adg vol5 n02 vall 313_320.pdf


ANNALS OF GEOPHYSICS, VOL. 45, N. 2, April 2002

313

A note on the topographic distortion
of magnetotelluric impedance

Filippos Vallianatos
Technological Educational Institute of Crete, Branch of Chania, Crete, Greece

Abstract
Magnetotelluric surveys are prone to interpretation errors in the presence of rough topography, which may need to
be compensated for. In the present work we assume that the surface of the Earth is simulated by a single-valued,
twice differentiable function f (x, y). By appropriately expanding the surface magnetic field, we obtain the distorted
magnetotelluric impedance tensor in terms of an expansion depending on the external radii of curvature of f (x, y)
at the observation point and on skin depth. Based on first principles, an analytic estimation of the topographic
corrections of the magnetotelluric impedance tensor is obtained.

1. Introduction

A problem commonly appearing in mag-
netotelluric (MT) surveys is topographic dis-
tortion (Schnegg et al., 1983, 1986; Chouteau
and Bouchard, 1988; Fischer, 1989), especially
in the case of measurements carried out in
mountainous areas (e.g., Draget et al., 1980;
Kurtz et al., 1986; Vallianatos, 1995). Inter-
pretation errors may occur if the distortion is
not taken into account, or compensated for.
Telluric currents flowing parallel to the surface
of the Earth, converge beneath valleys and
diverge beneath hills. Therefore the current

density J and the associated MT field tend to
increase under a valley and decrease under a hill.
The reduction of this distortion is therefore an
important step toward the correct detection and
interpretation of subsurface conductivity
distributions. Several studies exist on the effect
of topography on MT measurements, most of
which adopt a numerical modeling approach
(Jiracek, 1973; Harinarayana and Sarma, 1982;
Wannamaker et al., 1985, 1986; Jiracek et al.,
1986; Xu and Zhou, 1997). Under static con-
ditions, the topographic effects have also been
calculated analytically, using Schwarz-
Christoffel transformations (Thayer, 1975;
Harinarayana and Sarma, 1982; Harinarayana,
1986). In the present short work, we present
a formulation to analytically estimate the to-
pographic effect, to first and second order cor-
rection terms. The proposed approach is based
on the simulation of the earth surface by a single-
valued, twice differentiable function and an
appropriate expansion of the magnetic field. The
estimated corrections depend on frequency and
the external radii of curvature of the earth surface
at the observation point.

Mailing address: Dr. Filippos Vallianatos, Technologi-
cal Educational Institute of Crete, Department of Natural
Resources Engineering, Laboratory of Geophysics and
Natural Hazards, 3 Romanou Str., Chalepa, GR-73133
Chania, Crete, Greece; e-mail: fvallian@chania.teiher.gr

Key  words  magnetotellurics  topography



314

Filippos Vallianatos

normal to the surface is (Lipschutz, 1989;
Morse and Feshbach, 1963)

where

In order to generalize eq. (2.1), we note that

and (2.2)

Combining eqs. (2.1) and (2.2), gives

or equivalently (2.3)

where the sum over j is implied (Einstein
notation). In the case of a simple one dimensional
(1D) structure C gij = Zo Cij where Cij are the
elements of a «normalized» surface impedance
tensor. It is straightforward to see that for a flat
surface C

ij
=

ij
. We note that E// and have

only components in the tangent plane, so that the
normalized surface impedance tensor C, being a
surface tensor in curvilinear coordinates, is
represented by a 2 × 2 matrix. Hereafter, in order
to simplify the mathematical formulation,  we
assume one dimensional structure with a surface
profile function f (x, y).

2.  Mathematical formulation

As physical basis for the analysis of any
magnetotelluric problem, consider Maxwell’s
equations for an isotropic, source-free region.
In the frequency domain, with the constituent
relations D = E, B = µ H, J = E incorporated,
these are (see for example Rokityansky, 1982):

 × H = ( i )E = E

 × E = i µ H

H = 0     and E = 0.

By combining the curl equations and using the
vanishing divergences, it can be seen (Jackson,
1975) that both fields satisfy the homogeneous
wave equation (Helmholtz equation) (Wait,
1987)

where the propagation constant is k 2= i µ .
The MT impedance, can be expressed as

a rank 2 tensor. If x and y are orthogonal
measurement axes on the Earth surface, the
tangential to the surface electric field variations
E// = (Ex, Ey)

T is directly related to the variations
of the two magnetic fields H// = (Hx, Hy)

T by the
frequency domain relation

or (2.1)

where Zij are the elements of the impedance
tensor (Kaufman and Keller, 1981).

Next, consider an Earth interface (air-
Earth interface) described by a profile func-
tion f (x, y), which is assumed to be single
valued and twice differentiable. The vector

+ =2 2 0
E

H

E

H
k

=
Ex

Ey

Zxx Zxy

Zyx Zyy

Hx

Hy

=E Z H// //

ˆ , ˆ ,N z= ( ) ( )[ ]x y f x y

, , .( ) = + ( )( )[ ]x y f x y1 2
1

2

N̂ H= ×( )Hx
y

ˆ .N H= ×( )Hy
x

Ex

Ey

Zxy Zxx

Zyy Zyx
x

y

=
×( )
×( )

ˆ

ˆ

N H

N H

E C
i ij

g

j//,
ˆ= ×( )N H

N̂ H×



315

A note on the topographic distortion of magnetotelluric impedance

the equation for the next term

With a solution that vanishes at Re(w)→−∞
(Bronson, 1973; Morse and Feshbach, 1963)

where

Continuing in way, we expand the magnetic field
in terms that include the expression

The arbitrariness of the latter term is partially
removed by the remaining Maxwell equation
∇⋅H = 0.

After extensive algebra and differential
geometry, we can write

For z approaching f (x, y) from below, the
component of E tangential to the surface is

We introduce the coordinate variables u = x,
υ = y, w = [z – f (x, y)]/D where D(ω) = i/k (ω),
in which the Helmholtz vector equation for H is

Expanding H in powers of D(ω) (see Appendix)

we obtain the recursive relation

For n = 0, the solution that vanishes at Re
(w)→−∞ is h

(0)
 = a

(0)
 (u,υ) exp(Φw). Inserting

this in the above recursive relation, we obtain

E E N N E//
ˆ ˆ= − ⋅( )

0

H D hi= ( )
=

∞

∑ i
i

62V a= − ∇ ⋅∇∇ ⋅∇( )Φ f f f (0)

3 2 22W = ∇ − ∇ ⋅∇∇ ⋅∇ +[Φ Φf f f f

2+ ∇ ⋅∇( )f ]a(0).

∑ Di
i

iα( ).

N a a N+ × + ∇ × + ×Km ˆ ˆ(0) (0) Φ

f a N N a× ∇ ⋅∇( ) − ⋅∇ ×( )

ˆ ˆ(0) (0)

  ( )
∂

∂

2

2

2

w
− = +( ) ( )Φ Φh W V(1) expw w .

  
h a

W V V
(1) (1) exp= + − +















( )( )
2 4 42

2

Φ Φ Φ
Φw w w

i
=

−
∇ × − ⋅ ∇ ×( )[ ]{ }ˆ ˆ .H N N H1

σ ωµ

  
∂
∂

∂
∂

2

2

2 2 2 2
w

D f f
w

− − ∇ + ∇ ⋅∇ +






Φ Φ ( )

∂

∂

∂

∂υ
2 2

2

2

2

2
0D

u
H+ +



 =Φ ( )

  
( )∂
∂

2

2

2

w
h− =Φ ( )n

∂
∂

2 2 2f f
w

h
n-

= ∇ + ∇ ⋅∇( ) −( )Φ 1

( )∂
∂

∂
∂υ

2
2

2

2

2u
h n-− + ( )Φ .2

E
N

a N a//
ˆ

ˆ= −( ) × + × +






−σ ωεi 1
D

(0) (1)



316

Filippos Vallianatos

where

the mean surface curvature and S is the extrinsic
curvature tensor of the surface (Lipschutz,
1989). Finally, by applying vector calculus we
obtain at w = 0

(2.4)

where is the projection operator
in the tangent plane, which in matrix terms is
given as

Since the operator P acts in the same way as the
identity applied on    × H, eq. (2.4) could be
written as

(2.5)

where S = P f P is the extrinsic curvature
tensor of the surface (Lipschutz, 1989). A
comparison of eqs. (2.3) and (2.5) leads to the
conclusion that the normalized impedance tensor
C = (C

ij
) has the form C = I

-
 + DC (1), where C (1)

is simply the traceless part of the extrinsic
curvature tensor of the surface S, given by the
expression C (1) = S – K

m
P. We note that the

eigenvalues of C (1) are 0 and ±  where

Since the aforementioned first order correction
is expressed by terms that include the second
spatial derivatives of f (x,y), it is apparent that
the topographic profile with linear slope
introduces a topographic correction given by a
second order term.

Following the same procedure and keeping
second order terms in our approximations, we
reach an expression for the normalized im-
pedance tensor

where

According to the aforementioned mathematical
formulation, it is obvious that a crucial parameter
in our expansion is the dimensionless product

I N̂

E// =

I P P P N H
*

ˆ= + ( )[ ] ×( ) =
1

D
D f Km

I S P N Hˆ= + ( ) ×( )[ ]0Z D Km

=
1

2

1 1

R Rmin max
( ).

C I C I= + ( )[ ]D D Tr C(1) (1)   ( )
2

2

4

( )[ ] = =Tr C(1) 2 22

= ( )[ ] ( ) +Tr f f f2 2 4 22

+ ( )f f f Km
6 2 22 .

( )D
f

x xi j

2

  K R R Tr Sm = + = ( ) =
1

2

1 1 1

2
( )

min max

E N H//
ˆ= ( ) × +i

D
Km

1 1( )

P I N N= ˆ ˆ

f f f f= ( )1
2

2 2

P N Hˆ+ ×( )f

    
P = ( )1 1

1

2
2

2

2 2

2 2
2

2

2

2 2 2

( )
( )

f

x

f

x

f

y

f

x

f

x

f

y

f

y

f

y

f

x

f

y

2

2



317

A note on the topographic distortion of magnetotelluric impedance

the external radii of curvature of the Earth’s surface
at the observation point, leading to accurate results,
as the dimensionless product

is small.

The formulation does not make any assumption
about the type of topographic profile (i.e. slopes,
height). Applying the presented first principles
approach, we estimate the  correction terms as
functions of the lateral second derivatives of the
topographic profile. In this way we have obtained a
general expression for the topographic correction of
the surface MT impedance tensor up to first and
second order correction terms. The first order
correction C (1) term is given by the traceless part of
the extrinsic surface curvature tensor S, while the
second order correction is just a scalar given in terms
of trace of C (1).

where x
1
= x and x

2
= y. It is expected that when they

are small the retention of only the first few terms in
the expansion should yield inaccurate results.
Alternatively, we can say that our approximation
should be valid as long as k

T
D( ) is much smaller

than unity, where k
T
is the wavenumber that

characterizes the roughness of our terrain.

3.  Concluding remarks

In the present preliminary work we present a
formulation to analytically approximate the
topographic distortion of the magnetotelluric
impedance tensor up to first and second order
correction terms. The proposed approach is based
on the simulation of the Earth’s surface with a single-
valued, twice differentiable function and an
appropriate expansion for the magnetic field. The
estimated corrections depend on frequency and on

( )D
f

x xi j

2

Appendix

Consider a time-harmonic plane wave incident from free space into a homogeneous earth surface
described by a terrain function f(x,y). In formulating the problem one assumes that the topography acts
as a scatterer. Since the currents flowing parallel to the earth’s surface converge beneath valleys and
diverge beneath hills, it is straightforward to describe the scattering from the topography by a volume
distribution of currents induced inside the «effective topographic scatterer». Maxwell’s equations
corresponding to this representation of the electric field is (Verdin and Bostick, 1992; Habashy et al.,
1993)

where the wavenumber k2
b
= i µ

b
and the topography factor Q

T
(r) are the terms describing the scattering

of currents due to the presence of topography.
The solution to eq. (A.1) can be expressed with a dyadic Green’s function which is governed by the

appropriate boundary conditions and satisfies the equation

where I
-

is the identity dyadic. For a homogeneous background, the solution to eq. (A.2) is (Kong,
1986)

     , where g (r, r ) is the scalar, Green’s function which satisfies

  and is given by

× × =E r E r r E r( ) ( ) ( ) ( )k Qb T
2

× × = ( )G r G r I r r( ) ( ) 'kb
2

( ) + ( ) = ( )r r, ' , ' 'g r r k g r rb2 2 (r r, 'g )) = =
e eik Db r r

r r

r r r r

'

'4 4

'

'
.

(A.1)

(A.2)

G r r I r r, ,' '( ) = ( )+
1

2k
g

b



318

Filippos Vallianatos

Thus the solution to eq. (A.1) is given by (Yaghjlan, 1980; Habashy et al., 1993; Chave and Smith,
1994; Kong, 1986)

     (A.3)

where Vt is the effective volume related to the topography and supports the appearance of Q(r). Applying
Ampere’s law to (A.3) gives an analogous integral equation for the magnetic field

     (A.4)

We seek to obtain H as an expansion of D. In order to accomplish this we make the following change
in variables x = x + Dq1, y = y + Dq2. Introducing the dimensionless variables q1 and q2 the scalar
Green’s functions

     where

possesses the expansion

     (A.5)

where (see Morse and Feshbach, 1963; Nietro-Vesperinas, 1991)

( ) = ( ) + × ( ) ( )
i

d g QTH r H r r' r r r, ' '
µ

1
b ( EE r'( )

Vt

).

( ) =
D

g
e

r r
r r

r r

, '
'

'

4

= ( ) + ( ) + ( ) ( )( )[ ]R x x y y f x y f x y, ,
/

2 2
2 2

1 2

' ' ' '

  
, ' ,( ) = + ( )

r

g r r
D

e

r

D

r r
f q q

0 0 0
2

3 1 2

1
1

2

1 10 ( ) ++

2

0 0
2

4 1 2

0
2

0
3

0
4

3
2

1 2

0
3

0
4

4 1 2
3

1

2

1 1 1

8

1 1 1 1

8

1 1
+ + ( ) + + + ( ) + + ( ) + ( ) =D

r r
g q q

r r r
f q q

r r
h q q O D( ) ( ) ( ), , ,

==

=

1 0

0 0
D

e

r
K D

r

n
n

n

( )

r q q q q f x y q f x y q0 1 2 1
2

2
2

1 1 2 2
2

, , ,( ) = + + ( ) + ( )( )[ ]
1 2

b

b

T

Vt

I
k

d g QE r E r r r r r E r( ) = ( ) + + ( ) ( ) ( )' , ' ' '( )( )1
2



319

A note on the topographic distortion of magnetotelluric impedance

When the latter expansion is substituted into eq. (A.5) we obtain

(A.6)

From the preceding equation we see that the magnetic field can be expanded in powers of the

parameter D = i/k according to           .

f x y f x1 , /( ) =

f q q f q f q qi i ij i j3 1 2,( ) = ( )( )

g q q f q f qi i ijk i4 1 2
1
3

,( ) = ( ) qq q f q qj k ij i j( ) + ( )
1
4

2

h q q f q f q qi i ij i j( ) = ( ) ( )4 1 2
2

,  .

d
e

r
K D Qb

r

n
n

n

T

Vt

( ) = ( ) + × ( ) ( )+

=
0

1

0

0

'H r H r r r' E r'( ( ) ) .

D
n

n

n

( )

= 0

H = h( ))

f x y f y2 ,( ) =

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