19Magnetotelluric Impedance After Fourier transforming the E(t) and H(t) data into the frequency domain the MT surface impedance is calculated from:Ex(w) = Z(w) Hy(w)
20Note, that sinceEx(w) = Z(w) Hy(w)is a multiplication in the frequency domain, it is a convolution in the time domain.Therefore, this is a filtering operation, i.e.,Ex(t)Hy(t)Z(t)
21Apparent resistivity, ra and phase, f Apparent resistivity is the resistivity of an equivalent, but fictitious, homogeneous, isotropic half-spacePhase is phase of the impedancef = tan-1 (Im Z/Re Z)Take magnitude of complex number, multiply by frequency, another constant.App res most commonly used quantity in MT.
22surface electromagnetic impedance, Zs The goal of MT is the resistivity distribution, r(x,y,z), of the subsurface as calculated from thesurface electromagneticimpedance, Zsr7r1r2Dimensionality:r3One-DimensionalTwo-DimensionalThree-Dimensionalr4r5r6
24ra a |Z2| 1-D MT Sounding Curve ra x y z Log Period (s) Shallow Resistive LayerIntermediate Conductive LayerDeep Resistive LayerraPeriod (s)Logra a |Z2|
25Layered (1-D) Earth Ex Hy (Ack., Paul Bedrosian, USGS) 104 1000 100 Apparent resistivityImpedance Phase204060801011021031041003010-2Period (s)DegreesOhm-m500ExHy1000Longer period deeper penetration ( )mUsing a range of periods a depth sounding can be obtained(Ack., Paul Bedrosian, USGS)
26MT “Screening” of Deep Conductive Layer by Shallow Conductive Layer (Ack., Martyn Unsworth, Univ. Alberta)
27When the Earth is either 2-D or 3-D: Ex(w) = Z(w) Hy(w)NowEx(w) = Zxx(w) Hx(w) + Zxy(w) Hy(w)Ey(w) = Zyx(w) Hx(w) + Zyy(w) Hy(w)This defines the tensor impedance, Z(w)
283-D MT Tensor EquationEx and Ey depend on Hx, Hy. When it’s multidimensional, impedance has more than one value.
291-D, 2-D, and 3-D Impedance 1-D 2-D Assumes geoelectric strike 3-D No geoelectric assumptions[ ] is Tensor Impedance(Ack., Paul Bedrosian, USGS)
303- D MT DataMeasure time variations of electric (E) and magnetic (H) fields at the Earth‘s surface.Estimate transfer functions of the E and H fields.Subsurface resistivity distribution recovered through modeling and inversion.Impedance Tensor:App Resistivity & Phase:(Ack. Paul Bedrosian, USGS)Techniques - MT
312-D MT ra a |Z2| ra (Tensor Impedance reduces to two off- diagonal elements)yxzGeoelectricStrikera a |Z2|raPeriod (s)Log
32Boundary Conditions Map View E-Fields parallel to the geoelectric strike are continuous (called TE mode)E-Fields perpendicular to the geoelectric strike are discontinuous (called TM mode)TMTEMap ViewE-ParallelLog raE- PerpendicularLog Period (s)
33TE (Transverse Electric) and TM (Transverse Magnetic) Modes 2-D Earth structureDifferent results at MT1 (Ex and Hy)and MT2 (Ey and Hx)MT1TRANSVERSE ELECTRIC MODE (TE)TRANSVERSE MAGNETIC MODE (TM)Visualizing Maxwell’s Curl Equations(Ack., Martyn Unsworth, Univ. Alberta)
34MT Phase TensorDescribed as “elegant” by Berdichevsky and Dmitriev (2008) and a “major breakthrough” by Weidelt and Chave (2012)“Despite its deceiving simplicity, students attending the SAGE program often have problems grasping the essence of the MT phase tensor” (Jiracek et al., 2014)The MT Phase Tensor and its Relation to MT Distortion(Jiracek Draft, June, 2014)
35MT Phase TensorX and Y are the real and imaginary parts of impedance tensor Z, i.e., Z = X + iYIdeal 2-D, β=0Recommended β <3° for ~ 2-Dby Caldwell et al., (2004)
36MT Phase Tensor Ellipse Ellipses are traced out at every period by the multiplication ofthe real 2 x 2 matrix from a MT phase tensor, F(f) anda rotating, family of unit vectors, c(w), that describe a unit circle.2-D Tensor Ellipse p2D(w) is:
37Phase Tensor Example for Single MT Sounding at Taupo Volcanic Field, New Zealand (Bibby et al., 2005)1-D TP Tc 2-D TP Tc D TP
381-D TP D TP D TPTcTcPhase Tensor Determinations of Dimensionality (1-D. 2-D), Transition Periods (TP), and Threshold Periods (Tc)
40Geoelectric Section From Stitched 1-D TE Inversions (MT Sites Indicated by Triangles) WEConductive BasinElevation (m)Resistive BasementDistance (m)WEElevation (m)BasinBasement
412-D MT Inversion/Finite-Difference Grid M model parameters, N surface measurements, M>>NA regularized solution narrows the model subspaceIntroduce constraints on the smoothness of the model(Ack. Paul Bedrosian, USGS)Techniques - MT
42Geoelectric Section From 2-D MT Inversion (MT Sites Indicated by Triangles) WEConductive BasinElevation (m)Resistive BasementDistance (m)WEElevation (m)BasinBasement
50ReferencesBibby, H. M., T. G. Caldwell, and C. Brown, 2005, Determinable and non-determinable parameters of galvanic distortion in magnetotellurics, Geophys. J. Int., 163, Caldwell, T. G., H. M. Bibby, and C. Brown, 2004, The magnetotelluric phase tensor, Geophys. J. Int., 158, Heise, W., T. G. Caldwell, H. W. Bibby, and C. Brown, 2006, Anisotropy and phase splits in magnetotellurics, Phys. Earth. Planet. Inter., 158, Jiracek, G.R., V. Haak, and K.H. Olsen, 1995, Practical magnetotellurics in continental rift environments, in Continental rifts: evolution, structure, and tectonics, K.H. Olsen, ed., Jiracek, G. R., V. M Gonzalez, T. G. Caldwell, P. E. Wannamaker, and D. Kilb, 2007, Seismogenic, Electrically Conductive, and Fluid Zones at Continental Plate Boundaries in New Zealand, Himalaya, and California-USA, in Tectonics of A Continental Transform Plate Boundary: The South Island, New Zealand, Amer. Geophys. Un. Mono. Ser. 175,
51Palacky, G.J., 1988, Resistivity characteristics of geologic targets, in Investigations in Geophysics Volume 3: Electromagnetic methods in applied geophysics theory vol. 1, M.N. Nabighian ed., Soc. Expl. Geophys., 53–129. Winther, P. K., 2009, Magnetotelluric investigations of the Santo Domingo Basin, Rio Grande rift, New Mexico, M. S thesis, San Diego State University, 134 p. Wannamaker, P. E., D. P. Hasterok, J. M. Johnston, J. A. Stodt, D. B. Hall, T. L. Sodergren, L. Pellerin, V. Maris, W. M. Doerner, and M. J. Unsworth, 2008, Lithospheric Dismemberment and Magmatic Processes of the Great Basin-Colorado Plateau Transition, Utah, Implied from Magnetotellurics: Geochem., Geophys., Geosys., 9, 38 p.