Basics of the magnetotelluric method · The inverse of the real part of k is also known as the skin...

Chapter 1

Basics of the magnetotelluric method

The magnetotelluric (MT) method is a passive electromagnetic (EM) technique for which the

electric E and the magnetic B fields are measured in orthogonal directions on the earth’s surface.

The MT theory to be described, along with appropriate data inversion procedures, allows the

determination of the resistivity distribution in the subsurface, on depth scales ranging from

a few tens of meters to hundreds of kilometers (Tikhonov, 1950; Cagniard, 1953). The MT

method is passive in the sense that it utilises naturally occurring geomagnetic variations as

the power source. The periodicity of the source as well as the resistivity distribution of the

subsurface have influence on the depth of information retrieval.

1.1 Source of signal

The Earth’s time-varying magnetic field is generated by two different sources, which strongly

differ in amplitude and in their time-dependent behaviour. The primary cause is magneto-

hydrodynamic processes in the Earth’s outer core (McPherron, 2005). However, for the MT

method the external, small-amplitude geomagnetic variations are of special interest as they in-

duce eddy currents and secondary magnetic fields in the Earth due to their transient behaviour.

The geomagnetic fluctuations range between periods of 10−3 s and 105 s (or between frequen-

cies of 103 Hz and 10−5 Hz) depending on their origin (Vozoff, 1991). Meteorological activity

such as lightning discharges produce EM fields with periods shorter than 1 s. The signals trav-

elling within the waveguide are also known as spherics. Between 0.5 − 5 Hz lies the dead-band

at which the natural EM fluctuations have a low intensity. MT measurements in this frequency

range usually suffer from poor data quality.

Magnetotelluric measurements presented throughout this work have a period bandwidth

of 10 − 104 s. This period range is due to interactions between solar wind and the Earth’s

magnetosphere and ionosphere. Solar wind is a continuous stream of plasma, carrying a weak

1

1.2. Behaviour of EM fields 2

Table 1.1: Typical classification scheme for geomagnetic pulsation utilised in the MT method.

Continuous pulsations Irregular pulsations

Pc1 Pc2 Pc3 Pc4 Pc5 Pi1 Pi2

T in s 0.2-5 5-10 10-45 45-150 150-600 1-40 40-150f in Hz 0.2-5 0.1-0.2 0.022-0.100 0.007-0.022 0.002-0.007 0.025-1 0.002-0.025

magnetic field. Constant pressure of the solar wind onto the magnetosphere causes compressions

on the sun-directed side and a tail on the night-side. Due to changes in density, velocity and

strength of the solar wind, the Earth’s magnetosphere is subject to varying distortions and

changes in the magnetic field. On the day side of the Earth, soft x-rays and ultraviolet light

cause ionisation of air molecules in the ionosphere at altitudes of 80−120 km. Solar heat induces

thermal convection of the ionised air molecules and thus establishes large-scale electric currents

acting as magnetic field sources. The changes in magnetic field strength produce geomagnetic

pulsations with periods of up to 600 s, as shown in Table 1.1. Longer periods can be divided

into variations from quiet and disturbed days. Quiet days comprise, for example, the solar-quiet

(Sq), lunar (L) variations and the solar flare effect (sfe). However, on disturbed days sudden

storm commencements (SSC) cause an increase in ring currents and bay anomalies, which result

in higher amplitude variations than the Sq (Campbell, 1997).

1.2 Behaviour of electromagnetic fields

The propagation of EM fields can be described by a set of four relations, called the Maxwell

equations. Given a polarisable and magnetisable medium containing no electric and magnetic

sources, the following relationships hold at all times for all frequencies (Feynman et al., 2005):

∇ · B = 0, (1.1a)

∇ · D = , (1.1b)

∇× E = −∂B

∂t, (1.1c)

∇× H = j +∂D

∂t. (1.1d)

Equation (1.1a) states that magnetic fields, defined by the magnetic induction B (in T), are

always source-free, and no free magnetic poles exist. The electric displacement D (in C/m2) is

solely due to a electric charge density (in C/m3). Faraday’s law in Equation (1.1c) shows the

coupling of an induced electric field E (in V/m) in a closed loop due to a time varying magnetic

field B along the axis of the induced electric field Figure 1.1. Curls of magnetic fields with a

1.2. Behaviour of EM fields 3

− ∂B∂t

E

Figure 1.1: Faraday’s law ∇× E = −∂B

∂t .

j

H

Figure 1.2: Ampere’s law ∇× H = j.

magnetic intensity H (in A/m) are caused by electric current densities j (in A/m2) and time

varying electric displacements. This is expressed mathematically in Equation (1.1d), and is

also known as Ampère’s law Figure 1.2.

The expressions for Faraday’s and Ampere’s law are initially decoupled. However, in the

presence of a linear, isotropic medium the material equations can be introduced:

B = µH, (1.2a)

D = εE. (1.2b)

The symbols µ = µ0µr and ε = ε0εr denote the magnetic permeability and the dielectric

permittivity, respectively, with µ0 = 4π · 10−7 V s/A m – magnetic permeability of free space and

ε0 = 8.85 · 10−12 A s/V m – dielectric permittivity of free space. Variations in µr and εr are

assumed negligible compared to variations in the bulk conductivity σ of rocks.

Furthermore, currents in the presence of an electric field can only flow if the media has a

conductivity σ (in S/m) other than zero:

j = σE. (1.3)

Equation (1.3) is also known as Ohm’s law. Using the material equations (1.2) and Ohm’s law,

the Maxwell equations (1.1) take on the form:

∇ · B = 0, (1.4a)

∇ · E =

ε, (1.4b)

∇× E = −∂B

∂t, (1.4c)

∇× B = µσE + µε∂E

∂t. (1.4d)

1.3. EM induction in a homogeneous half-space 4

1.3 Electromagnetic induction in a homogeneous half-space

The Equations in (1.4) can be rearranged to yield an expression for extracting information

about the subsurface. Taking the curl of Equation (1.4d) yields:

∇× (∇× B) = ∇ · (∇ · B) −∇2B = ∇×(

µσE + µε∂E

∂t

)

= −µσ∂B

∂t− µε

∂2B

∂t2. (1.5)

Since ∇ · B = 0, Equation (1.5) simplifies to the equation of telegraphy, which describes the

diffusive wave characteristics in a lossy (σ > 0) medium:

∇2B = µ

(

σ∂B

∂t+ ε

∂2B

∂t2

)

. (1.6)

Assuming a time dependence eiωt for B (with ω = 2πf the angular frequency in Hz, and

i =√−1 the imaginary number), Equation (1.6) yields the Helmholtz equation in the frequency

domain:

(∇2 − γ2

)B = 0, (1.7)

where γ denotes the complex wave number

γ2 = iωµσ − ω2µε = k2 − κ2. (1.8)

The undamped wave part, travelling at a speed of c = 1√µε , is described by κ, whose wavelength

is λ = 2πκ = 2π

ω√

µε = cf . In a homogeneous medium such as Earth, the low frequencies utilised

in MT make the propagation constant k (in m-1) play a more important role since σ ≫ εω.

This notion is known as the quasi-static assumption and leads to the diffusion equation for the

B-field:

(∇2 − k2

)B = 0. (1.9)

Following the method used above, the diffusion equation for the E-field can be derived in

similar fashion1:

(∇2 − k2

)E = 0. (1.10)

Equations (1.9) and (1.10) are fundamental for understanding the MT method in comparison

to other geophysical techniques. The diffusive behaviour of electromagnetic waves in the fre-

quency range of MT place this method between geophysical exploration techniques governed

by the wave equation (ground penetrating radar, seismics) and the potential methods (gravity,

magnetics, DC resistivity). For a geographical coordinate system x, y, z, with x,y and z

1Here, the non-existence of current sources in the subsurface yields:∇ · j = ∇ · (σE) = σ∇ ·E + E · ∇σ = 0.Additionally, in a homogeneous halfspace and layered (1D) Earth there are no free charges (∇σ = 0), hence:∇ · j = σ∇ · E = ∇ · E = 0.


0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

z/δ

E0 /

E(z

)

Depth dependent behaviour of the E−field

absolute attenuationreal partimaginary part

Figure 1.3: The depth-dependent behaviour of the E-field. The damping term e−ikz =e−iυz e−υz is complex (Equation (1.12b)) and real and imaginary parts show different amplitudedecay. For the absolute value, the E-field attenuates to e−1 strength of the surface value atthe skin-depth δ (see Equation (1.14)).

pointing north, east and vertically downwards, respectively, Equations (1.9) and (1.10) are not

only dependent on the conductivity σ and angular frequency ω, but are also depth-dependent

B = B(z). Therefore, the diffusion equations have the following solution:

B = B0 e−ikz + B1 eikz, (1.11a)

E = E0 e−ikz + E1 eikz. (1.11b)

B0, B1, E0, E1 are the electromagnetic fields at the surface of the Earth. B and E have to

vanish for z → ∞, and therefore B1 = E1 = 0. Thus, the solution in Equation (1.11) simplifies

to:

B = B0 e−ikz = B0 e−iυz e−υz, (1.12a)

E = E0 e−ikz = E0 e−iυz e−υz. (1.12b)

with k2 = iωµσ.

k =√

i√

ωµσ =√

ωµσ e−iπ4 = (1 + i)

√ωµσ

2= (1 + i) υ. (1.13)


Equation (1.12) illustrates the superposition of two characteristics of the field. The fields vary

sinusoidally with depth due to the e−iυz term, but also show a depth dependant attenuation due

to e−υz term. The inverse of the real part of k is also known as the skin depth or penetration

depth δ.

δ =1

ℜk=

√2

ωµσ. (1.14)

Since the magnetic permeability µ does not vary substantially in the Earth (see Chapter 1.2

on page 2) Equation (1.14) can be approximated as a formula for the skin depth in [ m]:

δ(T ) ≈ 500√

Ta. (1.15)

where a = 1/σa is the apparent resistivity or the equivalent average resistivity of the uniform

half-space (in Ω m) and T = 2π/ω is the period in s. EM fields are attenuated to a value of e−1 of

their surface amplitude at a depth δ(T ) (Figure 1.3). This relation is crucial as Equation (1.15)

states that the skin depth is only due to the bulk conductivity of the overlying material and

the period range used.

1.3.1 Apparent resistivity and phase

Expanding out the curl operator in full notation, Equation (1.4c) shows the following relation-

ship between the components of the E- and B-fields:

∂Ez

∂y− ∂Ey

∂z= −iωBx, (1.16a)

∂Ex

∂z− ∂Ez

∂x= −iωBy, (1.16b)

∂Ey

∂x− ∂Ex

∂y= −iωBz. (1.16c)

Since the B-field propagates vertically downward in the z-direction, the induced E-field does

not have a z-component (Ez = 0). The plane-wave assumption states that the inducing mag-

netic field only has horizontal components due to large distance to the source and that it is

therefore valid for periods up to 105 s. This also implies that the Bz component is equal to

0. Therefore, ∂Ex∂y and ∂Ey

∂x have to be zero, too. The time derivative of, for example, the Bx-

component at the surface of the Earth is equal to the spatial derivative of the Ey-component

with respect to z. Equation (1.12b) is the solution to the spatial derivative of Ey:

−∂Ey

∂z= kEy0

e−kz = −iωBx = −iωBx0e−kz. (1.17)

The magnetotelluric impedance Z (in m/s) is the ratio of the electric and magnetic fields

measured at the surface:

Z(ω) =Ey0

Bx0

= − iω

k= − iω√

iωµσ= −

√ω

µσ

√i. (1.18)

1.4. EM induction in a 1-D Earth 7

The ratio of the orthogonal components of the fields yield the same Z with opposite sign.

Reordering Equation (1.18) gives the resistivity of the half-space in units of Ωm:

ρ =µ

ω|Z|2 . (1.19)

In the frequency domain, Z(ω) is complex and has an associated phase φ:

φ = arg Z = arg

√ω

µσ

√i = arg

√ω

µσeiπ

4 =π

4= 45°. (1.20)

Thus, the phase is constant at 45°, regardless of the resistivity of the underlying half-space.

Schmucker and Weidelt (1975) introduced the complex Schmucker-Weidelt transfer function

C, which is directly related to the inverse of the propagation constant k:

C :=1

k=

Z

iω. (1.21)

1.3.2 Anisotropic half-space

In an anisotropic media, the conductivity σ is represented by a second-rank, 3-D tensor

(Heise et al., 2006). Ohm’s Law can be expressed in 3-D form (see Equation (1.3) for the

isotropic case):

jx

jy

jz

=

σxx σxy σxz

σyx σyy σzz

σzx σzy σzz

Ex

Ey

Ez

, (1.22)

where σ is symmetric (σij = σji) and non-negative definite (Weidelt, 1999). With these prop-

erties, the tensor can be diagonalised and expressed by three principal conductivities σ1, σ2, σ3

and three rotation angles that relate the orientation of the tensor’s principal axes x′, y′, z′to the reference frame x, y, z. Following the notation of Pek and Santos (2006), the rotation

angles are referred to as Euler angles αL, αD, αS :

σ =

σxx σxy σxz

σyx σyy σzz

σzx σzy σzz

= Rz(−αS)Rx(−αD)Rz(−αL)

σ1 0 0

0 σ2 0

0 0 σ3

︸︷︷︸

σ(x′,y′,z′)

Rz(αL)Rx(αD)Rz(αS).

(1.23)

where Rx and Rz are elementary rotation matrices around the axes specified by the sub-

script. The Euler angles αS , αD, αL are the ’strike’, ’dip’ and ’slant’ angles of the anisotropy

(Pek and Santos, 2006).


x

σ1

σ2

· · ·

σn

y

z

z1

z2

zn−1

zn

σN

Figure 1.4: Schematic model of a one-dimensional resistivity model with σ varying in thez-direction.

1.4 Electromagnetic induction in a one-dimensional Earth

For the following considerations we assume a horizontally layered Earth, where the conductivity

σ only varies with depth along the z-axis (Figure 1.4). Each layer contains components of down-

going and up-going energy, e.g. the x-component of the E-field in the nth-layer can be expressed

as:

Exn = an(kn, ω)e−knz + bn(kn, ω)e+knz. (1.24)

The magnetic field in the orthogonal y-direction would equate to:

Byn =kn

iωExn . (1.25)

Therefore, the transfer function C for the nth-layer is:

Cn(z) =Exn(z)

iωByn(z). (1.26)

Wait (1954) noted that the transfer function can be analytically determined for each layer.

Starting from the lowermost layer N which is equal to a homogeneous half-space, Wait’s recur-

sive formula calculates each corresponding layer on top of the nth layer:

Cn(zn−1) =1

kn

knCn+1(zn) + tanh [kn(zn − zn−1)]

1 + knCn+1(zn) tanh [kn(zn − zn−1)](1.27)

The apparent resistivity a of the layered media can be expressed in terms of the complex

transfer function C (compare to Equation (1.19):

a(ω) = |C(ω)|2 µ0ω. (1.28)


Accordingly, the phase φ is given as the inverse tangent of the ratio of the real and imaginary

part of C:

φ = tan−1

(ℑC

ℜC

)

. (1.29)

Both the apparent resistivity and phase are coupled due to causality by a dispersion relationship

(Weidelt, 1972), also known as Kramers-Kronig relationship.

φ(ω) =π

4− ω

π

∞∫

0

1

ω′2 − ω2log

a(ω′)

0dω′ (1.30)

Thus, the phase of a homogeneous half-space or layered Earth can always be calculated from

the apparent resistivity except for a scaling factor 0. The scaling factor plays an important

role in multidimensional conductivity distributions, which cause the apparent resistivity to be

arbitrarily shifted. Equation (1.30) illustrates that the phase for a frequency ω is dependent

on the apparent resistivity for all frequencies ω′, with the influence of ρa(ω′) largest for ω′ = ω

due to the scaling term 1ω′2−ω2 . Therefore, the phase anticipates the behaviour of the apparent

resistivity with period; but it cannot determine its absolute position. In multi-dimensional

situations, the dispersion relations have to be adjusted according to the dimensionality of the

subsurface (Fischer and Schnegg, 1993).

In a two-layered model the phase will be above 45° if the top layer is less conductive than the

bottom layer (equivalent to σ1 < σ2 in Figure 1.4). Similarly, the phase will be below 45° for

σ1 > σ2. For an n-layered Earth, the apparent resistivity curve has an asymptotic behaviour

towards the shallowest and deepest layer at either end of the period range. The resolution of the

middle layers depends strongly on their thickness and their resistivity, and cannot necessarily

be determined. Furthermore, conductive layers are more easily sensed than resistive layers

(Spies and Frischknecht, 1991). If ρ1 and ρ2 (ρ2 > ρ1) are the resistivities of the layers, the

resolution of the resistive layer is√

ρ2/ρ1 times that of the conductive layer (Orange, 1989). It

should also be mentioned that the entire information about the layered Earth is contained in

the electric field E (see Appendix A.2 on page 107).

1.5 Electromagnetic induction in a two-dimensional Earth

Figure 1.5 illustrates a simplified two-dimensional (2-D) model with a vertical resistivity bound-

ary striking in the x-direction. The resistivity boundary may represent a geological fault or

dyke. Electric currents on both sides of the interface must be conserved following the equation

of continuity that the normal components of current density must be continuous on both sides.

Ohm’s law (Equation (1.3)) can then be used to connect the current density to the electric field.

For the y-component:

jy1= σ1 · Ey1

= σ2 · Ey2= jy2

= const. (1.31)


Ey1> Ey2

σ1 < σ2z

x

y

Ex

Hy

Hz

Hx

Ey

Ez

jy = σ1 · Ey1= σ2 · Ey2

E-polarisation B-polarisation

Figure 1.5: Two-dimensional resistivity model with a lateral contact striking in the x-direction.The resistivity boundary separates two regions of differing conductivity σ1 6= σ2. The E-field isdiscontinuous across the vertical contact because of current preservation. The E-polarisationand the B-polarisation modes arise out of the two-dimensionality.

The discontinuity in the resistivities causes a jump in the electric field normal to the bound-

ary (Ey-component in Figure 1.5). The tangential components of the E-field in both the x

and the z-direction are continuous: Et1 = Et2 . Due to the assumption of homogeneous mag-

netic permeability, the normal and tangential components of the magnetic field B are also

continuous. The 2-D assumption requires that the strike length is significantly longer than the

skin-depth (Equation (1.14)) and all variations of the fields parallel to strike are zero ( ∂∂x = 0).

Furthermore, the EM fields are orthogonal and can be decoupled into a component with the

E-field parallel to strike and the B-field parallel to strike. The two modes are referred to as

E-polarisation (transverse electric (TE)-mode) and B-polarisation (transverse magnetic (TM)-

mode), respectively. The E-polarisation can be described through the following relationships:

∂Ex

∂y=

∂Bz

∂t= iωBz,

∂Ex

∂z=

∂By

∂t= iωBy,

∂Bz

∂y− ∂By

∂z= µ0σEx.

E-polarisation. (1.32)

There is no discontinuous behaviour for the E-polarisation as the Ey-component does not exist

in this mode. The currents are flowing perpendicular to strike in case of the B-polarisation:

∂Bx

∂y= µ0σEz,

−∂Bx

∂z= µ0σEy,

∂Ez

∂y− ∂Ey

∂z= iωBx.

B-polarisation. (1.33)


At the ground-air boundary, Ez = 0. From the discussion above it is clear that the discontinuity

in the electric field is σ2/σ1 for the B-polarisation across the boundary. The apparent resistivity

a has an offset of (σ2/σ1)2 (Equation (1.28)). Thus, the B-polarisation achieves a sharper

resolution of the lateral resistivity boundary due to the jump in apparent resistivity. However,

the resistivities close to the boundary are estimated too low for the less resistive region and too

high for the more resistive region. The E-polarisation is more stable regarding the apparent

resistivity estimates.

During MT field data collection, the coordinates of measurement rarely coincide with the

strike of a 2-D structure. Therefore, the MT impedance tensor Z ′ contains non-zero compo-

nents, unless it is rotated by an angle α, which rotates it into a coordinate frame parallel and

perpendicular to the strike of the resistivity boundary.

Z ′ =

(

Z ′xx Z ′

xy

Z ′yx Z ′

yy

)

= R

(

0 Zxy

Zyx 0

)

RT = RZ2DRT , (1.34)

with

R =

(

cos α sin α

− sinα cos α

)

. (1.35)

In Equation (1.32) the ratio of the vertical to the horizontal magnetic field components depicts

the lateral resistivity boundary. The geomagnetic depth sounding (GDS) transfer function T

is a complex vector showing a relationship between the amplitude and phase of the horizontal

inducing field and the vertical anomalous induced field for a given frequency ω. The relationship

with T = (Tzx, Tzy) is generally as follows:

Bz = TzxBx + TzyBy. (1.36)

For a laterally-uniform resistive Earth (for example, above a sedimentary basin) and with a

horizontal source field, there is no anomalous induced vertical magnetic field Bz, and hence the

transfer functions are zero. By contrast, close to a boundary between low and high resistivity

structures (for example, at the boundary between seawater and land), there is a large Bz field

and the transfer functions can be of magnitude one or more.

Parkinson induction arrows are a graphical representation of the GDS transfer function com-

ponents Tzx and Tzy (Parkinson, 1962). Parkinson arrows have a real (in-phase) and a quadra-

ture (out-of-phase) part. Lengths of the real (Mr) and quadrature (Mq) arrows are given by

Mr =(ℜT 2

zx + ℜT 2zy

)1/2, (1.37a)

Mq =(ℑT 2

zx + ℑT 2zy

)1/2. (1.37b)


Table 1.2: Entries of the impedance tensor Z vs dimensionality and isotropic conductivitydistribution. The entries become increasingly independent for more complex dimensionality.Z‖ and Z⊥ denote the E-parallel and E-perpendicular impedance, respectively.

Dimensionality

1-D 2-D 3-D

tensor components Zxx = Zyy = 0 Zxx = −Zyy Zxx 6= −Zyy 6= Zxy 6= Zyx

Zxy = −Zyx Zxy 6= −Zyx

impedance tensor Z

(0 Zn

−Zn 0

) (0 Z‖

Z⊥ 0

) (Zxx Zxy

Zyx Zyy

)

Orientation of the arrows is similarly determined by:

θr = tan−1

(ℜTzy

ℜTzx

)

, (1.38a)

θq = tan−1

(ℑTzy

ℑTzx

)

. (1.38b)

In Equations (1.38), angles θr and θq are clockwise positive from the x-direction (usually ge-

omagnetic north). When plotted on a map, Parkinson arrows should point toward regions of

high conductance and away from resistive blocks. Depending on the sign convention for e±iωt

in the Fast Fourier transform 180° may have to be added to θq to effectively reverse quadrature

arrows (Lilley and Arora, 1982). In any case, 180° have to be added to θr. Induction arrows

are only associated with the E-polarisation in a 2-D Earth. The B-polarisation contains no

vertical magnetic field components, but does contain a vertical electric field within the Earth.

1.6 Electromagnetic induction in a three-dimensional Earth

In a three-dimensional environment all the components of the electric and magnetic field are

linked to each other. The impedance tensor or MT transfer function Z links the corresponding

horizontal components:(

Ex

Ey

)

=

(

Zxx Zxy

Zyx Zyy

)(

Bx

By

)

(1.39)

In MT surveys, the magnetic field is measured in all three directions and most often the

electric field in the two horizontal directions x and y. Hence, the full impedance tensor can be

calculated. Z is complex and can be separated into a real (X) and quadrature (Y ) part:

(

Zxx Zxy

Zyx Zyy

)

=

(

Xxx Xxy

Xyx Xyy

)

+ i

(

Yxx Yxy

Yyx Yyy

)

. (1.40)


σ1σ2

z

x

y

Figure 1.6: Three-dimensional resistivity model, where the impedance tensor Z is fully assignedand all components are independent of each other.

Z has both magnitude and phase.

aij (ω) =1

µ0ω|Zij(ω)|2 , (1.41)

φij = tan−1

(ℑZij

ℜZij

)

. (1.42)

In order to avoid confusion with Equation (1.28) on page 8, it should be noted that the

Schmucker-Weidelt transfer function C for a one-dimensional Earth equals the absolute val-

ues of the off-diagonal components of the impedance tensor scaled by Earth property values:

|Zyx1D| = |Zxy1D

| = iωµ0C. (1.43)

Only in cases where the conductivity distribution has a high degree of symmetry, the component

phases will be simply related to phase differences between the linearly polarised components

of the horizontal electric and magnetic fields (Caldwell et al., 2004). In other cases, where the

regional responses are distorted, the amplitude and phase of the components of the regional

impedance tensor are mixed among the components of the observed impedance tensor.

The impedance tensor Z gives essential information about the dimensionality of the subsur-

face (Table 1.2). If all four components are independent of each other, the subsurface is most

likely three-dimensional, unless the impedance tensor can be rotated to yield Zxx = −Zyy. A

geological scenario given in Figure 1.6 will usually produce a three-dimensional impedance ten-

sor. However, for long periods and a large skin depth the body will act as a local near-surface

inhomogeneity, which gives a non-inductive (galvanic) response. This produces a frequency-

independent distortion called static shift (Simpson and Bahr, 2005).

Chapter 2

Small-scale and regional resistivity

distribution

This chapter deals with the effects of small-scale and regional heterogeneities in the Earth’s crust

and their influence on the measured impedance tensor. Distortion and dimensionality analysis

are very crucial steps between time-series analysis and modelling of the impedance tensor data.

Here, I present recent distortion removal methods and categorise them according to their initial

assumptions. Following this, MT invariant analysis models are examined, which have become

increasingly popular. I will mostly concentrate on the phase tensor analysis method, as it will

be used throughout this thesis for dimensionality analysis.

A typical MT field survey involves measurement of the time-varying electric and magnetic

fields, which are recorded between a day and several months, depending on the maximum pe-

riod needed (longer recordings result in longer periods). The electric and magnetic fields are

Fourier-transformed into the frequency domain. Modern robust time series analysis codes are

capable of producing unbiased impedance estimates (Chave et al., 1987; Chave and Thomson,

1989; Egbert, 1997; Smirnov, 2003). Before forward modelling or inversion of the impedance

data can be attempted, it is crucial to have an understanding of the degree of distortion and

dimensionality of the data sets. Quite often, 2-D modelling (deGroot Hedlin and Constable,

1990; Rodi and Mackie, 2001) of the data is undertaken and needs to be justified beforehand

(Brasse et al., 2002; Wu et al., 2002; Jones et al., 2005). This chapter introduces several ap-

proaches that are utilised to analyse the MT impedance data.

As pointed out in Chapter 1.5 and Chapter 1.6, the impedance tensor has usually four non-

zero components. Even in the 2-D case, the coordinates of measurement will, in most cases,

not be in alignment with the strike of the substructure. Simple 1D analysis of data (Tikhonov,

1950; Cagniard, 1953; Wait, 1954) is not sufficient to fully explain the resistivity distribution

of the subsurface. For the 2-D case (Equation (1.35)), Swift (1967) proposed a rotation scheme

14

2.1. Perturbations of the electric and magnetic fields 15

to minimise the sum of the diagonal components of the impedance tensor. This leads to the

Swift-angle α:

α =1

4arctan

(

2ℜ [(Zxy + Zyx) · (Zxx + Zyy)]

(Zxx − Zyy)2 − (Zxy + Zyx)2

)

. (2.1)

The Swift-angle has a 90° ambiguity, i.e. a 90° rotation leads to an impedance tensor with

swapped off-diagonal components. Minimising the sum of the diagonal components does not

necessarily ensure that a 2-D structure has been found. Therefore, Swift (1967) introduced the

misfit measure κ:

κ =|Zxx + Zyy||Zxy − Zyx|

. (2.2)

If the Swift skew κ > 0.2 − 0.3, the underlying structure cannot be regarded as 2-D. However,

small skew values can also be observed along symmetry axes of 3-D structures, which means

that the skew value κ is not unique. A major disadvantage of the Swift analysis is that it

does not account for the influence of small scale inhomogeneities near the surface. Geological

bodies, which have smaller dimensions than the skin-depth of the shortest period used (see

Equation (1.15) on page 6) lead to galvanic distortion of the measured impedance tensor. In

this case, estimation of strike will lead to erroneous results, depending on the severity of the

distortion.

2.1 Perturbations of the electric and magnetic fields

In MT, small resistivity anomalies lead to perturbations of the E- and B-fields. Depending on

the depth and size of the anomaly and the frequency band used, the anomaly has an inductive

and/or galvanic effect which adds to the MT response of the background conductivity of the

medium. Following Chave and Smith (1994) and Utada and Munekane (2000), the E-field at

an arbitrary position r can be expressed by an integral equation (Hohmann, 1975):

E(r) = E0(r)−iωµ∑

j

∫

Vj

g(r, r′) δσj(r′)E(r′) dV ′+∇ 1

σ0∇·∑

j

∫

Vj

g(r, r′) δσj(r′)E(r′) dV ′.

(2.3)

In each scattering body Vj , a conductivity anomaly δσj(r′) = σj(r

′) − σ0(r′) perturbs the

background electric field E0 (with conductivity σ0). For a uniform Earth the dyadic Green

function g(r, r′) has an analytical expression:

g(r, r′) =ei√

iωµσ0|r−r′|

4π |r − r′| . (2.4)


3D regional structure

Thin layer with small-scale heterogeneitiesin

du

ctive sca

le le

ng

th

Figure 2.1: A typical subsurface structure encountered in MT. A regional 3-D structure isoverlain by 3-D heterogeneities, which may distort the impedance tensor. The regional structurefalls into the inductive scale length of the period range, while the small-scale features near thesurface cause galvanic distortion.

Equation (2.3) illustrates the inductive and galvanic contribution of each scattering body to

the background electric field E0. The second term on the right-hand side of Equation (2.3)

denotes the inductive component and the third term refers to the galvanic contribution. If the

scattering body is small and the frequency ω is low, the inductive influence is also small in

comparison to the galvanic term. The galvanic term is dependent on the conductivity contrast

and distance from each body (Groom and Bahr, 1992). For small volumes dV ′, E(r′) can be

approximated with E0(r′), thus simplifying Equation (2.3) to:

E(r′) = (I + α)E0(r) = CE0(r). (2.5)

Under these assumptions, the resulting electric field is a product of the regional electric field

E0 and a frequency-independent 2 × 2 real tensor C, with I the identity matrix. Applying

Faraday’s Law to Equation (2.3) yields an integral equation for the B-field:

B(r) = B0(r) + µ∇×∑

j

∫

Vj

g(r, r′) δσj(r′)E(r′) dV ′. (2.6)

The galvanic term vanishes due to the fact that ∇×∇ 1σ0

≡ 0. (see Chapter A.1). Similarly to the

case above, E(r′) can be approximated with E0. Therefore the magnetic field is a superposition

of the background magnetic field B0 and E0 multiplied with a frequency independent magnetic

distortion tensor D:

B(r) = B0(r) + DE0(r). (2.7)


Equations (2.5) and (2.7) are expressions for each of the electromagnetic fields. In practice

however, the impedance tensor Z is usually examined, and Equation (2.5) and (2.7) can be

combined using the relation E = ZB:

Z = CZ0 (I + DZ0)−1 . (2.8)

From Equation (2.8) it is apparent that magnetic distortion can potentially have a strong ef-

fect at short periods, while diminishing for longer periods (Chave and Smith, 1994). Magnetic

distortion is caused by small-scale elongated conductive structures, where electric currents are

channelled. However, if the site spacing near the elongated bodies is too large or the periods

used too long, the effect of the bodies has more of a perturbation character than an inductive

character. Numerical modelling of a small-scale 3-D body over a 2-D regional structure sug-

gests that the influence of the body in terms of the magnetic distortion is small, especially for

longer periods. The comparison with electric distortion shows that it is negligible at long periods

(e.g. T > 10 s for a 1 km cube) (Agarwal and Weaver, 2000). The situation is different for marine

MT data. The inductive ocean layer produces large current systems that distort the magnetic

field. The effect of bathymetry on the magnetic distortion tensor D has been shown using

analytical and numerical results (Baba and Seama, 2002; Schwalenberg and Edwards, 2004).

Recent contributions to modelling bathymetry in marine settings favours adaptive unstruc-

tured grid finite element methods, which are better suited to adapt to changes in bathymetry

(Franke et al., 2007a).

In most of the cases, especially for land surveys, magnetic distortion is considered not relevant

and galvanic distortion removal is undertaken prior to dimensionality analysis and modelling

(Ferguson et al., 2005; Hamilton et al., 2006; Harinarayana et al., 2006).

2.1.1 Galvanic distortion removal

Galvanic distortion removal techniques usually employ a presumption about the underlying

structure. They comprise a local 3-D surface anomaly above either a 2-D regional structure

(3-D/2-D scenario) (Groom and Bailey, 1989; Bahr, 1988; Lilley, 1998a; McNeice and Jones,

2001), or a 3-D regional structure (3-D/3-D scenario) (Ledo et al., 1998; Utada and Munekane,

2000). Galvanic distortion has a twofold effect on the impedance tensor, phase mixing (distor-

tion of the telluric orthogonality) and static shift (frequency-independent shift of the apparent

resistivity) (Ogawa, 2002).

The decomposition schemes by Bahr (1988) and Groom and Bailey (1989) illustrate the terms

phase mixing and static shift more closely. According to Bahr (1988), the electric and magnetic

fields at the surface of the Earth are connected in the following way,

E = RCZ2DRT B. (2.9)


The real operator C describes the local heterogeneity. A rotation defined by the rotation matrix

R, leads to a situation where the tensor elements in the same column of the impedance tensor

Z have nearly identical phases. The phase-sensitive skew determines whether this assumption

is correct or another modification introduced by Bahr (1991) becomes necessary. This modifi-

cation involves minimising the phase difference by introducing an exponential function, which

is basically just a complex parameter, into the 2-D impedance matrix.

Quite similar to Bahr’s approach is a decomposition scheme of a local 3-D current channelling

effect on top of the regional 2-D distribution (Groom and Bailey, 1989). Hence, the main model

in Equation (2.9) is still appropriate with the only difference that the distortion matrix C shows

behaviours of twist T , shear S and local anisotropy A, and is scaled by a scalar quantity g:

C = gTSA (2.10a)

with

T =1√

1 + t2

[

1 −t

t 1

]

, (2.10b)

S =1√

1 + e2

[

1 e

e 1

]

, (2.10c)

A =1√

1 + s2

[

1 + s 0

0 1 − s

]

, (2.10d)

where t, e, s are the twist, shear and anisotropy parameters, respectively. The Groom-Bailey

method is based on a least-squares fit to the observed impedance components allowing a sta-

tistical evaluation of the significance of the model. The parameters shear and twist contribute

to the telluric orthogonality, and therefore to the phase mixing. The site gain and anisotropy

parameter are responsible for the magnetotelluric static shift. McNeice and Jones (2001) ex-

tended the Groom-Bailey decomposition analysis to a multi-site, multi-frequency decomposition

scheme, keeping the twist and shear parameters site-dependent, but with the 2-D regional strike

being site-independent. This ensures a stable estimate of the regional strike and is therefore

considered superior to the single-site analysis of Groom and Bailey (1989).

None of the decomposition schemes can recover the anisotropy and site gain, since the system

of equations to be solved is underdetermined. This problem is widely known in magnetotellurics

and unless additional constraints are available, the absolute values of the apparent resistivity

cannot be determined uniquely. Several authors have contributed to implementing additional

constraints on MT responses in order to reduce static shift effects. Ogawa (2002) distinguishes

among three categories:

Invert for static shift Most 2-D inversion codes allow the joint inversion of MT impedances

and static shift factors as unknown variables, i.e. the multiplication of the apparent re-


sistivity curves is determined by the inversion procedure (Rodi and Mackie, 2001). The

usual procedure is to minimse the sum of all static shifts assuming a sufficient number of

stations and a random distribution of static shifts (deGroot Hedlin and Constable, 1990).

This approach is now widely used, however another possibility is to downweight apparent

resistivity data relative to phases, e.g. a small error floor for the phase data and large

error floors for ρa (Tauber et al., 2003).

Spatial filtering Continuous electric dipoles allow low-pass filtering of the electric field, and

thus reduce the inherent static shift in the E-field. Torres-Verdin and Bostick (1992)

reports on the electromagnetic array profiling (EMAP) technique which uses continuous

electric dipoles along a survey path. This setup reduces aliasing effects and also lends

itself to low-pass filtering. Uyeshima et al. (2001) utilised a network of telephone lines

of more than 10 km length in Japan to reduce the effect of small-scale heterogeneities.

Jones et al. (1992) make use of the the fact that, in a 2-D environment, the TE-mode

of the impedance tensor is mostly smoothly varying for long periods. The TE-mode is

not influenced by charge accumulations along the strike of the 2-D structure, as they

only contribute to the TM-mode. Recently, Tournerie et al. (2007) demonstrated that

geostatistical analysis using the cokriging method achieves good results for static shift

removal. This technique has been tested on synthetic COPROD-2S@ 2D MT data and

on a 3-D survey data set from Las Cañadas Caldera (Tenerife, Spain).

Constraints from other geophysical methods Galvanic distortion suffers from lack of res-

olution at shallow depths, which are determined by the lowest period used. In order to

obtain constraints from other methods, those methods have to be sensitive at the shallow

depth of the near-surface heterogeneities and not be influenced by static shift. Well-

logging data can be used to delineate conductive key layers. The key layers can be used

to tie in 1D-models of MT data and thus eliminate static shift Jones (1988). Geophysical

techniques conducted at the surface can also be utilised. Time-domain sounding obtains a

shallow 1D structure by use of the magnetic field only, which is unaffected by galvanic dis-

tortion and thus static shift free (Pellerin and Hohmann, 1990; Meju, 1996; Harinarayana,

1999). Spitzer (2001) presents a method in which DC resisitivity sounding can be used

to correct static shift affected MT impedances. The mentioned methods have in common

that the impedance tensor is corrected through constraints at the lower end of the period

band. Another method is to use geomagnetic station data in order to correct impedances

at longer periods. MT data collected in the Central Gawler Craton, South Australia,

allowed information retrieval from deep structures at the lithosphere-asthenosphere, due

to the resistive core of the craton (Maier et al., 2007). Therefore, data were comparable

with long period observatory data collected in Europe (Olsen, 1999), and could be static

shift corrected.


Other approaches for 3-D/2-D case exist, such as the Lilley angle analysis (Lilley, 1998a). It

analyses the real and quadrature part of the impedance tensor separately and essentially allows

independent rotation of both. In his general theorem, the independent rotation allows the

determination of a local electric and regional magnetic strike direction θe and θh, respectively.

These angles are commonly referred to as Lilley angles. Furthermore, the separation allows

an illustration of a 3-D matrix in a 2-D form. A split-after-twist and twist-after-split operator

yields the desired 3-D matrix starting from a 1-D matrix and also provides the definition of the

angles of the E- and H-splitting axis θe and θh, respectively.

The decomposition of the 3-D impedance matrix into a 2-D matrix with two principal

impedances and the additional information of the Lilley angles can be plotted in a Mohr circle

diagram, first introduced into the MT community by Lilley (1993). The two types of Mohr

circles for this decomposition scheme plot each of the Lilley angles at a time, allowing an imme-

diate illustration of constancy of these angles with period. In practice, the initial assumptions,

that θe and θh are period-independent and should agree in real and quadrature mode, are not

always fulfilled (Lilley, 1998b). Quite often, the electric field direction is constant through the

entire period range but the regional field is more or less period-dependent. Therefore, caution

is necessary when applying decomposition methods to general multi-dimensional conductivity

in not only the local but also the regional field.

The abovementioned decompositions assume a 2-D regional structure, a situation which is

not always fulfilled with real MT data. Ledo et al. (1998) showed that a decomposition of a

3-D regional structure is feasible as long as a period band can be found where the regional

structure is 2-D. A true 3-D/3-D decomposition scheme requires the combination of mag-

netic transfer functions and spatial derivatives of the impedance tensor using Faradays Law

(Utada and Munekane, 2000; Becken et al., 2008).

2.1.2 MT invariant models

A clear step forward is taken by more general approaches which require no 2-D assumptions of

the underlying structure. Szarka and Menvielle (1997) (referred to as S-M) introduced a set of

seven invariants that have been further developed to be clearly represented on a Mohr circle

diagram (Weaver et al., 2000) (referred to as WAL). While S-M have recently presented a field

test for their set of invariants (Szarka et al., 2005), WAL have attempted a simple interpretation

of MT data using their set (Weaver et al., 2003). It should be noted that both sets are very

similar and only differentiate slightly, so as to find a better graphical representation for the WAL

set. Table 2.1 displays all of the WAL invariants and their relationship to the impedance tensor

components. The MT invariant approaches have in common that both the local and regional

field can be 3-D and dimensionality results are obtained without pre-determined restrictions to

the underlying structure.


Table 2.1: An overview of the MT tensor invariants and a dependent invariant as introducedby Weaver et al. (2000). The parameters allow a thorough interpretation of both the local andregional field.

Invariant Dimensionality indication

I1 =(ξ24 + ξ2

1

)1/2I2 =

(η24 + η2

1

)1/2a A 1-D structure is implied if the following

invariants are zero: I3, I4, I5, I6. The apparent

resistivity is a = µ0(I21 + I2

2 )/ω and the phase

φ = arctan I2/I1.

I3 =(ξ2

2+ξ2

3)1/2

I1I4 =

(η22+η2

3)1/2

I2A 2-D structureb is present if I5 = I6 = 0, I3 6= 0

or I4 6= 0 and I7 = 0 or Q = 0.

I5 = s41c = ξ4η1+ξ1η4

I1I2A 2-D structure with a pure twist in the electric

fieldd if I3 6= 0 or I4 6= 0, I5 6= 0 and I6 = I7 = 0.

If Q = 0, a local in-phase distortion is present in a

1-D region or a 2-D region with identical E- and

B-polarisation phases with no recoverable strike

direction θ.

I6 = d41e = ξ4η1−ξ1η4

I1I2A 2-D structure with a local in-phase distortion

and strike angle θ.d

I7 = d41−d23

Q A 3-D structure is present if I7 6= 0.

Q =[(d12 − d34)

2 + (d13 + d24)2]1/2

The dependent invariant Q indicates whether I7 is

defined. If Q goes towards zero, I7 becomes

undefined.

a

[Zxx Zxy

Zyx Zyy

]

︸︷︷︸

Z

=

[ξ1 + ξ3 ξ2 + ξ4

ξ2 − ξ4 ξ1 − ξ3

]

︸︷︷︸

X

+i

[η1 + η3 η2 + η4

η2 − η4 η1 − η3

]

︸︷︷︸

Y

.

b 2-D structure has strike angle θ such that tan 2θ = −ξ3/ξ2 = η3/η2.c sij =

ξiηj+ξjηi

I1I2d Strike angle in this case is given by tan 2θ = d12−d34

d13+d24

e dij =ξiηj−ξjηi

I1I2

2.2. The phase tensor 22

2.2 The phase tensor

Caldwell et al. (2004) introduced another elegant method that did not require presumptions

about the underlying dimensionality. That means, the MT phase tensor is applicable where

both the regional conductivity distribution and the conductivity heterogeneity close to the

surface are 3-D. The tangent of the MT phase was defined in Equation (1.42) on page 13

as the ratio of the imaginary and real parts of the impedance tensor Z. This expression

allows a generalisation to the entire tensor, specifically its real and imaginary parts X and Y ,

respectively.

Φ = X−1Y . (2.11)

Here, Φ defines the real phase tensor and X−1 is the inverse of the real part of the impedance

tensor Z. In order to follow the considerations about the regional and galvanic distortion of

the electromagnetic fields, the phase tensor Φ can also be expressed in terms of the regional

field only.

Φ = X−1Y = (CXR)−1(CYR) = X−1R C−1CYR = X−1

R YR = ΦR. (2.12)

As expected, the phase behaves independent of the distortion C near the surface which affected

the observation of the impedance tensor.

Figure 2.2 illustrates the phase tensors with its 3 invariants, which are explained in detail

in Appendix A.3. The three coordinate invariants are the maximum phase tensor value Φmax,

the minimum phase tensor value Φmin, and the skew angle β. The skew angle defines the

asymmetry of the tensor, and deviates from the symmetry axis (dashed line in Figure 2.2).

The angle α expresses the tensor’s dependence on the coordinate system. However, α is not

coordinate invariant.

α =1

2tan−1

(Φxy − Φyx

Φxx + Φyy

)

. (2.13)

The phase tensor is fully defined with the angle α and the three coordinate invariants.

Φ = RT (α − β)

(

Φmax 0

0 Φmin

)

R(α + β). (2.14)

where R(α + β) is a rotation matrix, with its transpose or inverse RT (θ) = R(−θ):

R(α + β) =

(

cos(α + β) sin(α + β)

− sin(α + β) cos(α + β)

)

. (2.15)

The orientation of the major axis of the tensor is α − β. If β = 0, the tensor is symmetric.

2.2. The phase tensor 23

Φmin

Φmax

x

y

β

α

Figure 2.2: The MT phase tensor Φ graphically illustrated as an ellipse. The major and minoraxis of the ellipse are proportional to the principal values of Φ (namely Φmax and Φmin). αdefines the tensor’s dependence on the coordinate system (x, y). If the skew angle β = 0, theprincipal values are equal to the eigenvalues of the tensor and Φmax is aligned with the symmetryaxis (dashed line). The direction of the major axis Φmax is defined by α − β. Redrawn fromCaldwell et al. (2004).

2.2.1 Relationship to MT invariants

Weaver et al. (2003) have illustrated that the phase tensor agrees in detail with the invariants

introduced by WAL. It is clearly pointed out that the phase tensor is even preferable in the

sense that the often unnecessary information about the galvanic distortion in the scale smaller

than the minimum skin depth is neglected. In terms of the parameters introduced by WAL in

Table 2.1 the phase tensor can be expressed as:

Φ =

(

ξ1 − ξ3 −ξ2 − ξ4

−ξ2 + ξ4 ξ1 + ξ3

)(

η1 + η3 η2 + η4

η2 − η4 η1 − η3

)

ξ21 − ξ2

2 − ξ23 + ξ2

4

. (2.16)

Depending on the kind of information to be obtained one has to decide whether knowledge

of the local field is necessary. If it is, the seven invariants by WAL are preferred. However,

quite often it is more than enough to assess the dimensionality of depths where the electro-

magnetic fields are inductively coupled and are not stationary. This is also true for considering

2.3. Dimensionality Analysis 24

further interpretation techniques such as forward modelling or inversion procedures, in which

the knowledge about the dimensionality is very important.

2.3 Dimensionality Analysis

The coordinate invariants behave according to the symmetry and matrix entries of the phase

tensor. Therefore, the shape of the tensor ellipse is a visual aid in order to determine the

underlying conductivity structure.

2.3.1 One-dimensional case

In a 1-D situation we expect the phase tensor to have the same values for the minimum and

maximum phase.

Φ1D =

(

X−11D

Y1D 0

0 X−11D

Y1D

)

=(X−1

1DY1D

)I = tan(φ)I. (2.17)

I and φ are the identity matrix and the phase, respectively. In practice, a measure of ellipticity

is introduced, which is defined by:

ε =Φmax − Φmin

Φmax + Φmin

. (2.18)

If the value ε is smaller than 0.1 and the skew angle β around 0 the underlying structure is

regarded 1-D (Bibby et al., 2005). Effectively, this means that the absolute value of Φmin is

about 82% of Φmax. Graphically, the ellipse will turn into a near-circle, whose radius is propor-

tional to the conductivity of the period-dependent depth underneath the site of measurement.

The radii of the circles will increase with period if the conductivity increases with depth. Since

the underlying 1-D structure is independent of the coordinate frame, the angle α is undefined.

This means the orientation of the major axis cannot be determined. Furthermore, the phase

tensor has to be symmetric, and therefore β = 0.

An anisotropic conductivity distribution in an otherwise homogeneous half-space will lead to

a circle, where Φ = I. It should be noted, that the impedance tensor Z now has off-diagonal

elements with different amplitudes Zxy 6= Zyx (compare Table 1.2). One cannot recognize

an isotropic 1-D distribution overlying an anisotropic half-space (Caldwell et al., 2004). Phase

splits that occur only indicate a conductivity gradient, but cannot resolve whether the overlying

1-D distribution is anisotropic or not (Heise et al., 2006).


2.3.2 Two-dimensional case

Here, the regional impedance tensor takes a similar form to the impedance tensor of an

anisotropic impedance tensor, with the two off-diagonal elements being non-zero:

Z ′R =

(

0 Z‖−Z⊥ 0

)

. (2.19)

This expression for the regional impedance has already been rotated and is therefore denoted

with a prime (′). The expression for the 2-D impedance tensor in Table 1.2 is equal to Equa-

tion (2.19) with the only difference that Z in Table 1.2 has not been rotated. In Equation (2.19),

Z‖ and Z⊥ relate to the TE and TM impedances, respectively. The diagonal elements of the

2-D impedance tensor only vanish if the electric field is aligned parallel or perpendicular to the

geoelectric strike and is orthogonal to the magnetic field B. Since Z ′R is anti-diagonal, so must

be (X ′R)−1 and Y ′

R.

Out of these considerations it is clear that the phase tensor has to be diagonal and takes the

form:

Φ′2D =

(

X−1⊥ Y⊥ 0

0 X−1‖ Y‖

)

or Φ′2D =

(

X−1‖ Y‖ 0

0 X−1⊥ Y⊥

)

(2.20)

The principal values Φmax and Φmin now have different amplitudes and are defined by the ratios

X−1⊥ Y⊥ or X−1

‖ Y‖. The skew angle is again 0, since the phase tensor is symmetric. According

to Equation (2.14), the orientation of the major axis is defined by α′ = α + ζ. ζ rotates the

coordinate frame parallel or perpendicular to strike so that α′ = 0 and Equation (2.13) defines

an explicit expression for the direction of the strike axis. A 90° ambiguity in the determination

of the TE- and TM-modes remains (however the tensor’s main axis orientation is uniquely

defined), unless a priori information about the geological setting or vertical magnetic field data

are used. Furthermore, the azimuth of the major axis (α−β) is either parallel or perpendicular

to the direction of the real induction arrow (Equation (1.36)).

Galvanic distortion of a 2-D response leads to the same phase tensor as would exist without

distortion. The reason is that, regardless of the real distortion matrix C, there are still two

direction where the electric field will be linearly polarised for a linear polarised magnetic field

(Becken and Burkhardt, 2004). Moreover, the orientations of the principal axes of the phase

tensor are identical to the major and minor axes of both the regional real X and imaginary

Y parts of the impedance tensor. These axes define the strike direction of the regional con-

ductivity structure. If the coordinate system is rotated parallel or perpendicular to the strike

axis as well, then the diagonal elements of the impedance tensor are zero (Equation (2.19)).

Therefore, the angle α for both the real and imaginary parts of the impedance tensor are equal

to α′X = ±45° and α′

Y = ±45°. Thus, the angle between the major axis of X and Y always

has to be 0° or 90°.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

10

20

30

40

50

60

70

80

φ

tan−

1 (φ)

in d

egre

es

Behaviour of tan−1(φ)

Phase for unit circle

Figure 2.3: Relationship of Φ and its arctangent. If both the maximum and minimum phasevalue are equal to one, a plot showing the arctan of these values will both equal to 45°.

Again, it is important to mention that the third coordinate invariant β is zero for the 2-D case

and the phase tensor is symmetric. If this condition is not fulfilled the conductivity distribution

is 3-D. In practice it is sufficient to allow a scatter of approximately ±5°.

2.3.3 Three-dimensional case

The third invariant skew angle β is needed to fully describe the phase tensor in 3-D. It will have

a non-zero value as a measure of departure from the tensor’s symmetry. Figure 2.3 shows the

behaviour of tan−1(Φ). The phase angle of the maximum and minimum values give information

about the conductivity distribution in the subsurface, e.g. if the maximum and minimum phase

are both equal to 45°, the phase tensor is represented by a unit circle. Usually, the principal

values will split representing a 3-D environment. This means, in a coordinate system (x, y),

where the principal values are plotted as the phase tensor ellipse, the ellipse will become more

eccentric. In very strong 3-D cases and strong current channelling, the minor axis will finally

shrink to zero and the ellipse is represented by a line of length 2Φmax. Therefore, plots of the

skew angle β, the maximum and minimum phases, and the tensor ellipses provide a good basis

for interpretation of the conductivity distribution.

The above discussion has revealed that the three coordinate invariants are important means


to analyse the subsurface. Whereas β is a good dimensionality indicator, the principal axis

Φmax and Φmin indicate direction of maximum and minimum induction current. In the case

where the structure is 2-D, the principal axis will be aligned with the strike axis and one of

the principal axes will also be aligned with the real induction arrow. One can say that the

major axis shows the orientation of preferred induction current flow and is an indicator for the

geoelectric strike (Caldwell et al., 2004).

Chapter 3

Inversion routines

The aim of this chapter is to give the reader a brief overview of the 2-D and 3-D inversion

routines used throughout the thesis. A more comprehensive discussion of inversion routines

is given in Appendix E on page 123. The primary 2-D inversion code used extensively in

Chapter 4 and 5 is the non-linear conjugate gradient (NLCG) code by Rodi and Mackie (2001).

The 2-D smooth Occam inversion code by deGroot Hedlin and Constable (1990) was used to

verify the findings of the NLCG inversions (e.g. Figure 5.10 on page 68). Chapter 6 deals with

the 3-D inversion of a data set across the Gawler Craton and the 3-D data-space inversion by

Siripunvaraporn et al. (2005b) was used. The 3-D data-space inversion code is based on its 2-D

equivalent (Siripunvaraporn and Egbert, 2000).

3.1 Overview of inversion methods

The inverse models are discretised into M constant resistivity blocks m = [m1, m2, . . . , mM ]

and there are N observed data d = [d1, d2, . . . , dN ] with uncertainties e = [e1, e2, . . . , eN ]. The

data misfit functional ϕd is defined as:

ϕd = (d − F [m])TC−1

d (d − F [m]) , (3.1)

where F [m] denotes the model response and Cd the data covariance matrix, which is diag-

onal and contains the data uncertainties e. In principle, the inverse problem formulated in

Equation (3.1) will converge. However, the inverse problem is non-unique due to data errors,

spatially undersampled data, limited frequency bands and assumptions about dimensionality

(Parker, 1980). Therefore, a model structure functional ϕm is introduced:

ϕm = (m − m0)T

C−1m (m − m0) , (3.2)

with Cm the model covariance matrix, which characterises the magnitude and smoothness of

resistivity variation with respect to the prior model m0. The two functionals are combined to

28

3.2. The model-space Occam and NLCG methods 29

yield the unconstrained functional ϕλ with X∗ the desired level of misfit:

ϕλ(m) = ϕm + λ−1(ϕd − X2

∗) m,λ−−→ min. (3.3)

The Lagrange multiplier λ controls whether more weight is given to minimising the norm of

data misfit or the norm of the model (Tikhonov and Arsenin, 1977). For larger values of λ,

the data misfit becomes less important and more weight is given toward producing a smoother

model. In order to minimise Equation (3.3), the stationary points have to be found. Instead

of using Equation (3.3), the penalty functional Wλ(m) is differentiated with respect to m:

Wλ(m) = ϕm + λ−1ϕdm,λ−−→ min. (3.4)

For a constant value of λ, the stationary points of ϕλ(m) and Wλ(m) are the same. Therefore,

minimising Wλ(m) with varying λ gives the stationary points of ϕλ(m), and a value for λ can

be found such that the data misfit ϕd is equal to X2∗ .

3.2 The model-space Occam and NLCG methods

The MT inverse problem is non-linear (that means F (m)), and therefore iterative solutions to

solving Equation (3.4) are required. The model response F (m) is approximated by a Taylor

series expansion:

F (mk+1) = F (mk + ∆m) = F (mk) + Jk(mk+1 − mk). (3.5)

The subscript k is the iteration number and J =(

∂F

∂m

)

kis the N × M Jacobian or sensitivity

matrix. In order to find the stationary points, Equation (3.5) is substituted into (3.4), which

yields iterative solutions:

mk+1(λ) =(λC−1

m + Γmk

)−1JT

k C−1d Xk + m0, (3.6)

with Xk = d − F (mk) + Jk(mk − m0), and the M × M model-space cross-product matrix

Γmk = JT

k C−1d Jk (Siripunvaraporn and Egbert, 2000; Siripunvaraporn et al., 2005a).

The Occam inversion runs through two different stages before it reaches a final solution. The

Lagrange multiplier λ is used both as a step length control and a smoothing parameter. A

series of values for λ are used to minimise the misfit of each iteration (3.6) in phase 1. The aim

is to reduce the misfit to the desired level X2∗ . However, in the early iterations the misfit level

X2∗ will not be reached for a series of values for λ. For each iteration, the Occam code uses the

minimum misfit from the best λ as a basis for the next iteration. Once the desired misfit X2∗

is reached, phase two commences by keeping the misfit at the desired level and varying λ to

search for the smallest norm, i.e. the smoothest model. The downside of the Occam inversion

3.3. The data space method 30

is that the sensitivity matrix J must be calculated for each iteration (3.6). This process is both

time-consuming and requires much computer memory to store J.

The NLCG method used by Rodi and Mackie (2001) offers an improvement in speed due to

preconditioning and the computation of only the gradient g = ∂ϕ∂m

rather than the sensitivities

J (see Avdeev (2005)). Here, the Lagrange multiplier λ is kept fixed and a suitable value

for λ must be found for the inversion to find a model that both satisfies the data misfit and

model structure functional. In contrast to the Occam approach, the NLCG method requires

the solution of a univariate minimisation problem at each iteration. This is also referred to

as line search, which has been optimised by Rodi and Mackie (2001) to take only as long as

the computation of three solutions of the forward problem. The NLCG modelling approach

is therefore quicker than the Occam inversion, which requires the computation of the entire

sensitivity matrix at each iteration.

3.3 The data space method

Following the argument of Parker (1994), the iterative solution (3.6) can be reformulated as a

linear combination of rows of the smoothed sensitivity matrix CmJT :

mk+1 − m0 = CmJTk βk+1. (3.7)

Here, the unknown expansion coefficient vector of the basis functions[CmJT

k

]

jwith j =

1, . . . , N are called βk+1. Equation (3.7) can also be substituted into (3.4) and solved for

stationary points, which will give another set of iterative solutions:

βk+1 = (λCd + Γnk)−1

Xk, (3.8)

where Γnk = JkCmJT

k is the N × N data-space cross-product matrix. Similar to the Occam

approach, λ is used to minimise the misfits among the solutions of (3.8) to a desired misfit level,

followed by a search for the smallest norm. The solutions from both the model-space and the

data-space approaches are in theory identical (Siripunvaraporn et al., 2005a), however, depend-

ing on the model and data size M and N , respectively, the number of equations to be solved can

be significantly different. The data-space dimensions N×N are usually much smaller in practice

(especially in 3-D inverse problems), reducing the computational costs. Siripunvaraporn et al.

(2005a) note that the model covariance Cm is used in the data-space approach, but its inverse

C−1m is implemented in the model-space approach. The advantage of using the model covari-

ance itself is that a-priori information such the ocean can be readily included. It should be

noted that the model covariance Cm itself is never computed but only the product with the

sensitivity matrix CmJT is required.

Basics of the magnetotelluric method · The inverse of the real part of k is also known as the skin...

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