Magnetotelluric apparent resistivity—a comparative study of various definitions

Physics of the Earth and Planetary Interiors, 69 (1991) 8-13

8

Elsevier Science Publishers B.V., Amsterdam

Magnetotelluric apparent resistivity a comparative study of various definitions Yash Kant, R.P. Singh i

and N.K.

Goel

Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur 208 016, India

(Received 3 January 1991; revision accepted 17 May 1991)

ABSTRACT Kant, Y., Singh, R.P. and Goel, N.K., 1991. Magnetotelluric apparent resistivityIa comparative study of various definitions. Phys. Earth Planet. Inter., 69: 8-13. Apparent resistivity calculations have been carried out for three layered models using real, imaginary and absolute values of surface impedance. The apparent resistivity behaviours show characteristic features of the layered parameters using different definitions of apparent resistivity. It is concluded that the apparent resistivity behaviour using various definitions can be useful in qualitative interpretations in the field. The results have been illustrated graphically.

1. Introduction Electrical and electromagnetic methods do not c o n f o r m well with the response of lumped circuit elements such as resistance, inductance and capacitance. Inhomogeneities and stratification in the Earth's subsurface were found to give unexpected responses to incident electromagnetic waves. The overall response of the Earth's subsurface to magnetotelluric (MT) signals encompasses relatively much larger depths and is found to change significantly. The term ' a p p a r e n t resistivity' is generally used in electrical and electromagnetic methods, particularly in resistivity and M T methods for deducing layer parameters of the Earth. The characteristic behaviour of ' a p p a r e n t resistivity' arises primarily from the layered subsurface parameters, and at times is associated with characteristic oscillations and undulations. This has been discussed

1 Author to whom correspondence should be addressed.

as ' p a r a d o x e s ' (Satpathy, 1974) arising from interference effects (Morrison et al., 1969). The use and misuse of the term ' a p p a r e n t resistivity' has been discussed by Spies and Eggers (1986). In M T methods, the change in behaviour of apparent resistivity with frequency or period gives characteristic features which are used to deduce physical parameters of the layered Earth's subsurface. Cagniard (1953) for the first time defined the apparent resistivity in the processing of M T data, in the frequency domain, as ~ 1 !Z[2 Pa = co/t° ,

(1)

where ~0 = 2qrf, f is frequency, /% = 4 7 r x 10 -7 H e n r y m - 1 for most geological materials and Z is the surface impedance of the stratified Earth. In discussing alternatives to the basic definitions of the apparent resistivity, Spies and Eggers (1986) pointed out that the apparent resistivity is merely a normalizing procedure with little physical significance. Using the real a n d imaginary

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9

M A G N E T O T E L L U R I C A P P A R E N T RESISTIVITY

parts of the surface impedance, they further defined independent pairs of apparent resistivities as 2 [Real ( Z ) ] 2

(2)

2 [im(Z)]Z

(3)

p,, = ¢obt ~ and

Pa = cop,'---~

Further definitions are given in terms of real, imaginary and absolute values of the square of the impedance, i.e. 1 [ira(Z2)]

P" = ¢o!~---'o

p,, = ---~o 1221

(4)

(5)

and 1 [Real ( Z ) 2]

Pa = o.'#---o

effect, depends on the way the incident MT signal encounters the spatial distribution of the physical parameters of the inhomogeneously layered subsurface. For example, the definition of apparent resistivity of a two-layer subsurface in terms of either real or imaginary components of impedance will hold good only when Re ( Z ) << Im ( Z ) or Re ( Z ) >> Ira(Z). However, when Re ( Z ) = Im (Z), apparent resistivity is a measure of the combined response of the complex impedance represented by I z l 2 In the present work, we have carried out a comparative study of apparent resistivity behaviour for various three-layered models using the possible definitions as discussed above. These definitions show characteristic features of a particular model, which can be used as a rule of thumb in the interpretation of MT field data.

(6)

Definitions (1) and (5) are the same (Spies and Eggers, 1988). Definitions (1)-(4) are valid only in the sense that they yield the true resistivity in the case of a homogeneous half-space Earth model (Spies and Eggers, 1986). Definition (6), which uses the real part of the square of impedance, is not valid in the case of a homogeneous Earth model. Using the above definitions, Spies and Eggers (1986) have shown the variation of the normalized apparent resistivity with the normalized wave frequency for three-layer models. We have computed similar variations for different models, and have found that none of the four definitions show the characteristic features of a given subsurface contrast over the entire frequency range. Only one or two of the four valid definitions show the dominant characteristic features for a given subsurface model. In this frequency range, the responses of all the definitions are better delineated and may be helpful in the interpretation of field data. The apparent resistivity is basically the electromagnetic wave propagation response of the stratified subsurface. Therefore, the response depends on whether the signals first encounter a highly resistive or a highly conducting layer, and on the layer thickness. The apparent resistivity, in

2. Results and discussion

Using the above definitions (eqns. (1)-(4)) of apparent resistivity, we have made detailed computations for various three-layered Earth models. The variations of apparent resistivity as computed using these definitions are shown for varying periods of MT signals. In three-layered models, one may encounter four different types of model depending on the resistivity contrast of the layers. These models can be represented as K-, H-, Q- and A-type models. The details of apparent resistivity curves for (1)(4) are given below. 2.1. K-type models (Pl < P2 > [33)

The apparent resistivity defined by eqns. (1)-(4) behaves as shown in Fig. 1 for K-type models. The behaviours of apparent resistivity curves for (1), (2) and (4) are almost the same; only the curve for (3) shows a minimum in the lower period. The apparent resistivity variations shown using definitions (1)-(4) together (curves (1)-(4)) show the formation of two loops with a crossover point. The crossover point and the size of the loop vary with the model parameters. The apparent resistivity curves (first loop) show an increase in value

l0

¥. KANT

3.0

3.0 p~=lO

E

p2=500

E

pa=lO

~-m~500

m

p1=10 Q-mISO0 rn

~d=

--

d==tO

m

d==tO00

.....

E

p==500

E

pa=lO

I

2.0

~ d=

. . . . . D,

2.0

/ftoooo

I-CO Q£

/

~ ~.o

ET AL.

-=~.,,.,

<

/

t

i

~

".'.

;"// "'-.\',,~'~.

Ld

".

0.0 1.0 2.0 LOG PERIOD (See.)

Fsooo

~ 1.0 "-2000

q

-1.0

m

hi r~"

,"I D1, D2, D3, D4 /

q

-2.0

tO000

3.0

4-.O

Fig. 1. Variation of apparent resistivity curves obtained by various definitions with periods for K-type models.

with increase in period, whereas the apparent resistivity shown by curves for (1)-(4) (second loop) show a decreasing trend until they attain a constant resistivity value for the b o t t o m layer. The size of the loop depends on the resistivity contrast of the three-layered models. The curves for (1)-(4) show an almost constant value for a very small value of d E = 10 m (see Fig. 1). It is very clear from the illustrations that the analysis of apparent resistivity variations using definitions (1)-(4) together can give qualitative information on the existing subsurface model. In Fig. 2, we have c o m p a r e d the apparent resistivity behaviour using definition (3) and the conventional definition (1), for a three-layered K-type Earth model in which the thickness of the intermediate layer varies. The effect of the change in thickness of the intermediate layer is sharply distinguishable in the lower period on the curves for (3) c o m p a r e d with those obtained using the other definitions, especially the conventional definition (1) (Fig. 2).

-2.0

A p p a r e n t resistivity variations for H-type m o d els are shown in Figs. 3 - 5 . The apparent resistiv-

0.0 1.0 2.0 LOG PERIOD (Sec,)

3.0

4-.0

Fig. 2. Variation of apparent resistivity curves obtained by definitions (1) and (3) for K-type models with period for various values of d 2.

ity curves for (1) and (4) show very similar behaviour. The variations in the curves for (2) and (3) differ from those in the curves for (1) and (4). The resistivity and thickness contrast of the m o d 3.0 pl-50 ~-m~SO0 m

"" E

p,p= ~oo

~ 5000

D~..--;;;.;-~..?÷ ..."'...~::'5~ ..... /

-= 2.0

~,. 1.0 Z Ld n,,"

~ 0.0 D3

--p= = 2 ....... p= - 20

--'1.0

2.2. H-type models (Pl > P? < Ps)

-1.0

I ~ l l l ~ l

-2.0

-1.0

'll

J J I

J I

'

~'

~ I

0.0 1.0 2.0 LOG PERIOD (Sec.)

I

I

I,

I

3.0

I W J J I

4.0

Fig. 3. Variation of apparent resistivity curves obtained by various definitions for H-type models with period for two values of resistivity (2.0 and 20.0 fl m).

MAGNETOTELLURIC

APPARENT

11

RESISTIVITY

4.0

3.0

/It

~P

p1=I000 ~-rn ,~2000 m

p~=50 ~-m]1000 m E

p== 10

~d=

9 ~ ....... ::::::..-~

Ps=300

20

~, d=

p=-0.2

/":::" 2.0

....... ~ : 1 0 m 4==2000 m

(,q

",,:;',"" /"

.-'"

D

I.-Z ta.l n,"

I--

Z l.0 1.0

0

"I

0.0

....... de- 1000 m - d=-20000 m

,

0.0

,

,

I

-2.0

I

,

I

,

L

i

,

I

.--I \N

I

I

i

,

i

,

,

i

=

=

i

i

t

i

,

J

i

I

0.0 1.0 2.0 3.0 4.0 LOG PERIOD (See.) Fig. 4. Variation of apparent resistivity curves obtained by various definitions for H-type models with period for two values of d 2 (1 and 20 km).

-1.0

els affect the contrast of apparent resistivity curves for (1)-(4). The strong contrast of apparent resistivity corresponds to strong contrast of resistivity of models, as shown in Fig. 3 for resistivity values

3.0 p1-10 fl-m~2000

E I 2.0 E

p=-0.1

m

~d=

p=-200

J=

s"

/ >

pz=lO

, ,,-"

F-tO

~

E t= 3.0

.o".'°

."

n~'.-" /" "

-1.0 --2.0

l l l l l l I I l l l I I I l [ l l l l l l l l l l l l

-1.0

0.0 1.0 2.0 LOG PERIOD (See.)

I

3.0

4.0

Fig. 6. Variation of apparent resistivity curves obtained by various definitions for Q-type models with period, for two values of d 2 (10 and 2000 m).

( P l = 5 0 , P2=2.0, p3=300.0 fl m) and (P]-5 0 , 02 = 20.0, 03 = 300 fl m). Altering the thicknesses of the layers d a and d 2 yields similar apparent resistivity curves except for the shifting of the branch point (Figs. 3 and 4). As shown in Fig. 5, in the case of a thin intermediate layer, apparent resistivity curves for (3) show a distinct difference from those for (1), (2) and (4), whereas, in the case of a thick intermediate layer, the curve for (2) shows a distinct difference in the lower period. In the higher periods the curve for (3) shows distinct differences

1.0

from the other curves for greater thickness of the intermediate layer.

0.0

2.3. Q-type models (01 > 02 > P3)

W I---

~ 11.

~ -1.0 -2.0

-2.0

D a ~

....... de=tO m de-5000 m

,

,

,

,

I

-1.0

=

,

,

t

I

'

'

'

l

I

'

'

~

'

I

0.0 1.0 2.0 LOG PERIOD (See.)

~

~

I

~

~

I

3.0 . . . .

6.O

Fig. 5. Variation o f apparent resistivity curves obtained by various definitions for H-type models with period, for two values o f d 2 (10 and 5000 m).

Apparent resistivity curves for Q-type models are shown in Figs. 6 and 7. Apparent resistivity curves for (2) show distinct differences from the other curves, in the apparent resistivity contrast and thickness of the layers at lower periods. The large intermediate layer thickness results in a distinct feature in curves for (2) compared with those for the well-known conventional definition (1) and the other defintions (3) and (4) (Fig. 7).

Y. KANT ET AL.

12

3.0

4.0

DI

3.0

i

p,=1000 £~-rn~ 2000 m p==1o ~ d= p~=0.1

~%~.,::. ..,\

"N%, ',, N%.

2.0

1.0

,,,'. / ,',

U)

", ",.',,",e~-----~O00 m ",,','~ooo ',,'~5oo

\ \~ \ \ ~

p==50 p~=looo

~ E

~/,

,, , ,, _,- " ~ ,,/,."i" ~ ~f..~,i~v "'''~

t~

u

~

o

i/~

•

.< ~

-J 0.0

\

\

\ --1.0

i

-2.0

'

i

'

l

t

-1.0

i

i

,

§

\

\

\

I

R-

~,

"-.'.'-::..'-.. "-::"-';: ....

1000

',,.~--k------~ soo i

=

,

i

I

i

i

,

i

i

0.0 1.0 2.0 LOG PERIOD (Sec.)

'

'

i

i

I

3.0

i

,

,

i

I

-2.

4.0

Fig. 7. Variation of apparent resistivity curves obtained by definitions (1) and (2) for Q-type models with period, for d 2 values of 10, 500, 1000 and 2000 m.

2.4. A-type models (Pl < P2 < P3)

Figures 8-10 show the apparent resistivity behaviour for A-type Earth models. This behaviour,

-1.0

0.0

1.0

2.0

3.0

4.0

LOG PERIOD (Sec.) Fig. 9. Variation of apparent resistivity curves obtained by definitions (1) and (3) for A-type models with period, for various values of intermediate layer thickness (d 2 = 100, 1000, 2000 and 10000 m).

using definitions (1)-(4), is found to be almost the same, although the curve for (3) is qualitatively different from the other curves. Therefore, we 3.0

3.0 p,=lO Q-mtd~

.- " L ' S - ~

p== Ei E

,,,'),';;,!!,'is

0

~2000

p,3= 1 0 0 0

~ 2.0

2.0

I--

W

uJ ~ 1.0 D. .<

1.0

,2'

/'

Oj

. . . . . DI - -

0.0 -2.0

0.0

-1.0

0.0 1.0 2.0 LOG PERIOD (Sec.)

3.0

4.0

Fig. 8. Variation of apparent resistivity curves obtained by definitions (1) and (3) for A-type models with period, for various values of top layer thickness (d 1 = 100, 1000, 2000 and 10000 m).

,

-2.0

,

,

'

I

-1.0

t

~

=

[

I

0.0

'

i

r

,

I

1.0

'

'

i

,

I

2.0

'

i

~

[

I

3.0

'

D3

i

,

,

1

4.0

LOG PERIOD (Sec.) Fig. 10. Variation of apparent resistivity curves obtained by definitions (1) and (3) for A-type models with period, for various resistivity values of intermediate layer (P2 = 20, 50, 100 and 200 ~2 m).

13

M A G N E T O T E L L U R I C A P P A R E N T RESISTIVITY

have studied the apparent resistivity behaviour using definition (3) and compared it with the apparent resistivity behaviour using the conventional definition (1). In Figs. 8 and 9, we have shown the apparent resistivity behaviour for a three-layered A-type model with varying top (dr) and intermediate (d E) layer thicknesses. The effect of varying these thicknesses is clearly seen in the curves for (1) and (3); however, this effect is clearly different in the curves for (3) (Figs. 8 and 9). In Fig. 10, we have studied the change in behaviour of the curves for (1) and (3) with changes in the intermediate layer resistivity. The resistivity curves for (1) show realistic values of resistivity for the upper layer at lower periods. At higher periods they seem to attain the resistivity value of the bottom layer. The curve for (3) shows similar behaviour to that of the other curves at lower and higher periods. At intermediate periods it shows a minimum before reaching the resistivity of the bottom layer. The form of the minimum depends on the resistivity and thickness contrast of the layers.

3. Conclusions Detailed numerical results based on various definitions of apparent resistivity for various Earth models show similar or distinct behaviours in the broad band periods. In some of the models the set of apparent resistivity curves, or some of the curves defined by various definitions, show a characteristic variation from the curves based on other definitions for various three-layered models. This distinctive behaviour in the broad band or s h o r t e r / longer periods of apparent resistivity curves can

be used as an appropriate definition for a particular H-, K-, Q- or A-type three-layered model. The special features of apparent resistivity curves obtained by various definitions as discussed in this paper can be used as rules of thumb in the qualitative interpretation of M T data in the field. It is concluded that, together with the conventional definition of apparent resistivity given by Cagniard (eqn. (1)), other definitions (eqns (2)-(4)) may also be used to extend the interpretation tools of the M T method in deducing the resistivity and thickness of subsurface layers.

Acknowledgements T h e research work reported here was supported by a research grant from the D e p a r t m e n t of Science and Technology, New Delhi, awarded to one of the authors (R.P. Singh). We are grateful to the reviewers for their comments, which helped us to improve the original version of the paper.

References Cagniard, L., 1953. Basic theory of the magnetoteiluric method of geophysical prospecting. Geophysics, 18: 605-635. Morrison, H.F., Phillips, R.J. and O'Brien, D.R., 1969. Quantitative interpretation of transient electromagnetic fields over a layered half-space. Geophys. Prospect., 17: 82-101. Satpathy, B.N., 1974. A paradox in apparent resistivity measurements over a ground section with conductive substratum. Geophysics, 39: 93-94. Spies, B.R. and Eggers, D.E., 1986. The use and misuse of apparent resistivity in electromagnetic methods. Geophysics, 51: 1462-1471. Spies, B.R. and Eggers, D.E., 1988. Erratum on the use and misuse of apparent resistivity in electromagnetic methods. Geophysics, 53(12): 1637.