#### Transcript ERTH2020_(Magnetotelluric) - Exploration Geophysics at the

ERTH2020 Introduction to Geophysics The Electromagnetic (EM) Method Magnetotelluric (MT) ERTH2020 0 Magnetotelluric combination of magnetic and telluric* methods (Latin ‘tellūs’ = ‘earth’ “Earth current”) ERTH2020 1 Magnetotelluric ERTH2020 …other scientists Tikhonov (1950) and Rikitake (1951), Kato & Kikuchi (1950). 2 Induction Equivalent Circuits I I • I DC Resistivity • Induced Polarisation C L R • ERTH2020 Inductive EM 3 DC / IP ERTH2020 4 Magnetotelluric (Passive EM) 𝐻𝑧 𝐸𝑥 𝐸𝑦 𝐻𝑥 ERTH2020 𝐻𝑦 5 Goal 𝜕2 𝐅 − iωμσ𝐅 = 0 2 𝜕𝑧 1. 1D diffusion equation 2. → 𝑝= 2 ≈ 500 𝑇𝜌𝑎 ωμσ Skin Depth (Penetration Depth) ERTH2020 6 Overview DC Resistivity Induced Polarisation Method Direct Electrical Connection (galvanic) Passive EM Active EM No direct electrical connection (inductive) Induced primary magnetic field via loop Electrical potential Decay of electrical potential Ratio of E and H fields Secondary magnetic field Resistivity Resistivity & Chargeability Conductivity Conductivity Measured Injected DC current via electrodes Induced primary magnetic field via natural EM fields ERTH2020 (or its decay) 7 Contents • Introduction o o o o Maxwell Equations Induction Sources Example • EM theory o o o o o Divergence & Curl Diffusion equation 1D Magnetotelluric Skin Depth Apparent Resistivity & Phase • 2D MT Introduction o Example ERTH2020 8 Electromagnetic Induction Ampere’s Law (1826) 𝛻 × 𝐇 = J + 𝜕t 𝐃 Faraday’s Law (1831) 𝛻 × 𝐄 = −𝜕t 𝐁 (magneto) quasi-static approximation , i.e. separation of electrical charges occur sufficiently slowly that the system can be taken to be in equilibrium at all times J electric current density (A/m2) H magnetic field intensity (A/m) ERTH2020 e.g. B magnetic induction (Wb/m2 or T) E magnetic field intensity (V/m) http://farside.ph.utexas.edu/teaching/302l/lectures/node70.html http://farside.ph.utexas.edu/teaching/302l/lectures/node85.html 9 Electromagnetic Induction ERTH2020 Simpson F. and Bahr K, 2005, p.18 10 Electromagnetic Induction Plane Wave Source 𝐸𝑦 45∘ 𝐻𝑥 𝐸𝑦 𝐻𝑥 Faraday’s Law 𝛻 × 𝐄 = −𝜕t 𝐁 Primary field Ampere’s Law 𝛻×𝐇=J 𝐸𝑦 Ohm’s Law 𝐻𝑥 J = σ∙𝐄 ERTH2020 11 Magnetotelluric Sources ERTH2020 12 Magnetotelluric Sources Power spectrum: signal's power (energy per unit time) falling within given frequency bins ERTH2020 Simpson F. and Bahr K, 2005, p.3 13 Magnetotelluric Applications • Mineral exploration • Hydrocarbon exploration (oil/gas) • Deep crustal studies • Geothermal studies • Groundwater monitoring • Earthquake monitoring ERTH2020 Simpson F. and Bahr K, 2005, p.3 14 Magnetotelluric Example 2D-MT resistivity model • Locations of MT measurement sites, Mount St Helens and nearby Cascades volcanoes. • White and red dots show the locations of the magnetotelluric measurements; measurement sites shown in red were used for 2D inversion. • The east–west line (red) shows the profile onto which these measurements were projected. The coloured area shows the region of high conductances. (=conductivity X thickness) • The green-to-orange transition corresponds to a conductance of 3000 Siemens. ERTH2020 Hill et al., 2009 15 Magnetotelluric Example 2D-MT resistivity model after inversion the conductivity anomalies are caused by the presence of partial melt ERTH2020 Hill et al., 2009 16 EM Theory gradient divergence 𝛻𝑈 𝛻∙𝐅 curl 𝛻×𝐅 𝜕𝑥 𝑈 𝜕𝑦 𝑈 𝜕𝑧 𝑈 𝜕𝑥 𝐹𝑥 𝜕𝑦 ∙ 𝐹𝑦 𝐹𝑧 𝜕𝑧 𝜕𝑥 𝐹𝑥 𝜕𝑦 × 𝐹𝑦 𝐹𝑧 𝜕𝑧 𝜕𝒙 𝐹𝒙 + 𝜕𝒚 𝐹𝒚 + 𝜕𝒛 𝐹𝒛 𝜕𝒚 𝐹𝒛 − 𝜕𝒛 𝐹𝒚 𝜕𝒛 𝐹𝒙 − 𝜕𝒙 𝐹𝒛 𝜕𝒙 𝐹𝒚 − 𝜕𝒚 𝐹𝒙 𝜕𝑥 𝑈 𝑖 + 𝜕𝑦 𝑈 𝑗 + 𝜕𝑧 𝑈 𝑘 (vector) ERTH2020 (scalar) (vector) 17 Divergence (Interpretation) The divergence measures how much a vector field ``spreads out'' or diverges from a given point, here (0,0): • Left: divergence > 0 since the vector field is ‘spreading out’ • Centre: divergence = 0 everywhere since the vectors are not spreading out. • Right: divergence < 0 since the vectors are coming closer together instead of spreading out. is the extent to which the vector field flow behaves like a source or a sink at a given point. (If the divergence is nonzero at some point then there must be a source or sink at that position) ERTH2020 http://citadel.sjfc.edu/faculty/kgreen/vector/block2/del_op/node5.html 18 Curl (Interpretation) The curl of a vector field measures the tendency for the vector field to “swirl around”. (For example, let the vector field represents the velocity vectors of water in a lake. If the vector field swirls around, then when we stick a paddle wheel into the water, it will tend to spin.) • Left: curl > 0 (right-hand-rule thumb is up+) • Centre: curl = 0 everywhere since the field has no ‘swirling’. • Right: curl ≷ 0 since the vectors produce a torque on a test paddle wheel. describes the infinitesimal rotation of a vector field ( p.s. The name "curl" was first suggested by James Clerk Maxwell in 1871) ERTH2020 http://citadel.sjfc.edu/faculty/kgreen/vector/block2/del_op/node5.html & Wikipedia (Curl) 19 EM Theory Time-Domain Maxwell Equations (magneto-quasi-static) (Faraday) 𝜕𝐁 𝛻×𝐄=− 𝜕𝑡 1 𝜕𝐇 → 𝛻×𝐄=− μ 𝜕𝑡 (Ampere) 𝛻×𝐇=𝐉 Note the use of the constitutive relations: 𝐁 = μ𝐇 𝐃 = ε𝐄 𝐉 = 𝜎𝐄 Also note that generally 1 → 𝛻×𝐇=𝐄 𝜎 μ = μ 𝑥, 𝑦. 𝑧 𝜎 = 𝜎 𝑥, 𝑦. 𝑧 first order, coupled PDEs ERTH2020 20 EM Theory Time-Domain Maxwell Equations (magneto-quasi-static) to uncouple, take the curl (Faraday) 1 𝜕𝐇 𝛻×𝐄 =− μ 𝜕𝑡 1 𝜕 →𝛻× 𝛻×𝐄=− 𝛻×𝐇 μ 𝜕𝑡 (Ampere) 1 𝛻×𝐇 =𝐄 𝜎 1 →𝛻× 𝛻×𝐇= 𝛻×𝐄 𝜎 Second order, uncoupled PDEs ERTH2020 21 EM Theory Time-Domain Maxwell Equations (magneto-quasi-static) Second order, uncoupled PDEs 1 𝜕𝐄 → 𝛻 × 𝛻 × 𝐄 = −𝜎 μ 𝜕𝑡 Plane wave source sinusoidal time variation 𝐄 𝑡 = 𝐄0 𝑒 𝑖𝜔𝑡 1 𝜕𝐇 → 𝛻 × 𝛻 × 𝐇 = −μ 𝜎 𝜕𝑡 𝐇 𝑡 = 𝐇0 𝑒 𝑖𝜔𝑡 where 𝜔 = 2𝜋𝑓 the angular frequency and 𝑖 = −1 the imaginary unit • Complex numbers arise e.g. from equations such as 𝑥 2 + 1 = 0 → 𝑥 2 = −1 → 𝑥 = −1 ≡ 𝑖. • Complex numbers can also be written as 𝑧 = 𝑒 𝑖𝜃 • Generally complex numbers have a real and imaginary part and are written as 𝑧 = 𝑥 + 𝑖𝑦 where 𝑥 is the real part and 𝑦 the imaginary part. • 𝑧 = 𝑒 𝑖𝜃 = cos 𝜃 + 𝑖 sin 𝜃 • Compact way to describe waves ERTH2020 22 EM Theory Time-Domain Maxwell Equations (magneto-quasi-static) Second order, uncoupled PDEs Plane wave source sinusoidal time variation 1 𝜕𝐄 → 𝛻 × 𝛻 × 𝐄 = −𝜎 = −𝑖𝜎𝜔𝐄 μ 𝜕𝑡 𝐄 𝑡 = 𝐄0 𝑒 𝑖𝜔𝑡 1 𝜕𝐇 → 𝛻 × 𝛻 × 𝐇 = −μ = −𝑖μ𝜔𝐇 𝜎 𝜕𝑡 𝐇 𝑡 = 𝐇0 𝑒 𝑖𝜔𝑡 where 𝜔 = 2𝜋𝑓 the angular frequency and 𝑖 = −1 the imaginary unit • Complex numbers arise e.g. from equations such as 𝑥 2 + 1 = 0 → 𝑥 2 = −1 → 𝑥 = −1 ≡ 𝑖. • Complex numbers can also be written as 𝑧 = 𝑒 𝑖𝜃 • Generally complex numbers have a real and imaginary part and are written as 𝑧 = 𝑥 + 𝑖𝑦 where 𝑥 is the real part and 𝑦 the imaginary part. • 𝑧 = 𝑒 𝑖𝜃 = cos 𝜃 + 𝑖 sin 𝜃 • Compact way to describe waves ERTH2020 23 EM Theory Frequency Domain Diffusion Equations Second order, uncoupled PDEs 1 → 𝛻 × 𝛻 × 𝐄 + 𝑖𝜔𝜎𝐄 = 0 μ 1 → 𝛻 × 𝛻 × 𝐇 + 𝑖𝜔μ𝐇 = 0 𝜎 General equations for inductive EM ERTH2020 24 EM Theory 1D solution 𝜎 = (piecewise) constant, 𝜇 ≡ constant with vector identity 𝛻 × 𝛻 × 𝐅 = 𝛻 𝛻 ∙ 𝐅 − 𝛻 ∙ 𝛻 𝐅 =0 → 𝛻 𝛻 ∙ 𝐄 − 𝛻 ∙ 𝛻 𝐄 = −iωμσ𝐄 → 𝛻 2 𝐄 − iωμσ𝐄 = 0 → 𝛻 𝛻 ∙ 𝐇 − 𝛻 ∙ 𝛻 𝐇 = −iωμσ𝐇 → 𝛻 2 𝐇 − iωμσ𝐇 = 0 =0 Diffusion Equations (Frequency Domain) ERTH2020 25 EM Theory 1D solution 𝜎 = (piecewise) constant, 𝜇 ≡ constant 𝛻∙𝐄=𝟎 𝛻∙𝐇=𝟎 Divergence of Faraday’s law Divergence of Ampere’s law → 𝛻 ∙ 𝛻 × 𝐄 = −𝛻 ∙ → 𝛻 ∙ 𝛻 × 𝐇 = 𝛻 ∙ 𝐉 = 𝛻 ∙ σ𝐄 = 0 𝛻 ∙ σ𝐄 = σ𝛻 ∙ 𝐄 + 𝐄 ∙ 𝛻σ = 0 →𝛻∙𝐁=0 𝜕𝐁 𝜕 =− 𝛻∙𝐁 =0 𝜕𝑡 𝜕𝑡 (Gauss law for magnetism, i.e. no magnetic monopoles) → σ𝛻 ∙ 𝐄 = −𝐄 ∙ 𝛻σ →𝛻∙𝐄=0 𝛻σ = 0 Proof 𝛻 ∙ 𝛻 × 𝐅 = 0 via Cartesian coordinates ERTH2020 𝜕𝒙 𝜕𝒙 𝐹𝒙 𝛻 ∙ 𝛻 × 𝐅 = 𝜕𝒚 ∙ 𝜕𝒚 × 𝐹𝒚 = 𝐹𝒛 𝜕𝒛 𝜕𝒛 = 𝜕𝒙 𝜕𝒚 𝐹𝒛 − 𝜕𝒙 𝜕𝒛 𝐹𝒚 + 𝜕𝒚 𝜕𝒛 𝐹𝒙 − 𝜕𝒚 𝜕𝒙 𝐹𝒛 𝜕𝒚 𝐹𝒛 − 𝜕𝒛 𝐹𝒚 𝜕𝒙 𝜕𝒚 ∙ 𝜕𝒛 𝐹𝒙 − 𝜕𝒙 𝐹𝒛 𝜕𝒙 𝐹𝒚 − 𝜕𝒚 𝐹𝒙 𝜕𝒛 + 𝜕𝒛 𝜕𝒙 𝐹𝒚 − 𝜕𝒛 𝜕𝒚 𝐹𝒙 = 0 26 EM Theory 1D solution 𝜎 = (piecewise) constant, 𝜇 ≡ constant → 𝛻 2 𝐅 − iωμσ𝐅 = 0 ⇔ 𝜕2 𝐅 − iωμσ𝐅 = 0 𝜕𝑧 2 General solution for second-order PDE: 𝐅 = 𝐅1 𝑒 𝑖ωt−𝑞𝑧 + 𝐅2 𝑒 𝑖ωt+𝑞𝑧 decreases in amplitude with z 𝑧<0 increases in amplitude with z unphysical 𝑧>0 ERTH2020 Simpson F. and Bahr K, 2005, p.21 27 EM Theory 1D solution 𝜎 = (piecewise) constant, 𝜇 ≡ constant 𝐅 = 𝐅1 𝑒 𝑖ωt−𝑞𝑧 Taking the second derivative with respect to z 𝜕2 𝐅 = 𝑞2 𝐅1 𝑒 𝑖ωt−𝑞𝑧 = 𝑞2 𝐅 2 𝜕𝑧 → 𝑞= 𝑖ωμσ = 𝑖 ωμσ = 1 + 𝑖 ↔ 𝜕2 𝐅 − iωμσ𝐅 = 0 𝜕𝑧 2 ωμσ 2 = ωμσ/2 + 𝑖 ωμσ/2 Real part → ERTH2020 1 𝑝= = ℜ𝔢 𝑞 2 ωμσ Imaginary part Skin Depth (Penetration Depth) Simpson F. and Bahr K, 2005, p.22 28 EM Theory 1D solution 𝜎 = (piecewise) constant, 𝜇 ≡ constant 𝐅 = 𝐅1 𝑒 𝑖ωt−𝑞𝑧 Taking the second derivative with respect to z 𝜕2 𝐅 = 𝑞2 𝐅1 𝑒 𝑖ωt−𝑞𝑧 = 𝑞2 𝐅 2 𝜕𝑧 → 𝑞= 𝑖ωμσ = 𝑖 ωμσ = 1 + 𝑖 ↔ 𝜕2 𝐅 − iωμσ𝐅 = 0 𝜕𝑧 2 ωμσ 2 = ωμσ/2 + 𝑖 ωμσ/2 Real part → 1 𝑝= = ℜ𝔢 𝑞 2 ωμσ Imaginary part Skin Depth (Penetration Depth) For angular frequency 𝜔 for a half-space with conductivity 𝜎 ERTH2020 Simpson F. and Bahr K, 2005, p.22 29 EM Theory 1D solution 1 𝑝= = ℜ𝔢 𝑞 → 𝑝= ERTH2020 𝜇 ≡ constant𝜇 → 𝜇0 = 4𝜋 ∙ 10−7 𝜎 = (piecewise) constant, 2 ωμσ 2 = 4𝜋 ∙ 10−7 ωσ Skin Depth (Penetration Depth) 107 2 𝑇𝜌 = 4𝜋 ∙ 2𝜋 107 ≈ 40 107 𝑇𝜌 ≈ 500 𝑇𝜌 4𝜋 2 Simpson F. and Bahr K, 2005, p.22 & http://userpage.fu-berlin.de/~mtag/MT-principles.html 30 EM Theory 1D solution 𝜎 = (piecewise) constant, 𝜇 ≡ constant Real part 𝑞= Imaginary part ωμσ/2 + 𝑖ωμσ/2 and 1 𝑝= = ℜ𝔢 𝑞 2/ωμσ The inverse of q is the Schmucker-Weidelt Transfer Function 1 𝑝 𝑝 𝐶 = = +𝑖 𝑞 2 2 ..has dimensions of length but is complex The Transfer Function C establishes a linear relationship between the physical properties that are measured in the field. ERTH2020 Simpson F. and Bahr K, 2005, p.22 31 EM Theory 1D solution 𝜎 = (piecewise) constant, 𝜇 ≡ constant Schmucker-Weidelt Transfer Function 1 𝑝 𝑝 𝐶 = = +𝑖 with 𝑝 = 2/ωμσ 𝑞 2 2 We had with the general solution earlier 𝐸𝑥 = 𝐸1𝑥 𝑒 𝑖ωt−𝑞𝑧 → 𝜕𝐸𝑥 = −𝑞𝐸𝑥 𝜕𝑧 However Faraday’s law is 𝛻×𝐄 Therefore ERTH2020 𝑦 𝜕𝐸𝑥 = = −𝑖ωμ𝐻𝑦 𝜕𝑧 −𝑖ωμ𝐻𝑦 = −𝑞𝐸𝑥 → 𝐶 = 1 1 𝐸𝑥 1 𝐸𝑦 = =− 𝑞 𝑖ωμ 𝐻𝑦 𝑖ωμ 𝐻𝑥 Simpson F. and Bahr K, 2005, p.22 32 EM Theory 1D solution 𝜎 = (piecewise) constant, 𝜇 ≡ constant Schmucker-Weidelt Transfer Function 1 1 𝐸𝑥 1 𝐸𝑦 𝐶= = =− 𝑞 𝑖ωμ 𝐻𝑦 𝑖ωμ 𝐻𝑥 • 𝐶 is calculated from measured 𝐸𝑥 and fields 𝐻𝑦 (or 𝐸𝑦 and 𝐻𝑥 ) . • from 𝐶 the apparent resistivity can be calculated: with q = 𝑖ωμσ → 𝑞 apparent resistivity ERTH2020 2 𝑞2 1 = ωμσ → σ = or ρ = 2 ωμ ωμ 𝑞 → ρ = 𝐶 2 ωμ Simpson F. and Bahr K, 2005, p.22 33 EM Theory Apparent Resistivity and Phase apparent resistivity phase 𝜌𝑎 = 𝐶 2 ωμ 𝜙= tan−1 ℑ𝔪 𝐶 ℜ𝔢 𝐶 The phase is the lag between the E and H field and together with apparent resistivity one of the most important parameters in MT ERTH2020 Simpson F. and Bahr K, 2005, p.22 34 EM Theory Apparent Resistivity and Phase 𝜌𝑎 = 𝐶 2 ωμ −1 𝜙 = tan 𝐶= ℑ𝔪 𝐶 ℜ𝔢 𝐶 with 𝑝 𝑝 +𝑖 2 2 𝑝= 2/ωμσ For a homogeneous half space: ℑ𝔪 𝐶 = ℜ𝔢 𝐶 → 𝜙 = tan−1 1 = 45° • 𝜙 < 45° → diagnostic of substrata in which resistivity increases with depth • 𝜙 > 45° → diagnostic of substrata in which resistivity decreases with depth ERTH2020 Simpson F. and Bahr K, 2005, p.26 35 EM Theory ERTH2020 Simpson F. and Bahr K, 2005, p.27 36 2D-MT Introduction For this 2-D case, EM fields can be decoupled into two independent modes: • E-fields parallel to strike with induced B-fields perpendicular to strike and in the vertical plane (E-polarisation or TE mode). • B-fields parallel to strike with induced E-fields perpendicular to strike and in the vertical plane (B-polarisation or TM mode). ERTH2020 Simpson F. and Bahr K, 2005, p.27 37 2D-MT Introduction ERTH2020 Simpson F. and Bahr K, 2005, p.30 38 Numerical Modelling in 2D 2D solution 𝜎 = 𝜎(𝑥, 𝑧), 𝜇 ≡ constant 𝐸𝒙 → TE-mode (E-Polarisation) 𝛻 ∙ 𝛻𝐸𝒙 − 𝑖𝜔𝜇𝜎𝐸𝑥 = 0 Numerical schemes, e.g.: • Finite Differences • Finite Elements Escript Finite Element Solver (Geocomp UQ) ERTH2020 39 Numerical Modelling in 2D σ = 10-14 S/m σ = 0.1 S/m σ = 0.01 S/m Dirichlet boundary conditions via a single analytical 1D solution applied Left and Right; Top & Bottom via interpolation ERTH2020 40 Numerical Modelling in 2D Electric Field (Real) ERTH2020 Electric Field (Imaginary) 41 Numerical Modelling in 2D Apparent Resistivity at selected station (all frequencies) ERTH2020 42 Numerical Modelling in 2D σ = 10-14 S/m σ = 0.04 S/m σ = 0.1 S/m σ = 0.4 S/m σ = 0.001 S/m # Zones = 71389 σ = 0.2 S/m ERTH2020 # Nodes = 36343 43 Numerical Modelling in 2D Real Part 𝐸𝑥 ERTH2020 Imaginary Part 𝐸𝑥 44 Numerical Modelling in 2D Apparent Resistivity f = 1 Hz r = 25 Ωm Skin-depth 𝑝 ≃ 500 𝑇𝜌𝑎 ≃ 500 12 ≃ 1700 r = 10 Ωm r = 2.5 Ωm r = 1000 Ωm r = 2 Ωm ERTH2020 45 References Simpson F. and Bahr K.: “Practical magnetotellurics”, 2005, Cambridge University Press Cagniard, L. (1953) Basic theory of the magneto-telluric method of geophysical prospecting, Geophysics, 18, 605–635 Hill G J., Caldwell T.G, Heise W., Chertkoff D.G., Bibby H.M., Burgess M.K., Cull J.P., Cas R.A.F.: "Distribution of melt beneath Mount St Helens and Mount Adams inferred from magnetotelluric data", Nature Geosci., 2009, V2, pp.785 ERTH2020 46 Unused slides ERTH2020 47 EM Theory (Faraday) 𝜕𝐁 𝛻×𝐄=− 𝜕𝑡 → 𝛻 ∙ 𝛻 × 𝐄 = −𝛻 ∙ →𝛻∙𝐁 = 0 𝜕𝐁 𝜕 =− 𝛻∙𝐁 =0 𝜕𝑡 𝜕𝑡 (Gauss law for magnetism) via Cartesian coordinates 𝜕𝒚 𝐸𝒛 − 𝜕𝒛 𝐸𝒚 𝜕𝒙 𝜕𝒙 𝜕𝒙 𝐸𝒙 𝛻 ∙ 𝛻 × 𝐄 = 𝜕𝒚 ∙ 𝜕𝒚 × 𝐸𝒚 = 𝜕𝒚 ∙ 𝜕𝒛 𝐸𝒙 − 𝜕𝒙 𝐸𝒛 = 𝜕𝒙 𝐸𝒚 − 𝜕𝒚 𝐸𝒙 𝐸𝒛 𝜕𝒛 𝜕𝒛 𝜕𝒛 = 𝜕𝒙 𝜕𝒚 𝐸𝒛 − 𝜕𝒙 𝜕𝒛 𝐸𝒚 + 𝜕𝒚 𝜕𝒛 𝐸𝒙 − 𝜕𝒚 𝜕𝒙 𝐸𝒛 + 𝜕𝒛 𝜕𝒙 𝐸𝒚 − 𝜕𝒛 𝜕𝒚 𝐸𝒙 = 0 ERTH2020 48 EM Theory 𝜕𝐃 𝛻×𝐇=𝐉+ 𝜕𝑡 𝜕𝐃 𝜕 →𝛻∙𝐉+𝛻∙ =𝛻∙𝐉+ 𝛻∙𝐃 =0 𝜕t 𝜕𝑡 𝜕 →𝛻∙𝐉=− 𝛻∙𝐃 𝜕𝑡 (Ampere) however, the rate of change of the charge density ρ equals the divergence of the current density J Continuity equation 𝜕 𝜕 →𝛻∙𝐉=− 𝛻∙𝐃 =− ρ 𝜕𝑡 𝜕𝑡 ERTH2020 →𝛻∙𝐃=𝜌 (Gauss law) 49 2D-MT Introduction (Faraday) 𝜕𝒚 𝐸𝒛 − 𝜕𝒛 𝐸𝒚 𝛻 × 𝐄 = 𝜕𝒛 𝐸𝒙 − 𝜕𝒙 𝐸𝒛 𝜕𝒙 𝐸𝒚 − 𝜕𝒚 𝐸𝒙 𝜕𝒚 𝐸𝒛 − 𝜕𝒛 𝐸𝒚 𝐻𝒙 𝜕𝒛 𝐸𝒙 = = −𝑖𝜔𝜇 𝐻𝒚 −𝜕𝒚 𝐸𝒙 𝐻𝒛 (Ampere) 𝜕𝒚 𝐻𝒛 − 𝜕𝒛 𝐸𝒚 𝛻 × 𝐇 = 𝜕𝒛 𝐻𝒙 − 𝜕𝒙 𝐸𝒛 𝜕𝒙 𝐻𝒚 − 𝜕𝒚 𝐸𝒙 𝜕𝒚 𝐻𝒛 − 𝜕𝒛 𝐻𝒚 𝜕𝒛 𝐻𝒙 = −𝜕𝒚 𝐻𝒙 𝐸𝒙 → TE-mode (E-Polarisation) 𝐻𝒙 → TM-mode (B-Polarisation) 𝜕𝒛 𝐸𝒙 = −𝑖𝜔𝜇 𝐻𝒚 𝜕𝒛 𝐻𝒙 = σ𝐸𝒚 𝜕𝒚 𝐸𝒙 = 𝑖𝜔𝜇 𝐻𝒛 𝜕𝒚 𝐻𝒙 = −σ𝐸𝒛 σ𝐸𝒙 = 𝜕𝒚 𝐻𝒛 − 𝜕𝒛 𝐻𝒚 ERTH2020 𝐸𝒙 = σ 𝐸𝒚 𝐸𝒛 −𝑖𝜔𝜇𝐻𝒙 = 𝜕𝒚 𝐸𝒛 − 𝜕𝒛 𝐸𝒚 Simpson F. and Bahr K, 2005, p.28 50 Numerical Modelling in 2D 2D solution 𝜎 = 𝜎(𝑥, 𝑧), 𝜇 ≡ constant 𝐸𝒙 → TE-mode (E-Polarisation) 𝜕𝒛 𝐸𝒙 = −𝑖𝜔𝜇 𝐻𝒚 𝜕𝒚 𝐸𝒙 = 𝑖𝜔𝜇 𝐻𝒛 σ𝐸𝒙 = 𝜕𝒚 𝐻𝒛 − 𝜕𝒛 𝐻𝒚 𝜕𝒛 𝜕𝒛 𝐸𝒙 = −𝑖𝜔𝜇 𝜕𝒛 𝐻𝒚 𝜕𝒚 𝜕𝒚 𝐸𝒙 = 𝑖𝜔𝜇𝜕𝒚 𝐻𝒛 𝜕𝒚 𝜕𝒚 𝐸𝒙 + 𝜕𝒛 𝜕𝒛 𝐸𝒙 = 𝑖𝜔𝜇 𝜕𝑦 𝐻𝑧 − 𝜕𝑧 𝐻𝑦 = 𝑖𝜔𝜇𝜎𝐸𝑥 𝛻 ∙ 𝛻𝐸𝒙 − 𝑖𝜔𝜇𝜎𝐸𝑥 = 0 ERTH2020 Scalar PDE of 𝐸𝑥 51