Reflection characteristics of EM waves over 
homogeneous Earth model 

R A M E S H P . S I N G H ( * ) - T A R K E S H W A R L A L ( * ) 

Received on J u l y 10th, 1980 

ABSTRACT 

T h e o r e t i c a l a n a l y s i s h a v e been c a r r i e d o u t t o i n v e s t i g a t e t h e e f f e c t 
of s u b s u r f a c e e l e c t r i c a l p a r a m e t e r s s u c h a s r e s i s t i v i t y a n d d i e l e c t r i c cons-
s t a n t ; a n d a n g l e of i n c i d e n c e on r e f l e c t i o n c o e f f i c i e n t s f o r vertically a n d 
h o r i z o n t a l l y p o l a r i z e d e l e c t r o m a g n e t i c w a v e s . It is s h o w n t h a t t h e reflec-
tion c o e f f i c i e n t s f o r vertically a n d h o r i z o n t a l l y p o l a r i z e d e l e c t r o m a g n e t i c 
w a v e s a r e s t r o n g l y d e p e d e n t on w a v e f r e q u e n c y a n d a n g l e of incidence. 
At s m a l l angles of i n c i d e n c e t h e Ri, a n d Ris seen to go t h r o u g h a mini-
n u m v a l u e w h i c h is seen to d i f f e r f r o m o n e m a t e r i a l t o t h e o t h e r . Re-
f l e c t i o n c h a r a c t e r i s t i c s of e l e c t r o m a g n e t i c w a v e s d i s c u s s e d in t h i s p a p e r 
c a n b e h e l p f u l in t h e a c c u r a t e i n t e r p r e t a t i o n of g e o p h y s i c a l d a t a . 

(*) D e p a r t m e n t of Geophysics, 
B a n a r a s H i n d u University, 
V a r a n a s i - 221005, INDIA. 



250 R A M E S H P. S I N G H - T A R K E S H W A R LAL 

R I A S S U N T O 

S o n o s t a t e e s e g u i t e a n a l i s i t e o r i c h e p e r i n d a g a r e s u l l ' e f f e t t o dei p a r a -
m e t r i elettrici di s o t t o s u p e r f i c i e , c o m e la r e s i s t i v i t à , la c o s t a n t e d i e l e t t r i c a 
e l'angolo di incidenza sui c o e f f i c i e n t i di r i f l e s s i o n e p e r o n d e e l e t t r o m a -
g n e t i c h e p o l a r i z z a t e v e r t i c a l m e n t e e o r i z z o n t a l m e n t e . Si n o t a che i coeffi-
cienti di r i f l e s s i o n e p e r o n d e e l e t t r o m a g n e t i c h e p o l a r i z z a t e v e r t i c a l m e n t e 
e o r i z z o n t a l m e n t e , d i p e n d o n o s t r e t t a m e n t e dalla f r e q u e n z a d e l l ' o n d a e 
d a l l ' a n g o l o di i n c i d e n z a . 

A piccoli angoli d ' i n c i d e n z a \'R,, e l'i?,, p a s s a n o a t t r a v e r s o u n v a l o r e 
m i n i m o che si n o t a d i f f e r i r e da u n m a t e r i a l e a l l ' a l t r o . 

Le c a r a t t e r i s t i c h e della r i f l e s s i o n e di o n d e e l e t t r o m a g n e t i c h e di cui 
si è t r a t t a t o in q u e s t a n o t a , p o s s o n o e s s e r e di a i u t o p e r u n a più a c c u r a t a 
i n t e r p r e t a z i o n e di d a t i geofisici. 

INTRODUCTION 

T h e 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 a n d i n t e r a c t i o n w i t h 
v a r i o u s typesof e a r t h ' s m a t e r i a l h a s been extensively s t u d i e d . 
The r e f l e c t i o n a n d a t t e n u a t i o n of e l e c t r o m a g n e t i c waves f r o m the 
e a r t h ' s s u b s u r f a c e is c u m u l a t i v e l y a f f e c t e d by t h e s u b s u r f a c e ma-
terial a n d t h e i r electrical p r o p e r t i e s . T h e s t u d i e s on e l e c t r o m a -
gnetic field a t t e n u a t i o n a n d c h a n g e of r e f l e c t i o n c o e f f i c i e n t w i t h 
f r e q u e n c y , p o l a r i z a t i o n a n d angle of incidence play a n i m p o r t a n t 
role inrevealing t h e s u b s u r f a c e electrical p r o p e r t i e s ( S h i v p r a s a d 
a n d Stotz, 1972; Kozaki, 1970; Wait, 1971; Philippe, 1973; Lytle, 
1974; Suzuki et al., 1975; E v a n s , 1977; Singh a n d Singh, 1979) 
e m p l o y i n g t r a n s m i t t e r receiver s y s t e m s w i t h large s e p a r a t i o n s . 
These f e a t u r e of received e l e c t r o m a g n e t i c waves a r e p r i m a r i l y 
c o n t r o l l e d by the dielectric c o n s t a n t (sr) a n d c o n d u c t i v i t y (<r) of 
the s u b s u r f a c e layer. 

In this p a p e r , c o m p u t a t i o n s havebeen c a r r i e d o u t to see the 
e f f e c t of p o l a r i z a t i o n a n d angle of incidence f o r general value of 
dielectric c o n s t a n t a n d conductivity, of s u b s u r f a c e c o n s t i t u e n t s . 



R E F L E C T I O N C H A R A C T E R I S T I C S O F E M W A V E S E T C . 251 

T H E O R Y OF S U R F A C E W A V E PROPAGATION 

We c o n s i d e r the 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 t h r o u g h 
an i n t e r f a c e s e p a r a t i n g t h e e a r t h ' s a t m o s p h e r e f r o m t h e e a r t h ' s 
s u r f a c e . The p l a n e 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 n g t h r o u g h a 
m e d i u m ca n b e r e p r e s e n t e d as s u p e r p o s i t i o n of t w o waves 
( S c h m u c k e r a n d J a n k o w s k i , 1972): 

i) T r a n s v e r s e electric m o d e wave, 

ii) T r a n s v e r s e m a g n e t i c m o d e waves. 

T h e 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 of these waves a t low 
a n d high f r e q u e n c i e s a r e significantly d i f f e r e n t . T h e e l e c t r o m a -
gnetic wave p r o p a g a t i o n at high a n d low f r e q u e n c i e s a r e signi-
f i c a n t l y d i f f e r e n t . The r e f l e c t i o n c o e f f i c i e n t at low f r e q u e n c i e s 
is such t h a t the direct a n d s u b s u r f a c e r e f l e c t e d waves cancel o u t 
a n d p r o p a g a t i o n is p r i m a r i l y as s u r f a c e waves. H o w e v e r , in the 
high f r e q u e n c y e l e c t r o m a g n e t i c waves, the m a g n i t u d e of t h e si-
gnal p r i m a r i l y d e p e n d s on the relative p h a s e of t h e d i r e c t a n d 
g r o u n d r e f l e c t e d waves r e a c h i n g the receiving site. T h e w o r k i n g 
f o r m u l a f o r the a m p l i t u d e of the r e f l e c t i o n c o e f f i c i e n t is w r i t t e n 
as ( J o r d a n a n d B a l m a i n , 1969). 

= sin 6,- + {(e, •—cos
2 6,)2 + ( j / e 0 to)

2} '4 

sin 0,- — {(er — cos
2 6,)2 + (a/e„ to)2} * [ 1 ] 

a n d 

Rv = 

\ Er
2 sin 0, + f — — V s i n 2 0 , ^ . 

f \ So to / \ 

j sr
2 sin 0, + ( — - — ^ "sin2 0, I ' + 

I \ £„« 

+ | ( e r — cos
2 0, j + 

£u w 

[ 2 ] 



252 R A M E S H P. S I N G H - T A R K E S H W A R LAL 

RESULTS AND DISCUSSIONS 

The p o l a r i z a t i o n d e p e n d e n t r e f l e c t i o n c o e f f i c i e n t s a r e governed 
by the c o m p o s i t i o n of the s u b s u r f a c e m a t e r i a l a n d c h a n g e with 
c h a n g i n g f r e q u e n c y of the e l e c t r o m a g n e t i c wave a n d t h e i r angle 
of incidence as d e p i c t e d by e q t i a t i o n s [ 1 ] a n d [ 2 ] , T h e f r e q u e n c y 
d e p e n d e n c e of Ri, a n d Rv f o r s o m e p r e m i n e n t c o n s t i t u e n t s of t h e 

Fig. l a , b, c, d, e) - F r e q u e n c y v a r i a t i o n of Rv a n d Rh f o r 
soil, shale, d o l o m i t e , g a b b r o a n d q u a r t z i t e . 

e a r t h ' s s u b s u r f a c e a r e s h o w n in Figures ( l a , b, c, d, e). F r e q u e n c y 
d e p e n d e n c e values of dielectric c o n s t a n t a n d c o n d u c t i v i t y of soil, 
shale, d o l o m i t e , g a b b r o a n d q u a r t z i t e have b e e n t a k e n f r o m (Scott 





FREQUENCY IN Hz 

Fig. 1 d 

, I I I ! 
io 3 I04 I05 10® | 0 7 I08 

FREQUENCY IN H z 

Fig. 1 e 

M 
|RV| 

Q U A R T Z I T E 



R E F L E C T I O N C H A R A C T E R I S T I C S O F E M W A V E S E T C . 255 

et. al., 1967; K a t s u b e a n d Collett, 1976). The r e f l e c t i o n c o e f f i c i e n t 
is n e a r l y one at low f r e q u e n c i e s a n d is seen to d e c r e a s e r a t h e r 
d r a s t i c a l l y w i t h increasing f r e q u e n c y . The r e f l e c t i o n c o e f f i c i e n t s 
Ri, a n d Rv f o r soil, shale, d o l o m i t e , g a b b r o a n d q u a r z i t e s h o w si-
m i l a r v a r i a t i o n s w i t h f r e q u e n c y a l t h o u g h t h e i r m a g n i t u d e s a r e 
d i f f e r e n t . T h e Ri, v a r i a t i o n d e p i c t e d by t h e solid c u r v e s h o w s mi-
n i m u m v a r i a t i o n at 30° a n d a s y s t e m a t i c i n c r e a s e in v a r i a t i o n 
w i t h i n c r e a s i n g angle of incidence 60° a n d 90°. H o w e v e r , in t h e 
case of Rv, t h e m a x i m u m v a r i a t i o n is seen at 30° a n d t h e varia-
tion is seen to d e c r e a s e w i t h i n c r e a s i n g angle of i n c i d e n c e 60" 
a n d 90°. T h e angle of incidence c o n s i d e r a b l y i n f l u e n c e s Ri, a n d 
Rv in the higher f r e q u e n c y range. C o m p u t a t i o n a l r e s u l t s to s t u d y 
the v a r i a t i o n of Rv with angle of incidence at t w o f r e q u e n c i e s 
104 a n d 10" Hz a r e p r e s e n t e d in Fig. 2. The m a g n i t u d e of Rv a t 
104 Hz is seen to increase w i t h the i n c r e a s e in angle of incidence 
f o r soil, shale, g a b b r o , d o l o m i t e a n d q u a r t z i t e . H o w e v e r , at a 
h i g h e r f r e q u e n c y e.g. 106 Hz, t h e r e f l e c t i o n c o e f f i c i e n t R i s seen 
to b e h a v e d i f f e r e n t l y . It decreases w i t h the i n c r e a s e in the angle 
of incidence till a t t a i n s a m i n i m u m , w h e r e a f t e r it i n c r e a s e s again. 
At a given f r e q u e n c y the angle f o r m i n i m u m Rt. is seen to c h a n g e 
f o r soil, shale, g a b b r o , d o l o m i t e a n d q u a r t z i t e . 

F u r t h e r c o m p u t a t i o n s of Rv a n d Rn w i t h the angle of inciden-
ce f o r dielectric c o n s t a n t values 3, 81 a n d resistivity values 103, 
104 o h m m e t e r at t w o f r e q u e n c i e s 104 a n d 105 Hz h a v e b e e n car-
ried o u t a n d a r e p r e s e n t e d in Fig. 3. At 104 Hz, Ri, is seen to de-
c r e a s e slowly with t h e angle of incidence. T h e i n c r e a s e in resisti-
vity f r o m 103 o h m m e t e r to 104 o h m m e t e r d e c r e a s e t h e magni-
t u d e of Ri,. T h e change of dielectric value f r o m 3 to 81 does n o t 
i n f l u e n c e the Ri, values f o r 103 o h m m e t e r resistivity value whe-
reas f o r 104 o h m m e t e r , its m a g n i t u d e is l o w e r e d slightly w i t h 
increase in dielectric c o n s t a n t . The b e h a v i o u r of Rv v a r i a t i o n at 
104 Hz is d i f f e r e n t f r o m t h a t of Ri,. It decreases, w i t h the i n c r e a s e 
in angle of incidence, till it a t t a i n s a m i n i m u m , w h e r e a f t e r it 
increases to a t t a i n a s a t u r a t i o n value. H e r e again i n c r e a s e in re-
sistivity, d e c r e a s e s the m a g n i t u d e of Rv. F o r higher dielectric 
c o n s t a n t value a n d the s a m e resistivity, the m i n i m u m o c c u r s at 
s m a l l e r angle of incidence a n d its m a g n i t u d e is lower as compa-
r e d to t h a t f o r lower dielectric c o n s t a n t . A f t e r the m i n i m u m , a 



256 R A M E S H P. S I N G H - T A R K E S H W A R L A L 

A N G L E OF INCIDENCE 
Fig. 2 - V a r i a t i o n of Rr w i t h angle of i n c i d e n c e f o r soil, sha-

le d o l o m i t e , g a b b r o a n d q u a r t i z i t e . 





258 R A M E S H P. S I N G H - T A R K E S H W A R LAL 

c r o s s over a n d reversal in t h e m a g n i t u d e t a k e s place, b u t t h e va-
lues a r e closed to each o t h e r . T h e c h a r a c t e r i s t i c f e a t u r e s of Ri, 
a n d Rv, v a r i a t i o n s d i s c u s s e d a b o v e a r e r e t a i n e d at 105 Hz b u t t h e 
v a r i a t i o n in m a g n i t u d e is large f o r Ri„ w h e r e a s f o r Rv, the mini-
n u m m a g n i t u d e is m o r e o r less or equal f o r all cases. 

CONCLUDING R E M A R K 

The s h a r p v a r i a t i o n in Rv w i t h angle of incidence a n d appea-
r a n c e of a m i n i m u m at a specific f r e q u e n c y a r e r e m a r k a b l e fea-
t u r e s which c o n t r o l the m a g n i t u d e of the received signal at the 
receiving site. T h e r e f o r e , the m e a s u r e m e n t of wave a m p l i t u d e 
seem to p r o v i d e a p o t e n t i a l p r o b i n g tool f o r the s t u d y of s u r f a c e 
f e a t u r e s of the e a r t h ' s c r u s t a l t e r r a i n . The large s e p a r a t i o n bet-
ween the t r a n s m i t t e r a n d receiver is essential to m e a s u r e t h e 
a n g u l a r v a r i a t i o n of r e f l e c t i o n c o e f f i c i e n t s f o r each of these po-
larizations. This m e t h o d of e l e c t r o m a g n e t i c e x p l o r a t i o n is widely 
u se d by p r o f e s s i o n a l p r o s p e c t i n g agencies a n d it is c a p a b l e of 
m a p p i n g the c r u s t a l s t r u c t u r e in e x t e n d e d a r e a s . 

A C K N O W L E D G E M E N T S 

One of u s (RPS) is t h a n k f u l to CSIR, New Delhi f o r t h e 
a w a r d of S e n i o r R e s e a r c h Fellowship. We a r e g r a t e f u l to Dr. J. 
Singh, Visiting P r o f e s s o r f o r his keen i n t e r e s t in this w o r k . 



R E F L E C T I O N C H A R A C T E R I S T I C S OF E M W A V E S ETC. 259 

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