A novel approach to asynchronous MVP data ...

A novel approach to asynchronous MVP data interpretation based on elliptical-vectors 1

2

3

3

Igor Trofimov , Mikhail Kruglyakov , Sergey Korotaev , Orekhova Darya , 1

1

3

4

Irina Popova , Vitaly Shneyer , Yuri Scshors , Michael Zhdanov 1

Schmidt Institute of Physics of the Earth, 2Lomonosov Moscow State University, 3NIC Kurchatov Institute, 4Consortium for Electromagnetic Modeling and Inversion

1) Motivation

3) Elliptical vector and MVP

6) Interpretation of “North Pole” stations data

Interpretation steps We encountered two problems while interpreting the data from Arctic Ocean The dot product 1. Interpretation of major semiaxes to determine the place of equivalent “North Pole” stations. Namely, the asynchrony of the data and the impossibility The main MVP equation (1) can be considered as a dot product of two EPs sources — pair of vertical magnetic dipoles of using the plane wave as a source. To overcome these issues we came up with H = (Hx, Hy ) = (a, b, α, ϕ) and C = (A, B) = (m, n, β, ψ). 2. The selection of geoelectrical structures with respect to the minor semi axis the theory of EV-vector presented below. Hz = (C, D) = AHx + BHy = hHihCi cos γ, MVP problem Major semiaxis of experimental data at 24 hour period. (11) The MVP theory is based on the following linear relation between vertical Hz n b γ = β − α − i(arctgεν − arctgεh), εν = , εh = . and horizontal components (Hx, Hy ) of magnetic field. m a (1) From (8),(11) we get:

Hz = AHx + BHy

The coeficients A and B can be of two types: Hz = am((1 − εν εh) cos θ + i(εν − εh) sin θ)ei(ϕ+ψ), I Real in Wiese-Parkinson approach where θ = β − α. There are two extreme cases: I Complex in Schmucker approach I H is collinear to C: The traditional interpretation of A and B as “arrows”results in a lot of diffiI θ = 0, I εν = −εh culties in interpretation of asynchronous data, since this pair does not form a
2 I |Hz | = am 1 + ε . ν vector and (1) is not a dot product. I H is orthogonal to C: I

2) Elliptical-vectors

I I

(12)

π θ= , 2 εν = εh cos γ = 0, |Hz | = 0.

We propose a new approach for interpretation of any pair of complex numbers Note, that first case corresponds to E-polarization and the second one (X, Y ) as a vector in some space. corresponds to H-polarization. This allows us to relate the direction of major semiaxis m with the direction of From linear to circular basis geo-electric section heterogeneity. At the same time using the value of εν we Suppose, C = Xex + Y ey , (2) can estimate three-dimensionality of the section, assuming it to be 2d for εν = 0 where (ex, ey ) is some linear orthogonal basis. It is well known that in this Linear polarization case C is not a vector since it’s impossible to define it’s magnitude and If horizontal component of magnetic field is linear polarized, then εh = 0 and direction. However,in circular basis we obtain: we have C1 = X + iY Hz = am[cos θ + iεν sin θ]ei(ϕ+ψ) (13) Note, that their value is larger than 1. (3) Note, that linear polarization means not only 2d problem, but also cylindrical C2 = X − iY The interpreted data one and some others. The C = (C1, C2) is complex vector with the following parameters: Two-sources problem Major semiaxis X, km X, km I Magnitude p p To find A, B from (1) we need at least two magnetic fields from different 2 2 (4) sources. hCi = X + Y = C1C2   AHx1 + BHy1 = Hz1 I Direction X C1 + C2 (14) cos θ = = √  hCi 2 C1C2 AHx2 + BHy2 = Hz2 (5) Hence, if we want to model source by vertical magnetic dipole (VMD) we need Y i C1 C1 + C2 sin θ = θ = − ln = √ at least two dipoles. hCi 2i C1C2 2 C2 Hence, we obtain X = hCi cos θ,

4) VMD over the layered media

Y = hCi sin θ,

Suppose that sources of EM field are vertical magnetic dipoles and we have horizontally homogeneous media. Then from (14) we obtain :  1  Hz = Hρ1(A cos γ + B sin γ) (15)  2 2(A cos β + B sin β) H = H z ρ (6)

C = C(X, Y ) = hCi(cos θex + sin θey ) I

Additivity S(S1, S2) = C(C1, C2) + D(D1, D2)

I

S 1 = C 1 + D1 , Multiplication on a scalar:

S 2 = C 2 + D2

Y, km

Y, km Result modelling data

X, km

X, km

Minor semiaxis

where Hρ1, Hρ2 are the radial components of the magnetic field in the corresponding cylindrical coordinate system. The solution of (15) is the following W1 sin β − W2 sin γ (7) A=− sin(β − γ)

kC = K(K1, K2) K1 = kC1,

Minor semiaxis

K2 = kC2

W1 cos β − W2 cos γ B= sin(β − γ)

The expressions (4)–(7) justify that C is a vector. Elliptical-vector

(16)

Hz1 Hz2 where W1 = H 1 , W2 = H 2 . ρ ρ

The complex-vector C can be expressed in terms of 4 real components (a, b, α, ϕ) : p hCi = a2 − b2eiϕ (8)

b θ = α − iarctg . a These components are called I a — major semiaxis I b —- minor semiaxis I α — angle between direction of major semiaxis and X axis I ϕ — phase This notation is chosen because of historical reasons. Then C = (CX , CY ) = (a, b, α, φ) is called an elliptical vector (EV). Some features of the components of EV: I a ≥ 0 I a ≥ b ≥ −a X = 0 then b = 0 I if Im Y I Only α is changed under the rotation of the coordinate system

Y, km

Y, km

Using the asymptotic of Hρ and Hz as ρ → ∞ and (8) and (16) we obtain Black contours means borders of the well-known geological structures. that m → Zg n → 0, (17)

7)

Contacts where m is major semiaxis of (A, B), n is minor semiaxis, Z is Tikhonov impedance of layered media, g depends on the geometry, ∀ρg > q > 0. If you have any questions but I am not here you can find me at one of my other posters: In practice of 3d problems that means that in this asymptotic zone the major I On computing the volumetric integrals of the Green’s tensor of the layered semiaxis depends on the previous field, while minor semiaxis depends on local media at P3.2-259 heterogeneities. I On uniqueness in thin-sheet electromagnetic inversion at P3.1-018B z arg H Hρ

0

[email protected]

−1 −2

Feel free to take my card and email me

−3

Schmucker’s vectors

−4

The components of Schmucker’s vectors are expressed in terms of X, Y by the following: −5 CR = ReXex + ReY ey 0 0.5 1ρ 1.5 2 (9) Λ CI = ImXex + ImY ey The figure shows that for the homogeneous halfspace with wavelength Λ this Since the EV is the generalized form of complex-vector, the components of asymptotic zone begins near the ρ = 1.5Λ. Hence if ρ1, ρ2 > 1.5Λ, then n Schmucker’s vectors can be expressed in terms of the EV components: really depends only on the heterogeneities.  1 2 2 2 |CR| = (a cos ϕ) + (b sin ϕ) Main features of EV

5)

|CI | =



1

2 2 2 (b cos ϕ) + (a sin ϕ)

I

(10) b θR = α + arctg( tgϕ) a b θI = α + arctg( ctgϕ) a The formulas (10) are the reasons for well-known complicated behaviour of Shmucker’s vectors.

I I

I

Linear relation (1) became a dot product Only one component α depends on the coordinate system The ability to determine the three-dimensional nature of structures by ab Different EV’s components have different sensitivity with respect to the field sources and the local heterogeneities.

8) References [1] Berdichevsky M.N., Zhdanov M.S. Advanced theory of deep geomagnetic sounding. Amsterdam. Elsevier. 1984. 408 p. [2] Korn, G. A., Korn, T. M., Mathematical Handbook for Scientists and Engineers, Second Edition, Dover, New York, 2000. [3] Trofimov I.L. On the theory of magnetotelluric methods. Izvestia Physics of the Solid Earth. 1996. V.32. No 6. P. 515-521. [4] Fonarev G.A., Trofimov I.L., Shneyer V.S. Some Results of Magnetotelluric Research in the central Arctic // J. Geop. Res. 1973. V. 78. No 8. P. 1398-1400. [5] Parkinson W. D. Direction of Rapid Geomagnetic Fluctuation // Geophys. J. 2, 1959. [6] Schmucker, U., Anomalies of geomagnetic variations in the southwestern United States, Bull. 13, pp. 1-165, Scripps Inst. of Oceanogr., La Jolla, Calif., 1970.