MAGNETOTELLURIC ANALYSIS: USE OF INVARIANCES IN MOHR CIRCLES

1NGUYỄN THÀNH VẤN, 2LÊ VĂN ANH CƯỜNG

Department of Geophysics, University of Natural Sciences,VNU-HCM, Hồ Chí Minh City

Abstract: The magnetotelluric analysis is one of the methods used in the research on inhomogeneity of 2D and 3D electric environments, whose depths are about tens kilometers. Explaining magnetotelluric data is to get useful arbitrary parameters from a general magnetotelluric impedance tensor. Invariances are drawn from different methods to express the characteristics of 1D, 2D, and 3D models.


I. INTRODUCTION

The fundamental model of MT method is created by Tikhonov and Cagniard. In the model, MT wave transfers into a horizontally stratified earth bed, and the relation is expressed by:

 ;                       (1)

with Z: Tikhonov-Cagniard general impedance describing electrical conductivity distribution versus depth z.

                                   (2)

where:  and                                   (3)

: Unit vectors in Cartersian axes, direction  goes down.

General MT impedance Z is considered to be the relation between components  and .

In the 1950s, the experiment environment in which Berdichevsky and Cagniard established impedance Z is a horizontally electrical stratified earth case.

                                       (4)

However, because real environments are usually electric nonlayered, general impedance is considered as a tensor built from  and   in order to improve MT method.

Berdichevsky and Zdanov [1] said that there was an invariant relation between MT components, which expressed electrical conductivity distribution in the earth. It is an algebraic relation:

             (5)

where  is an arbitrary vector, which has properties of inducing field and affecting field,  and  are operators dependent on electrical conductivity distribution of environment and frequency. If they can be converted mutually,  is a matrix to define linear relation between components of field.

II. PROPERTIES OF GENERAL IMPEDANCE TENSOR

Its property is dependent on kinds of model. 1D structure, 2D structure and 3D structure are examined in turn.

- 1D structure: Electrical conductivity distribution only varies through the depth. The model of Cagniard is 1D structure. In this model, equations Zxx = Zyy = 0 and Zxy = −Zyx = Z are always right in arbitrary measured axes.

                                       (6)

 The components Zxy and Zyx of general impedance tensor are related with the change of electrical property versus depth, but Zxx and Zyy are related with the change of electric property versus horizon.

- 2D structure: Electrical conductivity distribution varies through depth and a horizontal axis such as x-axis or y-axis. The horizontal axis, which has conductance =const, is called as the homogenous axis. Polarized electromagnetic field is divided into two parts:

1. Parallel (//) or polarized electric field  (in case the electric field is polarized along homogenous axis).

2. Perpendicular () or polarized magnetic field  (in case the polarized magnetic field is perpendicular to homogenous axis).

 and  are respectively parallel and perpendicular components of general impedance tensor. Therefore, it has zero element on the diagonal.

                                 (7)

- 3D structure: Electrical conductivity distribution varies through depth and horizontal axes such as x-axis and y-axis. From diversity of these structures, people use symmetric cylinder structure in which general impedance tensor is simplest. Assuming that the direction of x-axis is the direction of symmetric axis, Zr and Zt are centripetal and tangential components of general impedance tensor. It means that the general impedance tensor has zero elements on the diagonal:

                                   (8)

3D symmetric cylinder structure and 2D structure are same in the shape of general impedance tensor.

III. MOHR CIRCLES METHOD

The Mohr circles method, most commonly met in the analysis of mechanical stress, is used to depict magnetotelluric impedance information, taking the real and quadrature parts of magneto-telluric tensors separately.

Tensor  =

Components of rotating tensor :

     (9)

where

              (10)

Complex A = Ar+iAq, where Ar and Aq are, respectively, real and quadrature. Therefore, we extract circle equations for real and quadrature parts from the components of the tensor  :                     

   (11)

where

                 (12)

                 (13)

A plot of Z’xx against Z’xy, as the measuring axes rotate, then describes a circle, where Z’xx and Z’xy are the values that would be measured with axes rotated   clockwise from those that gave the initial values of Zxx and Zxy. Similarly, the components Z’xy and Z’yy also form the relation through a circle equation.

Invariants with rotation of the measured axes: because Z1 and Z2 are invariants, the distance from the origin of axes to the circle center is invariant, which is called “central impedance” [3]. Denoting the central impedance by ZL, its value is given by:          

   (14)

 Other invariants are the values of the circle radii Rr and Rq.

 If the circle is offset from the Z’xy-axis, the offset can be expressed by an angle  or . The value of tan is showed:

                   (15)

  Now, we consider the relation between Mohr circles and the 1D, 2D and 3D structures.

- 1D structure: , the circle reduces to its central point, which is on the horizontal Z’xy-axis. 

- 2D structure: the circle is centered on the horizontal axis, and cut the horizontal Z’xy-axis at two values.

- 3D structure: the circle center moves off the axis, and the invariant  is different to zero. For highly anisotropic data (either 2-D or 3-D), the circumference of the circle approaches the origin of the axes (Lilley, 1993a).


Figure 1. A pair of real and quadrature circles for a tensor, showing the set of invariant.


From inspection of the previous figure (Fig. 1), one straight-forward set can be seen to be:

1. ZL, the distance from the origin of axes to the circle center, which gives the 1-D “scale” of a matrix, and is its 1-D kernel.

2. , an angle which gives a measure of the two dimensionality or anisotropy of a matrix, defined as:            

3., an angle which gives a measure of the three dimensionality of a matrix; ZL, , and  have values for both the real and quadrature matrices of  the tensor, and thus comprise six invariants.

A seventh invariant, which is a further parameter of three dimensionality, links the real and quadrature parts of a tensor and may be expressed as:  ; where the angle  is between the horizontal axis and the line linking the circle center to the observed point.  and  refer to the values of  for the real and quadrature parts of a tensor, respectively.

Example: 3D impedance tensor  

(a)

(b)

From the Fig. 2. the Mohr method shows the 3D structures of the environments. In Fig. 2. (b), although the quadrature Mohr circle shows 2D structure, the structure is 3D (because, in the real part, the invariant is different to zero).

IV. ASSESSMENT

Simons Spritz’s approach gives less information (6 parameters). The Eggers’s parameters are rather similar to La Torraca and Yee’s (8 parameters) but the beginning ideas to establish them are different. The principal axes of polarization ellipses of eigenvector pair (Ei, Hi) are perpendicular in Eggers’s method (the biorthogonal method). On the other hand, in the La Torraca and Yee‘s method, the principal axis directions for the and vectors are not at right angles as in the biorthogonal analysis. Because of invariant phases of eigenvalues of tensor  in Eggers’s method while phases of eigenvalues of tensor  in La Torraca and Yee’s method are variant (although they vary little), Eggers’s method is more effective. The Mohr circles construction (Lilley’s method) gives seven invariants of a tensor which explain the properties of 1D, 2D and 3D environment clearly and easily understandable. To research Mohr circles method further, we use it to analyze 3D model data order to draw helpful geological information.

Figure 2. Real (bold line)
and quadrature (thin line) Mohr circles in the 3 D examples.

- 3-D magnetotelluric modelling: A 3-layer model whose first layer contains 3D conductivity masses is researched at frequency 2700 Hz. The 3D conductivity masses are ellipse-shaped mass and sphere mass. The ellipse-shaped mass has major axis a = 50 km, minor axis b = 12.5 km, conductivity Sc of ellipse center and conductivity So of the ellipse outer. The sphere mass has radius r = 5 km and conductivity of the sphere mass St. The distance between the center of sphere mass and the major axis of ellipse-shaped mass is 7.5 km, and the distance between the center of sphere mass and the minor axis of ellipse-shaped mass is 5 km.



Figure. 4. Real (bold) and quadrature (thin) Mohr circles in the 3D model.


The 3D model: Resistivities of three layers are, respectively: =100; =1000000;=1;

Conductivity Sc of the ellipse center: Sc =10 CM;

Conductivity S0 of the ellipse outer:  S0 = 10 CM;

Conductivity St of the sphere mass:  St = 100 CM;

The depths of the first layer and the second layer:  h1 = 1 km; h2 = 200 km.

1. Analysis of the 3d model: 

- Mohr circles method:

At the measuring points 1-5, 7-11, 13, 15-26, 29, 31-33, 35-37, 39 and 40, the


Table 1. The set of invariant of the 3D model.

STT

1

0.000015

0.000203

18.03023

12.90871

1.859049

1.547317

2.125996

2

0.000015

0.000205

13.40122

9.593889

0.572939

0.720012

1.235592

3

0.000018

0.000271

16.13138

15.26543

0.502306

0.338899

1.079667

4

0.000013

0.000184

7.56186

4.486621

1.775164

1.290494

11.61628

5

0.000015

0.000209

19.8633

14.7104

1.162498

0.932863

0.992373

6

0.000015

0.000209

20.11097

16.15612

7.789108

6.256836

6.457061

7

0.000013

0.000183

5.066163

3.262733

2.015711

1.448568

31.09535

8

0.000015

0.000214

12.43617

10.04313

-3.56432

-3.24952

3.627354

9

0.000034

0.000516

21.54779

19.78241

-1.21088

-1.14076

0.030031

10

0.000016

0.000212

22.79522

17.28933

1.064586

0.874946

0.837864

11

0.000017

0.000229

28.482

23.23413

5.909015

4.77612

2.404132

12

0.000019

0.000265

29.36073

25.06058

9.340026

7.76614

2.746595

13

0.00002

0.000291

19.08616

16.07348

0.763432

0.495173

0.759273

14

0.000019

0.000274

23.11654

20.36288

-6.58996

-6.03368

0.752431

15

0.000021

0.000317

8.02095

6.979198

-3.84067

-3.52215

2.313074

16

0.000034

0.000508

22.26512

20.4194

-2.42938

-2.32048

0.174716

17

0.000016

0.000211

21.61166

16.49809

-0.36782

-0.1184

0.634357

18

0.000016

0.000215

28.10045

22.1313

-0.57066

-0.51298

0.060401

19

0.000017

0.000227

31.7643

26.14267

-0.75517

-0.77698

0.164174

20

0.000022

0.000306

25.92506

22.06239

-0.14975

-0.2335

0.257288

21

0.000028

0.000401

23.8888

21.41278

0.093607

0.028576

0.175762

22

0.000022

0.000317

22.00075

19.50587

-0.1715

-0.24801

0.485068

23

0.000022

0.000321

6.467167

5.682756

-1.06338

-1.00927

0.590644

24

0.000029

0.000439

15.3243

13.95562

-1.99235

-1.89829

0.109096

25

0.000035

0.000521

20.21577

18.54821

-2.22631

-2.14347

0.160561

26

0.000016

0.00021

24.91441

19.1862

-1.63166

-1.15395

1.016015

27

0.000017

0.000226

29.22115

23.64501

-7.06008

-6.02628

2.084077

28

0.000019

0.000262

27.33654

23.11352

-9.84447

-8.49831

2.557173

29

0.000021

0.000295

19.46554

16.59716

-0.8483

-0.89335

0.26702

30

0.00002

0.00028

24.92287

22.26862

5.900111

5.160111

1.135459

31

0.000021

0.000312

4.807794

4.528576

1.679112

1.480348

3.761664

32

0.000013

0.000514

43.99519

20.01832

-5.67866

-2.18032

36.65731

33

0.000015

0.000207

24.54191

19.07863

-0.64937

-0.02775

1.411758

34

0.000015

0.000205

20.42065

15.95597

-9.40079

-8.20674

4.797871

35

0.000013

0.000191

10.00696

7.048838

-3.77989

-3.97125

11.31574

36

0.000017

0.000245

21.83839

19.17688

2.25696

1.449625

3.142483

37

0.000035

0.000521

20.18581

18.5102

-3.11462

-2.99428

0.363877

38

0.000013

0.000178

16.64263

12.6226

-7.33216

-7.20092

1.191505

39

0.000014

0.000189

23.8599

18.13984

-2.361

-1.93338

1.287537

40

0.000013

0.000183

21.25694

16.7339

-2.25421

-1.8297

1.172901

41

0.000026

0.00039

21.73615

20.01137

-6.5198

-5.98825

0.047091

 


invariants are quite small (), radii of real and quadrature Mohr circles are nearly zeros. The centers of the Mohr circles are in the horizontal axes Zxy. Therefore, they show 1D, 2D structures of the model environment. To explain this, we consider locations of the measuring points expressing the affect of the sphere and ellipse-shaped anomalies. The points 5, 7-10, 18, 26 and 39 are in the axis of ellipse mass. The points 17, 19-24, 2, 13, 29, 35 and 40 are in the axis of sphere mass. The points 1, 3, 4, 11, 15, 16, 25, 31-33, 36 and 37 are far from two masses.

At the measuring points 6, 12, 14, 27, 28, 30, 34, 38, and 41, the invariants change onsiderably . The centres of the Mohr circles move off the horizontal axes Zxy and the  differ from zeros. The inhomogeneity of the environment is expressed clearly. Again, locations of the measuring points express the affect of the sphere and ellipse-shaped anomalies to explain the inhomogeneity. The points 6, 27, 34, 38, and 41 sit near the boundary of two anomaly masses. Especially, the 3D points 12, 14, 28, and 30 affected by sphere mass and the 2D points 13, 20, 22 and 29 create boundary of two anomaly masses ( in the Tab. 1, the bold and italic rows are used for the points 12, 14, 28 and 40; the bold rows are used for the points 13, 20, 22 and 29.).

When the biorthogonal method is applied to analyze the 3D model data, the result also makes the sphere mass outstanding. In the measuring points 12, 14, 28 and 30, the directions of the red bold  are directed toward the sphere mass.  In the measuring points 13, 20, 22, and 29, the linear polarized ellipses show 1D and 2D structures of the model. Therefore, the measuring points 12, 14, 28, 30, 13, 20, 22, and 29 create a boundary of two anomaly masses (Fig. 5.).


Figure 5. E-polarization ellipses for both eigenstates in the 3D model.
E1(red bold line); E2(blue line).


V.  CONCLUSION

By considering two real and quadrature elements, the Mohr circles method gives the useful approach to analyze the magnetotelluric data. With information gotten from the set of seven Mohr circle parameters, we have enough evidences to affirm an 1D, 2D and 3D environment, then, continue to measure or choose data to establish a reserve model problem. 

REFERENCES

1. Berdichevsky M.N., Dmitriev V.I., 1992. Magnetotelluric sounding of horizontally homogeneous media. Moscow (in Russian).

2. Berdichevsky M.N., Nguyen Thanh Van., 1991. Magnetovariational vector. Izv. Akad. Nauk USSR, Fizika Zemli, 3 : 52-62. Moscow.

3. Lilley F.E.M., 1998. Magnetotelluric tensor decomposition: Part I. Theory for a basic procedure. Geophysics, 63 : 1885 -1897. Part II. Examples of a basic procedure. Geophysics, 63 : 1898-1907.

4. Nguyễn Thành Vấn, 1996. The methods of assessment and classification of the geoelectrical inhomogeneities in magnetotelluric data processing. J. of  Sci. and Techn., 4 : 48-52. Hà Nội (in Vietnamese).

5. Nguyễn Thành Vấn, 2005. Magnetotelluric tensor: Decomposition and applications. Sci. and Techn. Dev. 8/8 : 26-34. VN Nat. Univ.-HCM City.

6. Nguyễn Thành Vấn, Lê Vân Anh Cường, 2008. Magnetotelluric analysis: Mohr circles. Sci. and Techn. Dev., 11/11 : 5-11. VN Nat. Univ.-HCM City.