forward modelling of electrical sounding experiments using convolution and a spreadsheet-stephen...
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8/10/2019 Forward Modelling of Electrical Sounding Experiments Using Convolution And A Spreadsheet-Stephen Sheriff.pdf
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Computers Geosciences
Vol. 18, No. 1, pp. 75-78, 1992 0098-3004/92 5.00 + 0.00
Printed in Great Britain. All rights reserved Copyright 1992 Pergamon Press Ltd
F O R W A R D M O D E L I N G O F E L E C T R IC A L S O U N D I N G
E X P E R I M E N T S U S I N G C O N V O L U T I O N A N D A
S P R E A D S H E E T
STEVEN D . SHERIFF
Departm ent of Geo logy, Universi ty of Mo ntana , Missoula , M T 59812, U.S.A.
(Received 23 July 1991; accepted 12 November 1991)
A bs tra ct- -F or man y geologic applications, programming a graphics interface or display takes much more
code th an the ac tual calculations. Usually, one can bypass this bottleneck by using the high-level language,
and b uilt in graphics, pr ovide d by comm ercial spreadsheets. A s an example, calculating expected app aren t
resistivities, using linear filtering and convolution, is rapid and convenient with a commercial spreadsheet.
The spread sheet 's immediate access to screen graphics facil i tates an intuitive grasp of electrical sounding
and saves much p rogramm ing t ime .
Key Words:
Electrical sounding, Spreadsheet, Geophysics, Convolution.
I N T R O D U C T I O N
B a s o k u r ( 1 9 9 0 ) p r e s e n t s a p r o g r a m f o r t h e d i r e c t , o r
i n v e r s e , i n t e r p r e t a t i o n o f e l e c t r i c a l s o u n d i n g e x p e r -
i m e n t s . B a s o k u r u s e s a n i t e r a t i v e m e t h o d t o a r r i v e a t
a s u b s u r f a c e m o d e l d i r e c t l y f r o m s u r f a c e m e a s u r e -
m e n t s o f a p p a r e n t r e s is t iv i ty . O t h e r d i r e c t a p p r o a c h e s
i n v o l v e l e a s t - s q u a r e s o r l i n e a r f i l t e r i n g ( G h o s h ,
1 97 1a ; Z h o d y , 1 98 9) . T h e s e a l l w o r k w e l l fo r m a n y
s i t u a t i o n s .
B a s o k u r ( 1 9 9 0 ) f o r s p a c e r e a s o n s d o e s n o t p r o v i d e
s u b r o u t i n e s f o r a g r a p h i c s i n t e r f a c e o r d i s p l a y o f t h e
r e s u l t s ; t h e p r o g r a m a l o n e i n c l u d e s m o r e t h a n 6 00
l in e s o f c o d e. M y e x p e r ie n c e w i th c o m m e r c i a l p r o -
g r a m s i s t h a t t h e g r a p h i c s i n t e r f a c e t a k e s m o r e c o d e
t h a n t h e g e o p h y s i c a l c a l c u l a t i o n s . G e n e r a l l y , t h e r e is
a n e a s i e r a n d f a s te r w a y t o p r o g r a m t h a t i s v a l u a b l e
p a r t i c u l a r l y f o r r a p i d l y s o l v i n g m a n y g e o l o g ic p r o b -
l e m s .
C o m m e r c i a l s p r e a d s h e e t s o f f e r a h i g h - le v e l p r o -
g r a m m i n g l a n g u a g e w i t h q u i c k a n d c o n v e n i e n t a cc e ss
t o a p r o f e s s i o n a l g r a p h i c a l d i s p l a y o f t h e r e s u l ts . T h i s
p a p e r s h o w s h o w t o p r o g r a m a s p r e a d s h e e t t o
c a l c u l a t e t h e e x p e c t e d a p p a r e n t r e s i s t iv i t y , u s i n g c o n -
v o l u t i o n , o v e r a n y h o r i z o n t a l l y l a y e re d m o d e l o f t h e
s u b s u r f a c e . T h e m e t h o d i s a p p l i c a b l e w i d e l y to
g e o l o g ic p r o g r a m m i n g s i t u at i o n s .
F O R W A R D M O D E L I N G O F A P P A R E N T
RESISTIV ITY
I n v e r s e m e t h o d s , s u c h a s B a s o k u r ' s ( 1 99 0 ), m a y b e
a g o o d s t a r t i f o n e o b t a i n s e l e c t ri c a l s o u n d i n g s f r o m
a g e o l o g i c a l ly u n k n o w n a r e a . H o w e v e r i n v er s e a p -
p r o a c h e s d o n o t i n c l u d e y e t a n i m p o r t a n t b u t f u z z y
p a r a m e t e r ; s u b s u r f a c e m o d e l s m u s t b e g e o l o g i c a l l y
r e a s o n a b l e . T h e o t h e r a p p r o a c h t o i n t e r p r e t i n g r e s i s -
t i v i t y d a t a i s t h e m o d e l i n g , o r f o r w a r d , m e t h o d .
F o r w a r d m o d e l i n g a l l o w s i n f o r m a t i o n o f v a r i a b l e
( a n d m a y b e q u e s t i o n a b l e ) v a l u e f r o m t h e l o c a l
a r e a a n d f r o m o n e ' s e x p e r i e n c e. A c c o r d i n g l y , i n
e x p e r i m e n t a l d e s i g n a n d i n t e r p r e t a t i o n , m o d e l i n g
i s a v a l u a b l e a p p r o a c h t o u n d e r s t a n d i n g e l e c t r i c a l
s o u n d i n g .
T h e f o r w a r d a p p r o a c h t o r e s i s ti v it y s o u n d i n g i n -
v o l v e s d r a w i n g a c r o s s s e c t io n o r m o d e l a n d c a l c u l a t -
i n g t h e e x p e c t e d a p p a r e n t r e s i s t i v it i e s . F o r e x a m p l e ,
u s e y o u r a v a i l a b l e w e l l c o n t r o l t o s k e t c h a m o d e l ,
t h e n c a l c u l a t e t h e e x p e c t e d r e s u l t s , a n d f i n a l ly d e s i g n
a n e f f i c ie n t f i e ld p r o g r a m t o t e s t t h e h y p o t h e s i s .
A l t e r n a t i v e l y , o n e c o u l d a d j u s t a m o d e l u n t i l t h e
t h e o r e t i c a l r e s u l t s f i t e x i s t i n g f ie l d m e a s u r e m e n t s .
F o r t h o s e u n f a m i l i a r w i th e l e c t r i c a l p r o s p e c t i n g t h e
a p p l i c a t i o n o f s p r e a d s h e e t p r o g r a m m i n g t o c o n v o l u -
t i o n m a y b e i n t er e s ti n g . S p r e a d s h e e t p r o g r a m m i n g i s
e f f i c i e n t , c o n v o l u t i o n i s a w i d e s p r e a d m a t h e m a t i c a l
m e t h o d . I n f a c t , g i v en a s y m m e t r i c a l f il t er , c o n v o l u -
t i o n a n d t h e g e o l o g ic o p e r a t i o n o f c r o s s - c o r r e l a t io n
a r e i d e n t i c a l . T h e r e s i s t i v i t y m e t h o d i t s e l f t a k e s
m i n i m a l i n t r o d u c t i o n , t h e n y o u a r e r e a d y t o p r o g r a m
y o u r s p r e a d s h e e t .
E l e c t r i c a l r e s i s t iv i t y s o u n d i n g s i n t r o d u c e a d i r e c t
c u r r e n t o n t o t h e g r o u n d v i a tw o e l e c t r o d e s . A s e c o n d
p a i r o f e l e c t r o d e s b e t w e e n t h e c u r r e n t e l e c t r o d e s
m e a s u r e s t h e e l e c tr i c a l p o t e n t i a l . T h e a p p a r e n t r e s i s -
t i v i ty , c a l c u l a t e d f r o m t h e s e m e a s u r e m e n t s a t t h e
s u r f a c e, i s a f u n c t i o n o f s u b s u r f a c e l a y e r i n g . I n
s o u n d i n g , o n e m o v e s t h e c u r r e n t e l e c t r o d e s f a r t h e r
a n d f a r t h e r a p a r t b e c a u s e w i d e r s p a c i n g s a l l o w
d e e p e r l a y e r s t o a f f e c t t h e s u r f a c e m e a s u r e m e n t .
T h r e e a r r a n g e m e n t s o f e le c t r o d e s a r e in u s e . F i r s t , t h e
W e n n e r a r r a y u s e s e q u a l l y s p a c e d e l e c t r o d e s , i t i s
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7 6 S . D . S r m R I~
c o n v e n i e n t f o r p r o fi l in g . S e c o n d , t h e d i p o l e - d i p o l e
a r r a y u s e s w i d e ly s e p a r a t e d p a i r s o f c l o s el y s p a c e d
c u r r e n t a n d p o t e n t i a l e l e c tr o d e s. T h e d i p o l e -
d i p o l e s e t u p i s c o n v e n i e n t f o r d e e p s o u n d i n g s . T h e
S c h l u m b e r g e r a r r a y , w i t h c l o s e l y s p a c e d p o t e n t i a l
e l e c t r o d e s a n d w i d e l y s p a c e d c u r r e n t e l e c t r o d e s i s t h e
m o s t u s e d a r r a n g e m e n t f o r s o u n d i n g e x p e r im e n t s in
g r o u n d w a t e r s t u d i e s .
C O M P U T I N G T H E A P P A R E N T R E S I S T I V I T Y
T h e m e t h o d f o r s o l v in g t h e o n e - d i m e n s i o n a l v a r i -
a t i o n o f r e s is t i v i ty w i t h d e p t h ( s o u n d i n g ) s t a r t s w i t h
t h e p o t e n t i a l o n t h e s u r f a c e :
p l 2 f f
( r ) = ~ r + K ( L ) J o ( L r ) d L
(1)
w h e r e
V ( r )
i s t h e p o t e n t i a l a t d i s t a n c e r f r o m a s o u r c e
t o a m e a s u r e m e n t , p i s t h e s u r f a c e r e s i s t iv i t y , I i s
c u r r e n t , L i s a d u m m y v a r i a b l e o f i n t e g r a t i o n , 3"0 i s a
B e s s e l f u n c t i o n ( o r d e r 0 ) , a n d
K ( L )
i s a cha r ac t e r i s t i c
f u n c t i o n o f t h e t h i c k n e s s a n d r e s is t iv i ty o f t h e s u b s u r -
f a c e l a y e r s . R e w r i t i n g E q u a t i o n ( 1 ) t o g i v e t h e a p p a r -
e n t r e s i s t i v i t y f o r a S c h l u m b e r g e r a r r a y ( O ' N e i l l ,
1975) yie lds :
o x
~ ( s ) = s 2 T ( L ) J ~ ( L s ) L d L .
(2)
T ( L )
= p [ 1 + 2 K ( L ) ] a n d i s c a l c u l a t e d e a si ly f o r a n
a r b i t r a r y n u m b e r o f l a y e r s a s s h o w n n e x t ; s i s o n e -
h a l f t h e s e p a r a t i o n o f t h e c u r r e n t e l e c tr o d e s. T h i s
i n t e g r a l is ti m e c o n s u m i n g t o e v a l u a t e b e c a u s e t h e
B e s s e l f u n c t i o n i n t h e i n t e g r a n d o s c i l l a t e s a n d c o n -
v e r g es s l o w l y f o r m a n y g e o l o g i c al l y r e a s o n a b l e
m o d e l s . T h e f o r w a r d p r o b l e m u s e d t o b e so l v e d b y
e v a l u a t i n g t h e s e i n t e g r a l s n u m e r i c a l l y , th e o n l y
m e t h o d i n u s e t o d a y i n v o l v e s a s u b s t i t u t i o n [ x = l n ( s )
a n d z = I n ( I / L ) ] t o c o n v e r t t h e m t o c o n v o l u t i o n
i n t e g r a l s s u c h a s :
Pa ( X) = T ( Z ) J l
(ex :)e2tx - z) dz . (3 )
T h i s i s a s t a n d a r d f o r m f o r l i n e a r f i lt e r in g . T h e
a p p a r e n t r e s i s ti v i ti e s a r e t h e o u t p u t , t h e r e s i st i v i ty
t r a n s f o r m f r o m t h e m o d e l i s t h e i n p u t , t h e r e m a i n i n g
t e r m i n t h e i n t e g r a n d i s t h e f i l t e r f u n c t i o n . O ' N e i l l
( 1 9 7 5 ) p r o v i d e s 2 0 a c c u r a t e c o e f f i c i e n ts ( T a b l e 1 ) f o r
t h e f il te r f u n c t i o n f r o m a S c h l u m b e r g e r a r r a y . W r i t -
i n g t h e l a s t e q u a t i o n i n a d i g i t a l f o r m s u i t a b l e f o r
p r o g r a m m i n g o r s p r e a d s h e e t u s e y ie l ds :
14
p a ( m ) = ~
b~Tm_~.
(4)
i = - 5
H e r e , t h e
bis
a r e t he f i l t e r coe f f i c i en t s . For each
c a l c u l a t e d v a l u e m , t h e r e s i s t i v i t y t r a n s f o r m T m
m u s t b e c a l c u l a t e d f o r e n o u g h s a m p l e s t o p r o v i d e
r o o m f o r f o u r t e e n l e a d i n g a n d f iv e t r a i l i n g f i lt e r
coe f f i c i en t s .
T h e r e s is t iv i ty t r a n s f o r m f o r a m o d e l c a n b e c a l c u -
l a te d a n u m b e r o f d i f fe r e n t w a y s ; a c o n v e n i e n t f o r m
( D a s a n d G h o s h , 1 9 7 4 ) i s :
T~ + p~_ ~ an h( L d i _ ~
T,_ I - (5)
1 + t anh( L d i t ) /P~ -
1
O n e s t a r t s w i t h t h e b o t t o m l a y e r i n t h i s r e c u r s i v e
r e l a t i o n a n d p r o c e e d s t o t h e t o p . L h a s u n i t s o f
i n v e r s e d i s t a n c e , P i a n d d~ a r e t h e r e s i s ti v i t y a n d
t h i c k n e s s o f t h e l a y e r , r e s p e c t i v e l y . F o r a t h r e e l a y e r
s i t u a t i o n s t a r t w i t h i = 3 t h e n T i = P 3 , th e r e s i s ti v i t y
o f t h e b o t t o m l a y er . N e x t c a l c u la t e th e t r a n s f o r m f o r
t h e s e c o n d l a y e r ( 7" 2) a n d t h e t o p l a y e r ( T ~ ):
P3 + P2 t anh ( L d2 )
/ 2
1 + P 3
tanh( Ld2) /p2
T 2 + P I t a n h ( L d l )
T~ - (6 )
1 + T 2 t anh( L d t ) / p~
T o u s e O ' N e i l l ' s (1 9 7 5 ) S c h l u m b e r g e r f i lt e r c o -
e f f i c i e n t s , c a l c u l a t e t h e r e s i s t i v i t y t r a n s f o r m f o r s i x
e v e n l y s p ac e d s a m p l e s p e r i n t e g e r p o w e r o f t e n. T h u s
T ~ i s ca l cu l a t ed s i x t i mes ( s i x va l ues o f L ) f o r each
o r d e r o f m a g n i t u d e o f e l e c tr o d e s p a c i n g .
P R O G R A M M I N G T H E S P R E A D S H E ET
F i g u r e 1 i l lu s t ra t e s o n e s e t u p f o r d o i n g t h e c o n v o -
l u t i o n o n a s p r e a d sh e e t . T h i s e x a m p l e i s f r o m B o r -
l a n d ' s Q u a t t ro P r o TM s p r ea d s h ee t r u n n i n g o n a n
I B M / M S - D O S TM c o m p a t i b l e c o m p u t e r . T h e t e c h -
n i q u e a p p l i e s t o a n y c o m p u t e r w i t h a s p r e a d s h e e t .
T h e n e c e s s a ry f o r m u l a e f o r c a l c u l a t i n g th e a p p a r e n t
r e s i s ti v i t y a r e s h o w n a t t h e b o t t o m o f F i g u r e 1 . T h e
f o l lo w i n g p a r a g r a p h s e x p l a i n th o s e f o r m u l a e .
M o s t s p r e a d s h e e t f e a t u r e s a r e i n t u it i v e a n d r a p i d l y
l e a r n e d . O n e c o n c e p t t h a t i s d i f f e r e n t f r o m o t h e r
p r o g r a m m i n g l a n g u a g e s i s r e l a t i v e a n d a b s o l u t e c e l l
a d d r e s s e s i n s p r e a d s h e e t f o r m u l a e . W h e n c e l l r o w s
Table 1 . O 'N ei l l ' s (1975)
filter coefficients; their sum
should be 1
b 5 0.003042
b 4 - 0 .001198
b_ 3 0.01284
b_ 2 0.0235
b t 0.08688
b0 0.2374
b I 0.6194
b 2 1.1817
b 3 0.4248
b4 - 3.4507
b 5 2.7044
b6 - 1.1324
b 7 0.3930
b 8 - 0 . 1 4 3 6
b9 0.05812
bl0 -0.02521
bll 0.01125
bl2
- 0.004978
b13 0.002072
bl4 -- 0.000318
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Forward modeling of electrical sounding experiments
Three la yer resist iv i ty exam ple. De pth and distance are
in m eters, resist iv i ty is in ohm-meters. Apparent
resist iv i ty is for the Schlumberger electrode arrangem ent.
from top to bottom :
Rho l = 100 d l = 10.0 ( th ickness o f layer 1)
Rho2 = 10 d2 = 20.0 ( th ickness of laye r 2)
Rho3 = 1000 ohm-meters
seed : 2 .556757
X inc : - 0 .166667
includes phase shi f t for O'Nei l l's c oeff ic ients
to get 6 samples per integer power o f ten
n L L d 2 > 2 3 0 ? t an h (L d 2)
15 1 .1E+00 2 .28E+01 1 .00E+00
16 7 .8E -O1 1 .6E+ 01 1 .00E+00
17 5 .3E -01 1 .1E+ 01 1 .00E+00
18 3 .6E -0 1 7 .2E +00 1 .00E+00
T2 Ld1>23 0tanh(Ld1) T1 b O'Nei l l convolve x
1 .0E+01 1 .1E+01 1 .00E +00 100 14 -0 .000318 10 0.0 1
1.0E+01 7 .8E+ 00 1 .00E +00 100 13 0 .002072 99.9 1
1.0E+01 5 .3E+ 00 1 .00E +00 100 12 -0 .004978 99.8 2
1.0E+01 3 .6E+ 00 1 .00E +00 100 11 0 .011250 99.4 3
exam ple formulas from the columns above:
L 1 0 ^ ( $ C $ 1 0 + ( A 4 3 $ C $ 1 1)
Ld2>230? @IF(B43*$F$7>230,230,B43*$F$7)
tanh(Ld2) (@EXP(C43)-@EXP(-C43)) / (@EXP(C43)+@EXP(-C43))
T2 ($D$8+$D$7*D43) / (1+($D$8*D43) /$D$7)
T 1 ( E 4 3 + $ D $ 6 * G 4 3 ) / ( 1(E43*G43)f$D$6)
Convo lve +H14*$J$43+H15*$J$44+H16*$J$45+H17*$J$46+ . .. +H34*$J$63
x 1 / ( 1 0 ^ ( 2 . 5 + A 4 3 $ C $ 1 1 ) )
Figure 1. Partial spreadsheet set up to calculate apparent resistivity over 3-layer model. Formulae at
bot tom are extracted from columns with similar labels. See text for discussion o f symbols and equations.
77
a n d / o r c o l u m n s a r e r e f e r r ed t o w i t h a n a s s o c i a t e d $
t h e $ s s p e c i f y t h a t t h e s e a r e a b s o l u t e r e f e r e n c e s. I f
y o u c o p y a n a b s o l u t e c e ll r e f e r e n c e to a n o t h e r c e l l i n
t h e s p r e a d s h e e t th e r o w a n d c o l u m n w i l l n o t c h a n g e .
W i t h o u t t h e $ s , th e a d d r e s s e s a r e r e la t i v e t o p o s i t i o n .
F o r e x a m p l e , i f w e r e f e r re d t o G 5 f r o m H 5 ( n o S s)
t h e m w e w o u l d b e r e f e r r i n g t o o n e c e l l l e f t o f H 5 .
C o p y t h e f o r m u l a i n H 5 ( w h i c h r e f er s t o G 5 ) t o t h e
c e ll K 9 a n d w h a t w a s G 5 n o w w o u l d b e d e s ig n a t e d
J 9 b e c a u s e J 9 i s o n e c e l l t o t h e l e f t o f K 9 . A l w a y s r e f e r
t o c o n s t a n t s i n a f o r m u l a w i t h a b s o l u t e a d d r e s s e s .
T o c a l c u l a t e t h e e x p e c t e d a p p a r e n t r e s i s t i v i t y f r o m
a t h r e e - l a y e r m o d e , p l a c e t h e l a y e r p a r a m e t e r s i n a
c o n v e n i e n t s p o t n e a r t h e t o p o f t h e s p r e a d s h e e t.
T h e s e m o d e l p a r a m e t e r s a r e t h e i m p u t t o t h e c a l c u -
l a t i o n , y o u c h a n g e t h e s e v a l u e s t o t r y a n e w m o d e l .
T h e m o d e l p a r a m e t e r s , t h e f i l t e r c o e f f ic i e n ts , t h e s e e d ,
a n d t h e i n c r e m e n t a r e u s e d w i t h a b s o l u t e a d d r e s s e s .
B e n e a t h t h e m o d e l p a r a m e t e r s ( F i g . 1 ) a r e t w o
c o n s t a n t s . T h e s e e d a c c o m m o d a t e s t h e p h a s e s h i ft o f
O ' N e i l l ' s ( 1 9 7 5) f i lt e r co e f f ic i e n ts ; i n p u t v a l u e s ( L s ) a t
s o m e p o s i t i o n x a r e a s s o c i a t e d w i th o u t p u t v a l u es
( a p p a r e n t r e s i s t i v i t y ) a t x + 6x. U s i n g 2 . 5 5 6 7 5 7 f o r
t h e s e e d w i l l p u t t h e f i r s t a p p a r e n t r e s i s t i v i t y v a l u e a t
a S c h l u m b e r g e r e l e c t ro d e o n e - h a l f - s p a c in g o f 1 m .
X i n c ( - 0 . 1 6 6 6 6 6 ) g i ve s t h e r e q u i si t e si x s a m p l e s p e r
i n t e g e r p o w e r o f t e n .
T h e l a b e l e d c o l u m n s i n F i g u r e 1 s t ep t h r o u g h t h e
c a l c u l a t i o n f o r t h e a p p a r e n t r e s i s t i v i t y , t h e y p r o g r e s s
f r o m l e f t t o r i g h t . F i g u r e 1 i n c l u d e s s o m e e x a m p l e
f o r m u l a e f o r t h e s e c o l u m n s . T h e f i r s t c o l u m n i s j u s t
t h e r o w n u m b e r , 3 8 r o w s p r o v i d e a p p a r e n t r e s i s ti v i-
t i es f o r e l e c t r o d e o n e - h a l f - s p a c i n g s f r o m 1 t o 1 00 0 m .
T h e s e c o n d c o l u m n , L , u s e s t h e s e e d , r o w , a n d X i n c
( u s e a b s o l u t e r e f e r e n c e s ) t o c a l c u l a t e L s f o r t h e
r e s i s t i v it y t r a n s f o r m . T h e t h i r d c o l u m n ( c r y p t i c a l l y
l a b e l e d : L d 2 > 2 3 0 ? ) t e s t s th e p r o d u c t o f L a n d t h e
t h i c k n e s s o f l a y e r 2 s o i t d o e s n o t g e t o u t o f r a n g e f o r
t h e e x p o n e n t i a t i o n o p e r a t o r ( @ E X P ) . T h e n e x t t w o
c o l u m n s c a l c u l a t e a h y p e r b o l i c t a n g e n t a n d t h e f i r s t
s t e p (T2 ) i n t h e r e s i s t i v i t y t r a n s fo rm [Eq . (6 ) ], r e sp e c t -
i v e ly . N e x t , t w o f a m i l i a r c o l u m n s y i e l d t h e r e s i s t i v i t y
t r a n s f o r m ( T 1 ) a t t h e s u r f a c e . Y o u o n l y h a v e t o t y p e
i n t h e f o r m u l a e ( b o t t o m o f F i g . 1 ) o n c e ; th e C O P Y
c o m m a n d w i ll m a p t h e m t o t h e o t h e r 3 7 ro w s .
T h e c o n v o l u t i o n f o r m u l a ( F i g . 1 ) a s s u m e s t h e 3 8
v a l u e s o f th e r e s i s t i v it y t r a n s f o r m a r e i n a c o l u m n
n e x t t o t h e f i l t e r c o e f f i c i e n t s (Ta b le 1 ). H o w e v e r , t h e
f i l te r c o e f f ic i e n ts m u s t h a v e a b s o l u t e a d d r e s s e s s o t h e y
c a n b e i n a n y c o n v e n i e n t s p o t . E n v i s i o n t h e f i l t e r
c o e f f ic i e n ts a s a o n e - d i m e n s i o n a l a r r a y ( o r c o l u m n )
n e x t t o t h e c o l u m n o f v a lu e s c o n t a i n i n g t h e r e s i s t i v i t y
t r a n s f o r m . T h e c o n v o l u t i o n i s t h e s u m o f p r o d u c t s
f o r e a c h f i lt e r c o e f fi c i en t w i t h t h e a d j a c e n t v a l u e o f
t h e r e s i s ti t v it y t r a n s f o r m . T h e o u t p u t v a l u e ( a p p a r e n t
r e s i s t i v i t y ) i s fi x e d n e x t t o O 'N e i l l ' s c o e f f i c ie n t b 0 . Fo r
e a c h o u t p u t v a l u e , t h e a r r a y o f f i lt e r c o e f f ic i e n ts
a d v a n c e s o n e s t e p a h e a d o f i ts p r e v i o u s p o s i ti o n . T h e
-
8/10/2019 Forward Modelling of Electrical Sounding Experiments Using Convolution And A Spreadsheet-Stephen Sheriff.pdf
4/4
7 8 S . D . S H E RI F F
R h o l =
R ho2 =
R ho3 =
100 dl = 10.0 ( thickness of l ayer 1)
I 0 d2 = 20 .0 (t h i ckness o f l aye r 2 )
1000 ohm- met e r s
Schlumber.qersoundinfl
10 ( 3 ( . . . . . . . . . . . . . . t ' . . . . . . . . . . . . . . . . . " . . . . . . .
' : . : : . ' ' i : . : . : . : . [ : . :. : . . ' ' : . :.
~ o . . . . . . . . . . . . . . i . . . . . . . . . . . . . . i . . . . . . . . . . . . . .
1 0 , . t. ; . . . . . . . . . . . . . . i . . . . . . . . . ' : . ~
:
. . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 . . . . . : . . . . . . . . . .
1 0 1 0 0 1 0 0 0
E l e c t r o d e s p a c i n g ( A I 3 ' 2; m e t e rs
Figure 2 . Com pleted spreadsheet se tup for interact ive mod -
eling. Rho's are resist ivi t ies assigned to layers, dl , d2 are
their thicknesses.
( F i g . 2 ) . E v e r y t i m e y o u a l t e r o n e o f t h e m o d e l
p a r a m e t e r s y o u g e t i n s t a n t g r a p h i c s- g r a t if i c a ti o n . O n
a 2 0 M H z 3 8 6 - 3 8 7 p e r s o n a l c o m p u t e r , t h e r e c a l c u la -
t i o n a n d g r a p h i n g t a k e s l e ss t h a n 1 s. T h i s s e t u p i s a n
e x c e l le n t t o o l f o r d e s i g n i n g s o u n d i n g e x p e r i m e n t s .
Y o u a l so c a n p l a y " w h a t i f " t o s e e i f s o u n d i n g w i ll
h e l p i n a p a r t i c u l a r s i t u a t i o n , i n t e r p r e t y o u r f i e l d
r e s ul t s, o r v e r if y a n o t h e r i n t e r p r e t a t i o n .
T h e r e a l ap p e a l o f p r o g r a m m i n g t h e p r o b l e m o n a
s p r e a d s h e e t i s i t s s i m p l i c i t y . T h e f o r w a r d c a l c u l a t i o n ,
w i t h g r a p h i c s , c a n b e s e t u p o n t h e s p r e a d s h e e t i n
a b o u t 2 0 m i n . B y u s i n g a b s o l u t e a n d r e l a ti v e a d -
d r es s es , t h e b u i l t - in m a t h f u n c t i o n s , a n d t h e l o g ic
o p e r a t o r s , a c o m m e r c i a l s p r e a d s h e e t i s a c o n v e n i e n t
a n d r a p i d h i g h -l e v e l p r o g r a m m i n g l a n g u a g e f o r a
w i d e v a r i e t y o f s c i en t if ic p r o b l e m s . T h e i m m e d i a t e
a v a i l a b i l i t y o f a w e l l d e s i g n e d g r a p h i c s s c r e e n r e -
m o v e s a t e d i ou s a n d t i m e - c o n s u m i n g st ep f r o m m a n y
g e o lo g ic p r o g r a m m i n g p r o b l em s .
f i rs t a p p a r e n t r e s i s t iv i t y , f o r a S c h l u m b e r g e r o n e - h a l f -
s p a c i n g o f 1 m , i s c a l c u l a t e d f o r t h e f i f t e e n t h r o w o f
t r a n s f o r m v a l u e s ( L = 1 . 1 3 9 6 ) . T h e l a s t v a l u e , f o r a
S c h l u m b e r g e r o n e - h a l f - s p a c i n g o f 1 0 00 m , i s a t r o w
33 ( L = 1 .1396 * 1 0 - 3 ) .
T h e f i n a l f o r m u l a i n t h e e x a m p l e c a l c u l a t e s t h e
e l e c t r o d e s p a c i n g a g a i n s t w h i c h t o p l o t t h e a p p a r e n t
r e s i st i v it i e s . T h i s l a s t f o r m u l a i s s i m i l a r t o t h e f o r m u l a
f o r L e x c e p t t h a t p a r t o f t h e s e e d r e s u l t i n g f r o m t h e
p h a s e s h i f t ( 0 . 0 5 6 7 5 8 ) i s o m i t t e d . P o s i t i o n y o u r
m e a s u r e d a p p a r e n t r e s i s t i v i t i e s , f r o m e x p e r i m e n t s i n
t h e fi el d , i n a c o l u m n a n d p l o t t h e m w i t h th e m o d e l ' s
a p p a r e n t r e s i s t iv i t i e s ( F i g . 2 ) . I f y o u w a n t t o e x t e n d
t h e c a p a b i l i t i e s o f t h e s p r e a d s h e e t , f i l t e r c o e f f i c i e n t s
f o r t h e W e n n e r e l e c tr o d e a r r a y a r e a v a i l a b l e i n G h o s h
( 1 9 7 1 b ) ; f o r d i p o l e - d i p o l e s o u n d i n g c o e f f i c i e n t s s e e
D a s a n d G h o s h ( 1 9 7 4 ) .
O n c e t h e s p r e a d s h e e t i s o p e r a t i n g p r o p e r l y , s e t u p
a g r a p h o f e l e c tr o d e s p a c i n g v e r s u s a p p a r e n t r e si s-
t i v it y a n d i n c l u d e a c o l u m n f o r d a t a . W h e n s a ti sf ie d
w i t h t h e g r a p h , i m b e d o r i n s e r t i t i n th e s p r e a d s h e e t
R E F E R E N C E S
Basokur , A . T . , 199 0, Microc omp uter program for the
di rect in terpre ta t ion of res is tivity so und ing data : C om-
puters & Geosciences, v. 16, no. 4, p. 587-601.
Das , U. C. , and G hosh, D . P. , 1974, The determ inat ion of
fi lter coefficients for the com pu tatio n of stan dar d curves
for dipole resist ivi ty sou nd ing over layered e arth b y
linea r digi tal f i ltering: G eoph ysical Prospecting, v. 22,
no. 4, p. 765-780.
Gho sh, D. P. , 1971a, The app l icat ion of l inear f il ter theory
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no. I , p. 192-217.
Gh osh , D . P ., 197 1b, Inverse fi l ter coefficients for the
com puta t ion of appa rent res is tivity s tan dard curves for
a ho rizon tal ly strat if ied ea rth: Ge ophy sical Prospecting,
v. 19, no. 4, p. 769-775.
O'N eil l , D. J. , 19 75 , Imp rove d l inea r coefficients for
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Zhod y, A. A. R. , 198 9, A new method for the autom at ic
i n t e r p r e t a t i on o f Sch l umber ge r and Wenner sound i ng
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