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 −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.

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