Basics of the Magnetotelluric Method

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. 3 5 The geomagnetic fluctuations range between periods of 10− s and 10 s (or between frequen- 3 5 cies of 10 Hz and 10− Hz) depending on their origin (Vozoff, 1991). Meteorological activity such as lightning discharges produce EM fields with periods shorter than 1 s. The signals travelling 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 ins 0.2-5 5-10 10-45 45-150 150-600 1-40 40-150 f 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) ∇· ∂B E = , (1.1c) ∇× − ∂t ∂D H = j + . (1.1d) ∇× ∂t 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 j − ∂t

E H

B Figure 1.1: Faraday’s law E = ∂ . Figure 1.2: Ampere’s law H = j. ∇× − ∂t ∇× 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 µ = 4π 10 7 V s/A m – magnetic permeability of free space and 0 · − ε = 8.85 10 12 A s/V m – dielectric permittivity of free space. Variations in µ and ε are 0 · − r r 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 = , (1.4b) ∇· ε ∂B E = , (1.4c) ∇× − ∂t ∂E B = µσE + µε . (1.4d) ∇× ∂t 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:

∂E ∂B ∂2B ( B)= ( B) 2B = µσE + µε = µσ µε . (1.5) ∇× ∇× ∇· ∇· − ∇ ∇× ∂t − ∂t − ∂t2 Since B = 0, Equation (1.5) simpliﬁes to the equation of telegraphy, which describes the ∇· diﬀusive wave characteristics in a lossy (σ > 0) medium:

∂B ∂2B 2B = µ σ + ε . (1.6) ∇ ∂t ∂t2 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) − − 1 The undamped wave part, travelling at a speed of c = √µε , is described by κ, whose wavelength 2π 2π c is λ = κ = ω√µε = f . 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 frequency 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. 1.3. EM induction in a homogeneous half-space 5

Depth dependent behaviour of the E−field 1 absolute attenuation real part 0.8 imaginary part

0.6

0.4 / E(z) 0

E 0.2

−0.2

−0.4 0 1 2 3 4 5 z/δ

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

ikz ikz B = B0 e− + B1 e , (1.11a) ikz ikz E = E0 e− + E1 e . (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 B = E = 0. Thus, the solution in Equation (1.11) simpliﬁes → ∞ 1 1 to:

ikz iυz υz B = B0 e− = B0 e− e− , (1.12a) ikz iυz υz E = E0 e− = E0 e− e− . (1.12b) with k2 =iωµσ.

i π ωµσ k = √i√ωµσ = √ωµσ e− 4 = (1 + i) = (1 + i) υ. (1.13) 2 r 1.3. EM induction in a homogeneous half-space 6

Equation (1.12) illustrates the superposition of two characteristics of the ﬁeld. The ﬁelds vary iυz sinusoidally with depth due to the e− term, but also show a depth dependant attenuation due υz to e− term. The inverse of the real part of k is also known as the skin depth or penetration depth δ. 1 2 δ = = . (1.14) k ωµσ ℜ r 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 T̺ . (1.15) ≈ a 1 p where ̺a = /σa is the apparent resistivity or the equivalent average resistivity of the uniform 1 half-space (in Ω m) and T = 2π/ω is the period in s. EM ﬁelds are attenuated to a value of e− 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 relationship between the components of the E- and B-fields: ∂E ∂E z y = iωB , (1.16a) ∂y − ∂z − x ∂E ∂E x z = iωB , (1.16b) ∂z − ∂x − y ∂E ∂E y x = iωB . (1.16c) ∂x − ∂y − z 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 magnetic ﬁeld only has horizontal components due to large distance to the source and that it is 5 therefore valid for periods up to 10 s. This also implies that the Bz component is equal to ∂Ex ∂Ey 0. Therefore, ∂y and ∂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 kz kz = kE e− = iωB = iωB e− . (1.17) − ∂z y0 − x − x0 The magnetotelluric impedance Z (in m/s) is the ratio of the electric and magnetic ﬁelds measured at the surface: E iω iω ω Z(ω)= y0 = = = √i. (1.18) B − k −√iωµσ − µσ x0 r 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 Ω m: µ ρ = Z 2 . (1.19) ω | | In the frequency domain, Z(ω) is complex and has an associated phase φ:

ω ω π π φ = arg Z = arg √i = arg ei 4 = = 45°. (1.20) µσ µσ 4 r r 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: 1 Z C := = . (1.21) k iω

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 σxx σxy σxz Ex

jy = σyx σyy σzz  Ey , (1.22) jz  σzx σzy σzz  Ez              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:

σxx σxy σxz σ1 0 0

σ = σ σ σ  = Rz( αS)Rx( αD)Rz( αL)  0 σ 0  Rz(αL)Rx(αD)Rz(αS). yx yy zz − − − 2 σzx σzy σzz   0 0 σ3        ′ ′ ′  σ(x ,y ,z ) | {z } (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). 1.4. EM induction in a 1-D Earth 8

x σ1 z1 σ2 z2 · · · zn 1 − σn zn σN z Figure 1.4: Schematic model of a one-dimensional resistivity model with σ varying in the z-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:

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

The magnetic ﬁeld in the orthogonal y-direction would equate to: k B = n E . (1.25) yn iω xn Therefore, the transfer function C for the nth-layer is:

Exn (z) Cn(z)= . (1.26) iωByn (z) 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:

1 knCn+1(zn) + tanh [kn(zn zn 1)] Cn(zn 1)= − − (1.27) − kn 1+knCn+1(zn) tanh [kn(zn zn 1)] − − The apparent resistivity ̺a of the layered media can be expressed in terms of the complex transfer function C (compare to Equation (1.19):

̺ (ω)= C(ω) 2 µ ω. (1.28) a | | 0 1.5. EM induction in a 2-D Earth 9

Accordingly, the phase φ is given as the inverse tangent of the ratio of the real and imaginary part of C: C φ = tan 1 ℑ . (1.29) − C ℜ Both the apparent resistivity and phase are coupled due to causality by a dispersion relationship (Weidelt, 1972), also known as Kramers-Kronig relationship.

∞ π ω 1 ̺a(ω′) φ(ω)= 2 2 log dω′ (1.30) 4 − π ω′ ω ̺0 Z0 − 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 1 due to the scaling term ω′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 thep 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 boundary 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:

j = σ E = σ E = j = const. (1.31) y1 1 · y1 2 · y2 y2 1.5. EM induction in a 2-D Earth 10

E-polarisation B-polarisation j = σ1 E = σ2 E y · y1 · y2

x Ex Hx

y Hy Ey z Hz Ez σ1 < σ2

Ey1 > Ey2

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 σ = σ . The E-ﬁeld is 1 6 2 discontinuous across the vertical contact because of current preservation. The E-polarisation and 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 boundary (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 magnetic 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:

∂E ∂B x = z =iωB , ∂y ∂t z  ∂Ex ∂By  = =iωBy,  E-polarisation. (1.32) ∂z ∂t   ∂Bz ∂By = µ0σEx. ∂y − ∂z    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: ∂B x = µ σE , ∂y 0 z  ∂B x = µ σE ,  B-polarisation. (1.33) 0 y  − ∂z  ∂Ez ∂Ey  =iωBx. ∂y − ∂z     1.5. EM induction in a 2-D Earth 11

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 2 ̺a has an offset of (σ2/σ1) (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 components, 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.

Zxx′ Zxy′ 0 Zxy T T Z′ = = R R = RZ2DR , (1.34) Zyx′ Zyy′ ! Zyx 0 ! with

cos α sin α R = . (1.35) sin α cos α! − 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 = ( , ) is generally as follows: Tzx Tzy B = B + B . (1.36) z Tzx x Tzy y 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 components and (Parkinson, 1962). Parkinson arrows have a real (in-phase) and a quadra- Tzx Tzy ture (out-of-phase) part. Lengths of the real (Mr) and quadrature (Mq) arrows are given by

1/2 M = 2 + 2 , (1.37a) r ℜ Tzx ℜ Tzy 1/2 M = 2 + 2 . (1.37b) q ℑ Tzx ℑ Tzy 1.6. EM induction in a 3-D Earth 12

Table 1.2: Entries of the impedance tensor Z vs dimensionality and isotropic conductivity distribution. The entries become increasingly independent for more complex dimensionality. Z and Z denote the E-parallel and E-perpendicular impedance, respectively. k ⊥ Dimensionality 1-D 2-D 3-D tensor components Z = Z = 0 Z = Z Z = Z = Z = Z xx yy xx − yy xx 6 − yy 6 xy 6 yx Z = Z Z = Z xy − yx xy 6 − yx 0 Zn 0 Z Zxx Zxy impedance tensor Z k Zn 0 Z 0 Zyx Zyy − ⊥

Orientation of the arrows is similarly determined by:

1 zy θr = tan− ℜ T , (1.38a) ℜ Tzx 1 zy θq = tan− ℑ T . (1.38b) ℑ Tzx

In Equations (1.38), angles θr and θq are clockwise positive from the x-direction (usually geomagnetic north). When plotted on a map, Parkinson arrows should point toward regions of iωt high conductance and away from resistive blocks. Depending on the sign convention for e±

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:

E Z Z B x = xx xy x (1.39) Ey! Zyx Zyy! By!

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:

Z Z X X Y Y xx xy = xx xy +i xx xy . (1.40) Zyx Zyy! Xyx Xyy! Yyx Yyy! 1.6. EM induction in a 3-D Earth 13

σ2 σ1

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

Z has both magnitude and phase.

1 2 ̺aij (ω)= Zij(ω) , (1.41) µ0ω | |

1 Zij φ = tan− ℑ . (1.42) ij Z ℜ ij 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 values of the oﬀ-diagonal components of the impedance tensor scaled by Earth property values:

Z = Z =iωµ C. (1.43) | yx1D | | xy1D | 0 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 subsurface (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 Z = Z . A xx − yy geological scenario given in Figure 1.6 will usually produce a three-dimensional impedance tensor. 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 period 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 approaches 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:

1 2 [(Zxy + Zyx) (Zxx + Zyy)] α = arctan ℜ 2 · 2 . (2.1) 4 (Zxx Zyy) (Zxy + Zyx) ! − − 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: Z + Z κ = | xx yy|. (2.2) Z Z | xy − yx| 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):

1 E(r)= E (r) iωµ g(r, r′) δσ (r′) E(r′) dV ′+ g(r, r′) δσ (r′) E(r′) dV ′. 0 − j ∇σ ∇· j j Z 0 j Z XVj XVj (2.3)

In each scattering body V , a conductivity anomaly δσ (r ) = σ (r ) σ (r ) perturbs the j j ′ j ′ − 0 ′ background electric ﬁeld E0 (with conductivity σ0). For a uniform Earth the dyadic Green function g(r, r′) has an analytical expression:

′ i√iωµσ0 r r e | − | g(r, r′)= . (2.4) 4π r r | − ′| 2.1. Perturbations of the electric and magnetic ﬁelds 16

Thin layer with small-scale heterogeneities inductiv e s cal e l engt h 3D regional structure

Figure 2.1: A typical subsurface structure encountered in MT. A regional 3-D structure is overlain by 3-D heterogeneities, which may distort the impedance tensor. The regional structure falls into the inductive scale length of the period range, while the small-scale features near the surface 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 E and a frequency-independent 2 2 real tensor C, with I the identity matrix. Applying 0 × Faraday’s Law to Equation (2.3) yields an integral equation for the B-field:

B(r)= B (r)+ µ g(r, r′) δσ (r′) E(r′) dV ′. (2.6) 0 ∇× j j Z XVj The galvanic term vanishes due to the fact that 1 0. (see Chapter A.1). Similarly to the ∇×∇ σ0 ≡ 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) 2.1. Perturbations of the electric and magnetic ﬁelds 17

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:

1 Z = CZ0 (I + DZ0)− . (2.8)

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

T E = RCZ2DR B. (2.9) 2.1. Perturbations of the electric and magnetic ﬁelds 18

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 modification 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

1 1 t T = − , (2.10b) √1+ t2 "t 1 # 1 1 e S = , (2.10c) √1+ e2 "e 1# 1 1+ s 0 A = , (2.10d) √1+ s2 " 0 1 s# − 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- 2.1. Perturbations of the electric and magnetic ﬁelds 19

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 resolution 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 distortion 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. 2.1. Perturbations of the electric and magnetic fields 20

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 magnetic 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. 2.1. Perturbations of the electric and magnetic fields 21

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

Invariant Dimensionality indication

2 2 1/2 2 2 1/2a I1 = ξ4 + ξ1 I2 = η4 + η1 A 1-D structure is implied if the following invariants are zero: I3,I4,I5,I6. The apparent 2 2 resistivity is ̺a = µ0(I1 + I2 )/ω and the phase φ = arctan I2/I1.

2 2 1/2 2 2 1/2 (ξ2 +ξ3 ) (η2 +η3 ) b I3 = I4 = A 2-D structure is present if I5 = I6 = 0, I3 = 0 I1 I2 6 or I = 0 and I = 0 or Q = 0. 4 6 7

I = s c = ξ4η1+ξ1η4 A 2-D structure with a pure twist in the electric 5 41 I1I2 ﬁeldd if I = 0 or I = 0, I = 0 and I = I = 0. 3 6 4 6 5 6 6 7 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.

e ξ4η1 ξ1η4 I = d = − A 2-D structure with a local in-phase distortion 6 41 I1I2 and strike angle θ.d

d41 d23 I = − A 3-D structure is present if I = 0. 7 Q 7 6

Q = (d d )2 + (d + d )2 1/2 The dependent invariant Q indicates whether I is 12 − 34 13 24 7 deﬁned. If Q goes towards zero, I7 becomes undeﬁned.

a Z Z ξ1 + ξ3 ξ2 + ξ4 η1 + η3 η2 + η4 xx xy = +i . Zyx Zyy ξ2 − ξ4 ξ1 − ξ3 η2 − η4 η1 − η3 Z X Y b 2-D structure has strike angle θ such that tan 2θ = −ξ3/ξ2 = η3/η2. | {z } | {z } | {z } c ξiηj +ξj ηi sij = I1I2 d d12−d34 Strike angle in this case is given by tan 2θ = d13+d24 e ξiηj −ξj ηi dij = 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.

1 Φ = X− Y . (2.11)

1 Here, Φ defines the real phase tensor and X− 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.

1 1 1 1 1 Φ = X− Y = (CXR)− (CYR)= XR− C− CYR = XR− 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.

1 1 Φxy Φyx α = tan− − . (2.13) 2 Φ + Φ xx yy The phase tensor is fully deﬁned with the angle α and the three coordinate invariants.

Φmax 0 Φ = RT (α β) R(α + β). (2.14) − 0 Φmin! where R(α + β) is a rotation matrix, with its transpose or inverse RT (θ)= R( θ): − cos(α + β) sin(α + β) R(α + β)= . (2.15) sin(α + β) cos(α + β)! − 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

Φmax

Φmin α

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

ξ ξ ξ ξ η + η η + η 1 − 3 − 2 − 4 1 3 2 4 ξ2 + ξ4 ξ1 + ξ3 ! η2 η4 η1 η3! Φ = − − − . (2.16) ξ2 ξ2 ξ2 + ξ2 1 − 2 − 3 4 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 electromagnetic 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.

1 X1D− Y1D 0 1 Φ 1D I I 1D = 1 = X1D− Y = tan(φ) . (2.17) 0 X1D− Y1D! I and φ are the identity matrix and the phase, respectively. In practice, a measure of ellipticity is introduced, which is deﬁned by:

Φmax Φmin ε = − . (2.18) Φmax + Φmin 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 proportional 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 Z = Z (compare Table 1.2). One cannot recognize xy 6 yx 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. Dimensionality Analysis 25

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:

0 Z ZR′ = k . (2.19) Z 0 ! − ⊥ 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 k ⊥ 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 ZR′ is anti-diagonal, so must 1 be (XR′ )− and YR′ . Out of these considerations it is clear that the phase tensor has to be diagonal and takes the form: 1 1 X− Y 0 X− Y 0 Φ2D = ⊥ ⊥ or Φ2D = k (2.20) ′ 1 ′ k 1 0 X− Y ! 0 X− Y ! k k ⊥ ⊥ The principal values Φmax and Φmin now have different amplitudes and are defined by the ratios 1 1 X− Y or X− Y . The skew angle is again 0, since the phase tensor is symmetric. According ⊥ ⊥ k k 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 conductivity 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 = 45° and α = 45°. Thus, the angle between the major axis of X and Y always X′ ± Y′ ± has to be 0° or 90°. 2.3. Dimensionality Analysis 26

Behaviour of tan−1(φ) 80

Phase for unit circle 40 ) in degrees φ ( −1 tan 30

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 φ

Figure 2.3: Relationship of Φ and its arctangent. If both the maximum and minimum phase value 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 1 behaviour of tan− (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 2.3. Dimensionality Analysis 27 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:

T 1 ϕ = (d F [m]) C− (d F [m]) , (3.1) d − d −

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

T 1 ϕ = (m m ) C− (m m ) , (3.2) m − 0 m − 0

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: ∗

1 2 m,λ ϕλ(m)= ϕm + λ− ϕd X 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:

1 m,λ W (m)= ϕ + λ− ϕ min. (3.4) λ m d −−→

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

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 (m )= F (m + ∆m)= F (m )+ J (m m ). (3.5) k+1 k k k k+1 − k F The subscript k is the iteration number and J = ∂ is the N M Jacobian or sensitivity ∂m k × matrix. In order to ﬁnd the stationary points, Equation (3.5) is substituted into (3.4), which yields iterative solutions:

1 1 m − T 1 mk+1(λ)= λCm− + Γk Jk Cd− Xk + m0, (3.6) with X = d F (m )+ J (m m ), and the M M model-space cross-product matrix k − k k k − 0 × m T 1 Γk = Jk Cd− 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 ∂ϕ preconditioning and the computation of only the gradient g = ∂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 T linear combination of rows of the smoothed sensitivity matrix CmJ :

m m = C JT β . (3.7) k+1 − 0 m k k+1 C JT Here, the unknown expansion coeﬃcient vector of the basis functions m k j with 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:

n 1 βk+1 = (λCd + Γk )− Xk, (3.8)

where Γn = J C JT is the N N data-space cross-product matrix. Similar to the Occam k k m k × 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, depending 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 1 Cm− is implemented in the model-space approach. The advantage of using the model covariance 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 T sensitivity matrix CmJ is required.