matrix

~(i)

the

(i+l)th

n o t needed.

o

recursive

within

until

the

The m a t r i x

are

known',

the c y c l e ,

the

(i+l)th

starting

of

dependence o f

is

T(i+l)

t (i+l) eo

reached

is

ization

are o b t a i n e d

gating

of

is

(6) "

the r e f l e c t i o n the

layer

coefficient coefficient

and p r o c e e d i n g

values

coefficient

the upward p r o p a g a t i n g

in a straightforward

wave can t h e n be c a l c u l a t e d

wave

way from eq.

using

the by

are

•

components

of

obtained

t ( i +I) ee

The i n i t i a l

T (0) = 1 and T (0) = T ( 0 ) = O. The t r a n s m i s s i o n ee oe eo ~ ( n + l ) ionosphere is simply T = The f i e l d

can be

and the r e s u l t

the t r a n s m i s s i o n

from the bottom o f

z = zn

~(i+l)

iono-

llt ioo 1, oe

and the v a l u e s

height

the

the v a l u e s

lamel,

exp(_ik (i+1 e )~Zi+l )

the

level

layer.

o

)6Zi+l)

(5)

from the bottom o f

and ~ ( i + l )

To c o m p l e t e

formulas

can be c a l c u l a t e d

upwards,

matrix

elementary ~(i)

t h e phase s h i f t s

and R ( i + l ) ,

the form

o)

coefficient

0

matrices

in

t(i+l) ee

when T ( i ) ,

exp(-ik i+l

T(i+l) =

(2)

•

t(i+l) eo

transmission

the bottom

calculated field

the

from eq.

it iloo tfoe il u0zo]ox

=

u(i+1)(zi ) ey Here ~ ( i + l

429

the r e f l e c t i o n

matrix

for

of

T oo (0) = the whole

any i n c i d e n t

(I).

polar-

The downward p r o p a -

coefficients.

DISCUSSION The above method has now been used s u c c e s s f u l l y old

formalism

bility

is

has t u r n e d

evidently

to t h e a m p l i t u d e s the old that

is,

faulty field

one, in

characteristic continuity

of

of

ordinary

and

magnetic

field

of

of

critical

wave.

relevant errors

for

the

coefficients

transmission

coupling

where t h e

improved

unlike

proceeds

wave p r o p a g a t e s .

are a t t e n u a t e d

sta-

are p r o p o r t i o n a l

I n the new v e r s i o n , coefficients

a condition

and wave p o l a r i z a t i o n s

extraordinary needed. fail

The reason

cases

Therefore

rather

in

upwards; the

than ampli-

the calculation.

modes merge.

is

the

in which the

at

out

Therefore

encountered

b u t an a d d i t i o n a l t h e methods conditions.

where the two

T h i s means

by Budden ( 1 9 6 1 ,

sharp boundaries

waves,

in critical

is

become e q u a l .

Then, as p o i n t e d

the wave f i e l d s

necessarily

be u n s t a b l e .

The t r a n s m i s s i o n

caused by r o u n d i n g

the course indices

to

the upward p r o p a g a t i n g

the d i r e c t i o n

In the case refractive

here

of

the calculation

amplitudes in

out

as f o l l o w s .

in several

p.

c a n n o t be s a t i s f i e d non-progressive

presented

both

that 67),

the

by mere electro-

i n Paper

The same must be t r u e

the

I and

for

all

430

Short Paper

existing

procedures based on the c o n t i n u i t y

of the o r d i n a r y and e x t r a o r d i n a r y back,

since c r i t i c a l

coupling

waves.

of the wave f i e l d s This i s ,

and the concepts

however, not a s e r i o u s draw-

is a r a t h e r s p e c i a l

situation.

When comparing the computing times of the old and new v e r s i o n s , no e s s e n t i a l d i f f e r e n c e s are found.

Instead,

new v e r s i o n ,

values

since a l l

when c a l c u l a t i n g small,

it

and t h i s

will

If

s t o r e the r e f l e c t i o n

i n c r e a s e the t o t a l

ACKNOWLEDGEMENTS - The a u t h o r i s g r a t e f u l unstability

coefficients

the t r a n s m i s s i o n c o e f f i c i e n t s .

i s a d v i s a b l e to

data f i l e ,

the need of core memory i s much g r e a t e r of the r e f l e c t i o n

in the

must be at hand

the a v a i l a b l e memory is too

coefficients

t e m p o r a r i l y in a

computing time to s o m e d e g r e e . to Mr. L. Jalonen who p o i n t e d out the

in Paper I .

REFERENCES Altman,

C.

and Cory, H. (1969a).

t r o m a g n e t i c wave p r o p a g a t i o n . Altman, C.

The s i m p l e t h i n - f i l m Radio S c i .

and Cory, H. (1969b).

The generaTized t h i n - f i l m

e l e c t r o m a g n e t i c wave p r o p a g a t i o n . Budden, K.G.

(1961).

optical

method in e l e c -

4, 449.

Radio S c i .

optical

method

in

4, 459.

Radio Waves in the lonosphere.

Cambridge U n i v e r s i t y

Press,

Cambridge. Chessel, C . I .

(1971a).

coefficients

for

The numerical c a l c u l a t i o n

thin

highly

E a r t h ' s magnetic f i e l d . Chessel, C . I .

(1971b).

tion Inoue,

ionised layers

J. atmos, t e r r .

Results

transmission coefficients

thin

to s p o r a d i c - E r e f l e c t i o n s . Y.

and H o r o w i t z ,

S. (1966).

including

and t r a n s m i s s i o n the e f f e c t

of the

Phys. 33, 1515.

of numerical

for

of r e f l e c t i o n

highly

J. atmos,

calculation

of

reflection

i o n i s e d l a y e r s and t h e i r terr.

Numerical

and

applica-

Phys. 33, 1803.

solution

of f u l l

wave

equation

w i t h mode c o u p l i n g . Radio S c i . I , 957. Jalonen, L. (1981). Q u a s i - p e r i o d i c f r e q u e n c y dependence of Es- and E - l a y e r echo a m p l i t u d e s caused by mode c o u p l i n g .

J. atmos, t e r r .

Phys. 43,

1285.

I

Jalonen,

L.,

Nygr~n, T. and Turunen,

T. (1981).

echoes it~ the presence of a s p o r a d i c E - l a y e r . Nygr~n, T.

(1981).

coefficients ified

ionosphere.

Nygr6n, T . ,

A s i m p l e method

and f i e l d s

Jalonen,

for

Planet.

Properties

for obtaining

reflection

stratified

(1971).

Space S c i .

L. and Turunen,

Intermode

collisionless

ionospheric gyroPhys. 43,

T.

coupling

ionosphere.

1065.

and t r a n s m i s s i o n

an e l e c t r o m a g n e t i c wave in a h o r i z o n t a l l y

strat-

29, 5 2 l . (1981).

Anomalous m u l t i p l e

fo E in high gain ionograms recorded at Sodankyl~. 151. Wang, T . - I .

of

J. atmos, t e r r .

t r a c e s below

J. atmos, t e r r .

at the ion w h i s t l e r

Phys. 43,

frequencies

J. geophys. Res. 76, 947.

in a