Time Series Generation

1

Chapter 1

INTRODUCTION

Magnetotelluric Method The magnetotelluric method is a frequency domain electromagnetic tool that utilises

natural variations in the Earth’s magnetic field as a source. Variations in the earth’s

natural magnetic field supply frequencies ranging from nearly DC to several kilohertz,

thus giving one the ability to study the electric substructure of the earth to great

depths. The large frequency range also means that the method is not hampered by the

presence of conductive overburden or sampling frequencies that do not allow for deep

penetration.

A major advantage of the MT method is that it simultaneously measures the electric

and magnetic fields in two perpendicular directions. This provides useful information

about electrical anisotropy in an area. Other advantages include the wide frequency

range at which data can be sampled. It is also considerably cheaper than for example

deep reflection seismic surveys.

Cagnaird (1953) and Tikhonov (1950) developed the theory underlying the

magnetotelluric method independent of each other in the 1950’s. They both observed

that the electric and magnetic fields associated with telluric currents that flow in the

Earth as a result of variations in the Earth’s natural electromagnetic field, should relate

to each other in a certain way depending on the electrical characteristics of the Earth.

The ratio of the horizontal electric field to the orthogonal horizontal magnetic field

gives the electromagnetic impedance. This is measured at a range of frequencies

providing information about the resistivity of the earth as a function of frequency (and

therefore depth). The theory of the magnetotelluric method is based on the assumption

that the source is a natural electromagnetic plane wave propagating vertically

downward into a layered Earth. Sources: Natural MT signals come from a variety of sources, but in the frequency

range of interest (~0.001–104 Hz), the atmosphere and magnetosphere are the main

source regions. The higher frequency component mainly emanates from

meteorological activities such as lightning. Variations in the Earth’s magnetic field

linked to solar activity are responsible for a low frequency field.

Object: Object of my project is to generate two dimensional magnetotelluric data and

then process this synthetic data to calculate the impedance tensor. For synthetic data

genaretion, I used algorithm given by T Ernst et al in his paper Comparison of two

techniques for magnetotelluric data processing using synthetic data sets, 1999-2000.

Further processing of synthetic data was done by basic method of Magnetotelluric

data processing. For calculation of impedance tensor, I used least square estimation

method.

2

Chapter 2

Synthetic MT Time Series Generation

Algorithm: In this algorithm we can synthesis all three horizontal MT field vectors, the main

electric field E, the main magnetic field H, and the additional (remote) magnetic field

R. But I am interested only in synthesis of electric field E and magnetic field H.

synthesis of electric and magnetic field have following steps:

The discrete values of field vectors introduced in the spectral domain on the log

frequency grid {l+1: ln l+1 = ln (l + Δ)} can be presented in following common

way:

( ) ( ) ( ) j = x, y (2.1)

Where U= E, or H and A () is a common frequency amplitude dependence (real

function). Usually A () is taken in the form of A () = such an approximation is

natural for geomagnetic variations, and for period from 100 to 10000 s the power

index is estimated for middle latitudes in the range from -1.0 to -1.5. The introduction

of the equidistant grid provides better simulation of broad band variation ensembles.

The coefficients BUj express specific properties of particular field components and

may be written in the form:

( ) ( )

( ) ( ) (2.2)

where SUj is a magenetotelluric signal, NUj is the additive noise, and δUj is a coefficient

used to introduced large (but rare) outliers. The signal components SUj are related by

transfers operators:

(

)

(

)

(2.3)

The impedance operator Ẑ will be treated in the 2D structure typical for many

practical applications:

( ) ( ) (2.4)

(

) (2.5)

Where Ŵ() is a rotation operator for the clockwise right co-ordinate system, and Z1,

Z2 are the arbitrary scalar 1D impedances (taken, for example, for two chosen layered

structures).

3

The MT signal random structure is primarily introduced in spectral complex

amplitudes of magnetic fields SHx, SHy:

( ) ( )

(2.6)

( ) ( ) ( ) ( ) (2.7)

Here ( ) is the random quantity equally distributed at the segment [a, b], 0 ≤ (1-A)

≤ 1 is the random amplitude factor for SHx, SHy, 0 ≤ P ≤ 1 is the correspondent random

phase factor (in the particular case of A = 0 the spectral amplitudes become

determinated and only phases stay random), and 0 ≤ H ≤ 1 is the linear relation factor

between SHx and SHy. In accordance with (3), other spectral components take the form:

, j=x, y (2.8)

The Noise elements NUj in formula (2) are also determined by generation of equally

distributed random noise values. Each of these noise components consists of the

following parts: individual part NUj and one or two correlated one NEH, NHR:

( )

, (2.9)

( )

(2.10)

Here values , and NEH are randomly selected as ( ) ( ).

Coefficients are determined from the impedence matrix,

√| |

| |

Coefficients give the relative noise to signal level in each component (for

example, = 0 means the absence of random noise in the field Ex), and non-zero

coefficients c1 provide an addition of correlated noise components NEH into ensemble

of {E, H} fields, respectively.

The frequency domain outliers, as already mentioned, are introduced by the

coefficients Uj in relation (2). The desired number outliers are directly assigned for

each component Uj and their frequencies are selected randomly. For these selected

frequencies we have

( ) (with constants individual fro each

component), while for other frequencies .

4

At the last stage, we have to transform spectra Uj, defined on the frequency grid {l}.

Into the time domain by means of the inverse Fourier transform. Here arrays {Uj (l)}

may be interpreted as the approximated by the simplest rectangular formula:

( ) ( ), (

)

Where values

are the centers of the intervals ( ) and ( ) on the

log-frequency scale, correspondingly.

The following integral transform is further used:

( )

∑ ( ) ∫ ( )

(2.11)

With parameter s = +1 defining the proper variant of the transform. The integral in

relation (11), ignoring the A () variation on the interval (

) as compared with

the exponent, takes the simplified form and then it can easily solved.

The MT fields calculated in the time domain can also be supplied with trend and pulse

error, Pulse s may have rectangular or exponential form with amplitude attenuation ,

duration and starting moments selected on the random basis (either independently for

each field component , or simultaneously in the previously mentioned ensembles of

{E,H}). Trends are considered linear and also have random coefficients.

The algorithm described above forms the mathematical basis for all the

procedure of synthetic time series generation.

5

Chapter 3

Processing of MT data

In processing of MT data we remove noise and make data such it can use for further

calculation, like estimation of impedance tensor or input for inversion. In contrast, the

transfer functions of one site are a very small dataset, typically described by the

complex impedance tensor at 30–50 evaluation frequencies.

We next describe a simple but flexible processing scheme that can be applied either

automatically or with user-selected time windows. It can be operated in conjunction

with a least-square estimation of the transfer function, or incorporate robust

processing or remote reference processing of noisy data.

Calculating the Fourier coefficients of one segment of a single time series

Suppose, for example, that a digital time series that represents the Hx-component has

been cut into intervals of N data points, each of which can be denoted as xj, where

j=1,N . Then, by applying the trend removal and the cosine bell, and the discrete

Fourier transformation, we obtain N/2 pairs of Fourier coefficients am, bm which are

combined as complex Fourier coefficients

Where

∑

∑

Where m =1……. N/2.

In the following, we anticipate that all energy above the Nyquist frequency, NY, has

been removed from the analogue time series by an anti-alias prior to digitisation.

Therefore,

( )

and the discrete Fourier transformation recovers the entire information contained in

the digital time series xj, j=1, N.

Evaluation frequencies:

The choice of evaluation frequencies (or periods) is somehow arbitrary, but two

conditions apply:

1) The evaluation frequencies (or periods) should be equally spaced on a

logarithmic scale. For example, if we choose 10 s and 15 s as evaluation

periods, then we should also choose 100 s and 150 s (rather than 100 s and 105

s).

6

2) Ideally, we should have 6–10 evaluation frequencies per decade – more are

unnecessary.

A suggested frequency distribution is:

√ ⁄

⁄

√ ⁄

If the sampling interval is 20 s and therefore, fmax =0.25 Hz. The lowest frequency

(longest period) is determined by the window length. Typically, we might use a

window consisting of 512 data points. The first line in the raw spectrum should be

ignored, as it is affected by the cosine bell. Averaging is performed in the frequency

domain. At higher frequencies, more information is merged into the same evaluation

frequency in order to meet the requirement that the evaluation frequencies are equally

spaced on a logarithmic scale. (It could also be stated that, at short periods, we have

more independent information per period of the resulting transfer function). The

degree of averaging and the form of the spectral window is a compromise between the

principle of incorporating as much data as possible (because then the number of

degrees of freedom, which determines the confidence intervals of the transfer

functions, will be larger) and not smoothing too much (because mixing different

evaluation periods that are associated with different penetration depths reduces

resolution of individual conductivity structures). In other words, the aim is to reach a

trade-off between data errors and resolution.

We next provide an overview of a possible procedure for estimating the power and

cross spectra. For the kth evaluation frequency, we need to know the number of raw

data to be merged:

Where fs is a sampling frequency; Cr is a parzen radius; N is total number of data point

in spectra; ft is an evaluation frequency.

7

Chapter 4

Theory of estimation of Impedance Tensor Six different estimates of the magnetotelluric impedance tensor elements may be

computed from measured data by use of auto-power and cross-power density spectra.

Two of the six estimates are relatively unstable in the one-dimensional case when the

incident fields are unpolarized. For the remaining four estimates, it is shown that two

are unaffected by random noise on the H signal, but are biased upward by random

noise on the E signal. The remaining two estimates are unaffected by random noise on

the E signal, but are biased downward by random noise on the H signal. The

magnetotelluric sounding method for the determination of subsurface electrical

conductivity profiles as proposed by Cagniard is based upon the assumption of a

horizontally stratified layered earth model. For an anisotropic or laterally

inhomogeneous earth, the impedance becomes a tensor quantity.

Consider the equation

where Ex, Hx, and Hy may be considered to be Fourier transforms of measured electric

and magnetic field data. If one has two independent measurements of Ex, Hx, and Hy at

a given frequency, denoted by Ex1, Hx1, and Hy1 & Ex2, Hx2, and Hy2 respectively,

|

|

|

| (4.1.1)

And

|

|

|

| (4.1.2)

Provided,

(4.1.3)

Equation (4.1) simply states that the two field measurements must have different

source polarizations. If the two have the same polarization, they are not independent.

Since any physical measurement of E or H will include some noise, it is usually

desirable to make more than two independent measurements, and then to use some

type of averaging that will reduce the effects of the noise. Suppose one has n

measurements of Ex, Hx, and Hy, at a given frequency. One can then estimate Zxx and

Zxy in the mean-square sense; that is, one may define:

8

∑( )(

)

where is the complex conjugate of , etc., and then find the values of and

that minimize . Setting the derivatives of with respect to the real and

imaginary parts of to zero yields:

∑ ∑

∑

(4.3)

Similarly, setting the derivatives of with respect to the real and imaginary parts of

to zero yields:

∑ ∑

∑

(4.4)

Notice that the summations represent auto power and cross-power density spectra.

Equations (4.3) and (4.4) may be solved simultaneously for and . This solution

will minimize the error caused by noise on Ex. It is possible to define other mean-

square estimates that minimize other types of noise.

There are four distinct equations that arise from the various mean-square

estimates. In terms of the auto-power and cross-power density spectra, they are

(4.5)

(4.6)

(4.7)

(4.8)

Strictly speaking, equations (4.5) through (4.8) are valid only if ,

, etc.,

represent the power density spectra at a discrete frequency .

In practice, however, the Zij are slowly varying functions of frequency; consequently,

, etc., may be taken as averages over some finite bandwidth.

Estimation of impedance tensor from power density spectra:

Consider again equations (4.5) through (4.8). Under certain conditions, these

equations are independent, so that any two of them may be solved simultaneously for

Zxx and Zxy. Since there are six possible distinct pairs of equations, there are six ways

to estimate Zxx and Zxy. For example, the six estimates for Zxy are:

( )(

) ( )(

)

( )(

) ( )(

) (4.9)

9

( )(

) ( )(

)

( )(

) ( )(

) (4.10)

( )(

) ( )(

)

( )(

) ( )(

) (4.11)

( )(

) ( )(

)

( )(

) ( )(

) (4.12)

( )(

) ( )(

)

( )(

) ( )(

) (4.13)

( )(

) ( )(

)

( )(

) ( )(

) (4.14)

( )(

) ( )(

)

( )(

) ( )(

) (4.15)

( )(

) ( )(

)

( )(

) ( )(

) (4.16)

( )(

) ( )(

)

( )(

) ( )(

) (4.17)

( )(

) ( )(

)

( )(

) ( )(

) (4.18)

( )(

) ( )(

)

( )(

) ( )(

) (4.19)

( )(

) ( )(

)

( )(

) ( )(

) (4.20)

10

where denotes a measured estimate of . It turns out that two of these

expressions tend to be relatively unstable for the one-dimensional case, particularly

when the incident fields are unpolarized. For that case, ,

, , and

tend toward zero, so that equations (4.11) and (4.12) become indeterminant.

The other four expressions are quite stable and correctly predict ⁄ , for

the one-dimensional case, provided the incident fields are not highly polarized.

These same remark true for the Zyx. In each case there are six way to estimate

Zij, two of which are unstable for one-dimensional models with unpolarized incident

fields. Also, in each case, the four other estimates are quite stable for any reasonable

earth model, provided the incident field are not highly polarized.

11

Chapter 5

Generated Synthetic Time Series

Structure and different parameters for generation of synthetic time series:

For synthetic time data generation above algorithm was followed step by step with

minor changes. Our aim is to generate synthetic data set which contains four basic

component two horizontal electric and two horizontal magnetic fields.

Total duration of each data set is one day long with sampling interval 20 s, i.e.

a total 4320 samples in each field component. The spectrum of 78 harmonics spread

uniformly along the log-period scale in the interval of 80-42752 s was used in the

simulation of the magnetic field in the frequency domain. Amplitudes of the magnetic

spectra have both a regular part with power dependence on frequency and a random

factor. These regular parts of magnetic field spectra were generated with power factor,

, typical for middle latitudes. The random amplitude factor A was usually

chosen as 0.5, and the totally random phases, P = 1, were simulated, having uniform

distribution on the segment [-]. The electric field spectra were calculated from

created magnetic spectra by means of 2D impedance tensor . It was then transformed

to the final structure by the coordinate system rotation. The rotation angle was in this

case is 15 degree clockwise.The time domain outlier with random amplitudes and

random durations were not added in this synthetic data generation. For data generation

we give value of several parameters as input, which are following:

Further, I have to provide impedance of layered earth model. I assume a 2, four

layered earth model, which resistivity and layer thickness are as:

First layered model parameter:

Depth Resistivity (ohm/m)

0 100

1000 500

6000 1000

50000 1200

Second layered model parameter:

Depth Resistivity (ohm/m)

0 500

1000 100

6000 500

50000 1000

12

From these earth models we generate impedance tensor for 2D body, which is input

impedance Zxy and Zyx for synthetic data generation and plot of this impedance are

shown in Figure 5.1.

Figure 5.1: Input Impedance vs. Time

Using these parameters, I generate time series having 4320 sample with 20 s sampling

interval. 200 sample of each component of synthetic data were shown in following

Table 5.1 and Table 5.2.

13

Table 5.1: 200 Sample of Synthetic MT time series data (Ex and Ey);

Amplification factor = 1000

Ex Component Ey Component 0.0652 0.9323 0.1854 -0.2223 0.2749 -0.1158 -2.2381 0.5696 -0.6633 -1.9241

-0.2362 0.6312 0.188 0.3715 0.4815 0.7809 -1.4933 -1.0232 -1.1688 -0.3748

1.3287 0.1989 -0.251 0.7706 -1.7992 -1.3095 -0.4739 -0.2394 -1.0527 3.2854

1.3821 -0.1235 0.5311 -1.2089 -0.309 -1.8688 -1.3738 0.7498 1.2093 1.4078

-0.2613 0.6355 1.859 0.177 -0.1261 -0.3628 -0.9105 -2.5833 0.1745 0.9713

-0.0071 0.4914 0.017 -0.8078 -1.7327 0.0061 0.1245 -1.1712 0.2389 3.1287

-0.4611 0.415 -0.6516 -0.8535 0.9566 1.3869 0.0755 0.9406 0.1782 -1.5171

-1.4148 -0.6861 -0.0409 -0.307 -1.605 1.0499 1.1935 2.4816 0.3192 1.8989

0.9608 1.6295 2.6734 1.3564 -2.0408 -0.9739 -1.7501 -0.7696 -2.0801 3.1805

-1.3011 1.4213 0.3266 1.223 -0.5605 2.38 -2.7971 0.0686 -2.4185 0.8676

-0.5226 -1.2163 -0.3689 -0.6933 0.6705 0.195 0.7206 0.971 1.48 -0.35

-1.8394 0.2728 -0.4649 -0.3921 1.6764 -0.7427 1.2755 0.2784 1.8969 -1.777

0.4722 -0.064 -0.6067 -1.015 -0.8551 -1.1213 0.1126 2.1378 1.585 0.5593

3.2041 -1.3346 -0.9689 0.9621 1.3992 -4.7089 1.5362 0.5703 -0.1137 -1.854

-1.0004 -0.0124 -1.4621 2.2126 0.9919 1.3181 0.4995 2.0398 -2.6944 -0.134

1.5991 -2.3228 -0.0916 0.9059 -1.9442 -0.5492 3.4322 0.2559 -0.8073 3.1758

-0.1981 -1.6372 -1.0167 -0.1165 -0.5937 0.85 2.1551 0.3146 -2.5504 0.2224

0.0835 -0.0545 -0.6988 -0.2622 1.1029 -0.691 -0.194 0.8973 0.7188 -2.3225

0.5211 0.5655 0.2037 0.5276 1.2713 -1.6683 -1.5809 -0.851 0.4904 -0.7742

-0.0542 -1.1974 0.3093 -0.3619 -0.0536 1.4182 2.2617 1.0467 0.9058 0.6882

1.2044 0.7087 -2.0952 0.2966 -0.0768 -0.5796 -1.5331 3.9276 0.1984 0.3282

1.1542 -1.2454 -0.1105 -1.5203 -0.6021 -1.1606 1.4406 1.4633 1.1122 1.6968

0.2033 0.1145 -0.3604 0.2376 -0.5202 -1.3804 -0.2158 0.8922 -1.5248 0.8893

0.5622 2.4499 0.0005 -0.9463 2.2182 -0.2969 -2.3102 -0.0163 0.9007 -2.4239

-0.2446 -0.255 -0.0239 -0.9991 0.2749 0.2776 0.6845 -1.0324 1.1776 -1.1853

-0.0769 0.7508 -1.4326 -0.4654 -0.5028 0.3021 -0.7125 1.2366 1.4722 1.692

-0.3963 -1.1382 0.3485 1.0408 -0.0165 0.371 1.8773 -1.3381 -1.4791 0.0842

1.4467 -0.803 -0.2439 1.8846 -0.7787 -1.423 3.167 -0.4733 -1.5106 0.4322

-0.1814 -0.5256 -0.9249 0.839 0.0545 0.2079 -0.2628 1.9487 -1.7964 0.3398

-1.2943 -0.8256 0.899 0.9411 0.0418 2.9359 1.8249 -0.5918 0.2049 -0.1667

-0.298 1.4822 0.7887 -0.1611 -0.9595 1.4864 -0.2945 -2.0192 1.2205 0.3162

-1.2919 -0.1055 0.2243 0.6162 0.0816 1.7038 0.0383 0.0434 -1.1831 -0.7786

-0.4905 0.243 0.3365 2.2866 -0.1121 1.6235 -0.4042 0.1024 -3.4 -0.3628

1.4985 -0.0852 0.7827 0.9851 -0.2614 -0.6772 0.1421 0.6895 -0.5848 -0.1784

0.2517 -0.0496 -0.9823 0.8684 0.9257 -0.6216 -1.0674 2.6105 -0.85 -1.1175

-0.6506 1.0354 0.3337 -0.1736 2.1901 -0.6739 -1.4555 -0.9413 -0.104 -1.5713

-0.0748 -2.3068 1.5421 0.0812 1.8465 -0.6599 3.1818 -2.877 -0.1846 -2.4385

0.8642 0.4991 -2.2197 1.9772 0.8019 -1.2695 0.061 3.213 -3.9735 -1.4417

1.2298 0.1267 -2.2682 0.1113 1.1713 -1.5976 -0.4379 3.2847 -0.1609 -1.4036

1.9333 0.5691 1.5221 1.763 0.1549 -2.5791 -0.5625 -2.5566 -2.0963 -0.107

14

Table 5.2: 200 Sample of Synthetic MT time series data (Hx and Hy)

Hx Component Hy Component -0.032 1.4601 0.4539 -0.518 0.5639 -0.0448 1.1075 1.7275 -1.1457 0.1194

0.0369 1.2069 -0.1817 0.2767 0.1574 0.0553 1.229 0.0615 -0.0441 0.7444

0.1994 0.1367 0.4799 -0.1319 -1.3727 0.1488 0.0419 0.7086 0.2296 -0.8443

0.9327 -0.2388 0.7549 -0.3574 -0.8777 1.3531 -0.0673 1.8012 -0.4993 -0.889

0.269 -0.6629 -0.4147 -0.3711 -0.6736 0.4542 0.0715 0.4087 -0.7873 -0.3627

0.1764 0.3061 -0.0873 0.0188 -1.7386 0.0981 0.8855 -0.0573 -0.6349 -1.9104

-0.7596 -0.1743 -0.5252 0.7476 -0.4735 -0.9365 0.0023 -0.9509 0.5514 -0.0043

0.159 0.6694 -1.6593 0.1548 0.1023 -0.1717 0.6478 -1.8529 -0.0731 -0.2931

-0.1143 1.1566 -1.5902 -0.4493 -0.2084 -0.1224 1.5174 -0.9504 0.0161 -0.4936

-0.7812 0.5151 -0.5856 0.5519 -0.5376 -1.1641 0.1185 -0.4189 0.8487 -0.5188

-0.7544 -0.517 -0.3806 -0.7748 -0.0548 -1.2756 -1.1024 -0.0128 -0.1867 -0.113

-0.9398 -0.3219 -0.0907 -1.453 2.0759 -2.1838 -0.7688 -0.1705 -1.1596 2.1042

0.3212 -0.6972 -0.6736 -1.427 0.9125 0.6236 -1.0597 -0.4998 -2.0651 0.9557

2.048 -0.8971 -0.6996 0.5351 1.7356 2.9151 -1.9145 -1.0309 0.3533 2.0123

0.0324 -0.8963 -2.0053 -0.2552 0.3503 0.1764 -1.1508 -2.4588 -0.0719 0.6574

0.3077 -0.247 -1.1558 0.1013 -1.2033 1.2793 -0.2771 -0.9554 0.4514 -1.218

-0.2308 -0.9756 -1.4467 1.1716 -0.3865 0.5166 -1.5929 -1.9137 1.123 -0.3802

-0.7975 0.316 -0.6292 0.1589 0.9184 -0.9593 0.0351 -0.876 0.1234 1.2731

-0.4028 0.6761 -0.6273 -0.1152 0.0944 -0.6672 0.348 -0.7272 0.1439 0.9005

-0.0887 -0.2087 -0.7135 -0.4634 -0.6595 0.2373 -0.0615 -0.561 0.0153 0.5089

-0.6331 0.3994 0.1253 -0.2364 -0.2379 -0.5877 0.915 -0.2425 0.3556 0.3676

1.1065 -0.714 0.6127 0.1862 0.5288 1.5533 -1.0146 0.9135 0.2806 1.1496

1.8116 -0.5077 1.5275 -0.2219 -0.0594 1.5632 -0.7477 1.6201 -0.8231 0.0416

0.3644 0.7855 1.8041 -0.7631 0.395 0.2091 1.9019 1.5636 -1.2972 0.7414

0.8187 0.2828 1.5113 -1.2098 0.3869 0.3158 0.4996 1.2326 -1.7917 -0.2768

0.3872 1.6846 1.4862 -1.5577 0.0889 -0.0105 2.0433 1.4075 -1.9495 0.2343

-0.6863 0.5966 0.7119 -0.7672 0.7393 -1.7297 0.828 0.1346 -0.5917 0.9432

0.345 -1.6761 0.7725 -0.1584 -0.2942 -0.2669 -1.7188 0.8089 0.9607 -0.5016

-0.2421 -0.607 0.2363 -0.5619 -0.3542 -0.3129 -0.9328 -0.0589 -0.1907 -0.6554

-1.5326 0.4685 -0.5101 -0.1284 -0.5961 -2.3942 0.1995 -0.5689 1.1495 -1.0115

-1.2669 0.4229 0.6323 0.1153 -1.6281 -1.4422 0.5989 1.2435 1.3062 -1.8907

-0.766 0.2899 -0.3615 -0.2145 -0.6842 -0.8384 0.4472 0.0117 0.6337 -0.5541

-1.0771 0.6566 -1.2958 -0.1632 -0.1298 -1.1161 0.7119 -0.989 0.9681 -0.4627

-0.1789 -0.3771 -0.3147 -0.9887 0.1433 0.7765 -0.377 -0.1413 -0.4376 0.0372

0.672 -0.7866 -0.7956 -0.28 0.9374 0.5952 -1.3343 -0.2605 0.4622 1.288

1.5729 -0.6979 0.912 -0.7637 0.6426 1.1147 -0.6255 1.421 -0.6508 0.77

0.1605 -1.703 1.3981 -0.6067 0.3013 -0.4 -2.1341 1.4358 -1.2631 1.0759

0.5928 -1.1855 -0.8411 1.5074 0.2093 0.2866 -0.977 -1.0258 1.5767 0.9487

1.4084 0.495 -2.0326 0.7349 -0.5715 1.1342 0.7313 -2.7617 0.7799 -0.2582

1.4409 1.4001 -0.2953 0.5829 0.9844 1.2194 2.2953 -0.2062 1.0654 1.4437

15

Figure 5.2: Plot of Synthetic Time Series

0 1 2 3 4 5 6 7 8

x 104

-4

-2

0

2

4x 10

-3

Time (sec)

Am

pli

tud

e(m

V/m

)

Time Series of Ex

0 1 2 3 4 5 6 7 8

x 104

-6

-4

-2

0

2

4

6x 10

-3

Time (sec)

Am

pli

tud

e(m

V/m

)

Time Series of Ey

0 1 2 3 4 5 6 7 8

x 104

-4

-2

0

2

4

Time (sec)

Am

pli

tud

e(n

T)

Time Series of Hy

0 1 2 3 4 5 6 7 8

x 104

-4

-2

0

2

4

Time (sec)

Am

pli

tud

e(n

T)

Time Series of Hy

16

Chapter 6

Estimated Impedance Tensor

In processing step, first we divide the whole data into five segments with 20%

overlapping. Then I take Fourier transform of each segment and decide 16 evaluation

frequencies, equally spaced in logarithmic scale .Then I apply Parzen windows,

parzen radius is 0.5, on each spectra. Plot of one parzen window for evaluation

frequency ft= 256 Hz was plotted in figure 6.1. At each evaluation we generate parzen

window. And we use this window in averaging of spectra at evaluation frequencies.

Then I estimate cross power and auto power density of spectra and using least square

method, I calculate six impedance tensor Zxy and Zyx for each segment.

Figure 6.1: Plot of Parzen Window in frequency domain

Calculated Zxy and Zyx for each segment and for each equation is presented in

following table.

0 50 100 150 200 250 300 350 400 450 5000

0.2

0.4

0.6

0.8

1

Frequency(Hz)

Am

pli

tud

e

Parzen Window

17

Estimated Zxy for each segment and each equation: Table 6.1: Zxy using equation (4.9)

Zxy from segment1

Zxy from segment2

Zxy from segment3

Zxy from segment4

Zxy from segment5

Mean Zxy

0.0039 0.0008 0.0012 0.0017 0.0012 0.0018

0.0023 0.0019 0.0009 0.0019 0.0031 0.002

0.0021 0.0024 0.0027 0.0018 0.0025 0.0023

0.0023 0.0039 0.0025 0.0021 0.0018 0.0025

0.0047 0.0039 0.0029 0.0037 0.0021 0.0035

0.0024 0.0036 0.0035 0.0031 0.0025 0.003

0.0057 0.0046 0.0051 0.0034 0.0061 0.005

0.0013 0.0011 0.0017 0.0014 0.0039 0.0019

0.0026 0.0028 0.004 0.0017 0.0032 0.0029

0.0012 0.0017 0.0022 0.0014 0.0011 0.0015

0.003 0.0032 0.0036 0.0035 0.0036 0.0034

0.0033 0.0043 0.0048 0.0036 0.0052 0.0042

0.0035 0.0039 0.0035 0.0036 0.0037 0.0036

0.0027 0.003 0.004 0.0033 0.0024 0.0031

0.0031 0.004 0.0038 0.0036 0.0041 0.0037

0.0023 0.0031 0.0032 0.0026 0.0043 0.0031

Table 6.2: Zxy using equation (4.10)

Zxy from segment1

Zxy from segment2

Zxy from segment3

Zxy from segment4

Zxy from segment5

Mean Zxy

0.0038 0.0008 0.0012 0.0017 0.0011 0.0017

0.0023 0.002 0.0017 0.002 0.003 0.0022

0.0021 0.0025 0.0028 0.0021 0.0026 0.0024

0.0026 0.0059 0.0027 0.0023 0.0019 0.0031

0.0118 0.0142 0.0049 0.0046 0.0048 0.0081

0.0025 0.0036 0.0033 0.0027 0.0027 0.003

0.0044 0.0031 0.0058 0.0025 0.0104 0.0052

0.0022 0.0021 0.0041 0.0025 0.0117 0.0045

0.0031 0.003 0.0048 0.0018 0.0036 0.0033

0.0025 0.0047 0.0037 0.0029 0.0024 0.0032

0.0041 0.0038 0.0041 0.0044 0.0042 0.0041

0.0051 0.0062 0.0084 0.0041 0.0082 0.0064

0.0071 0.0076 0.0081 0.0074 0.0077 0.0076

0.0068 0.0068 0.0066 0.0058 0.0062 0.0064

0.0058 0.0063 0.0071 0.0054 0.0071 0.0063

0.0036 0.0037 0.0044 0.0034 0.0056 0.0042

18


Zxy from segment1

Zxy from segment2

Zxy from segment3

Zxy from segment4

Zxy from segment5

Mean Zxy

0.0038 0.0008 0.0012 0.0017 0.0013 0.0018

0.0023 0.0022 0.0017 0.002 0.0031 0.0023

0.0021 0.0028 0.003 0.0021 0.0029 0.0026

0.0029 0.0162 0.0033 0.0029 0.0021 0.0055

0.0226 0.0072 0.0068 0.0062 0.006 0.0098

0.0026 0.0043 0.0038 0.003 0.003 0.0033

0.0059 0.0035 0.0078 0.0027 0.0088 0.0057

0.0047 0.0022 0.0434 0.0031 0.0039 0.0115

0.0034 0.0032 0.0054 0.0021 0.0041 0.0037

0.0028 0.0048 0.0038 0.0033 0.0027 0.0035

0.0047 0.0043 0.0044 0.0052 0.0047 0.0047

0.0079 0.0085 0.0139 0.0059 0.0212 0.0115

0.0127 0.0131 0.017 0.014 0.0131 0.014

0.0145 0.0288 0.0329 0.0128 0.0117 0.0201

0.0094 0.0093 0.0138 0.0088 0.0131 0.0109

0.0062 0.005 0.007 0.0057 0.0093 0.0066


Zxy from segment1

Zxy from segment2

Zxy from segment3

Zxy from segment4

Zxy from segment5

Mean Zxy

0.0036 0.0008 0.0012 0.0017 0.001 0.0017

0.0023 0.002 0.0016 0.002 0.003 0.0022

0.0021 0.0025 0.0028 0.0022 0.0025 0.0024

0.0022 0.0049 0.0027 0.0023 0.0018 0.0028

0.0082 0.0236 0.0054 0.0046 0.0057 0.0095

0.0031 0.0051 0.0042 0.0034 0.0039 0.004

0.0036 0.0029 0.0048 0.0024 0.0066 0.0041

0.0012 0.0013 0.002 0.0019 0.0042 0.0021

0.0042 0.0044 0.0064 0.0019 0.004 0.0042

0.0025 0.0036 0.0031 0.003 0.0027 0.003

0.0051 0.0048 0.0048 0.006 0.0048 0.0051

0.0086 0.0137 0.0108 0.007 0.0136 0.0107

0.0213 0.0188 0.029 0.0364 0.0175 0.0246

0.0496 0.2819 0.0316 0.0159 0.0152 0.0788

0.0198 0.0118 0.0404 0.0152 0.0155 0.0205

0.0054 0.0049 0.0059 0.0049 0.0073 0.0057

19


Zxy from segment1

Zxy from segment2

Zxy from segment3

Zxy from segment4

Zxy from segment5

Mean Zxy

0.0036 0.0008 0.0012 0.0018 0.001 0.0017

0.0023 0.002 0.0011 0.0018 0.0032 0.0021

0.0021 0.0027 0.0028 0.0019 0.0025 0.0024

0.0018 0.0037 0.0022 0.0019 0.0018 0.0023

0.0149 0.0276 0.0083 0.0059 0.0139 0.0141

0.0032 0.0064 0.005 0.0039 0.0037 0.0044

0.0048 0.0034 0.0052 0.0027 0.0059 0.0044

0.0008 0.0009 0.0013 0.0014 0.0022 0.0013

0.0023 0.0029 0.0048 0.0009 0.0041 0.003

0.0019 0.0025 0.0027 0.0025 0.002 0.0023

0.0056 0.0051 0.0051 0.0075 0.0053 0.0057

0.0033 0.0069 0.0025 0.0025 0.0051 0.004

0.0129 0.0125 0.0114 0.0088 0.0171 0.0125

0.0074 0.0037 0.003 0.0059 0.0152 0.007

0.0088 0.0092 0.0072 0.0069 0.0137 0.0091

0.0047 0.0059 0.006 0.0064 0.0119 0.007


Zxy from segment1

Zxy from segment2

Zxy from segment3

Zxy from segment4

Zxy from segment5

Mean Zxy

0.0036 0.0008 0.0012 0.0017 0.0009 0.0017

0.0023 0.0019 0.0015 0.0019 0.0029 0.0021

0.0021 0.0023 0.0025 0.0018 0.0024 0.0022

0.0013 0.001 0.0016 0.0015 0.0016 0.0014

0.0012 0.0005 0.0021 0.0021 0.0016 0.0015

0.0016 0.0013 0.0016 0.0016 0.0014 0.0015

0.0015 0.0016 0.0013 0.0017 0.0013 0.0015

0.0006 0.0005 0.0004 0.0006 0.0002 0.0004

0.0006 0.0007 0.0009 0.0006 0.0011 0.0008

0.0011 0.0007 0.0011 0.0012 0.0012 0.001

0.002 0.0019 0.0018 0.0016 0.0019 0.0018

0.0012 0.0017 0.0008 0.0016 0.0013 0.0013

0.0014 0.0015 0.0012 0.0013 0.0014 0.0014

0.0012 0.0012 0.0012 0.0013 0.0014 0.0012

0.0013 0.0015 0.0011 0.0014 0.0013 0.0013

0.0008 0.0008 0.0008 0.0008 0.0007 0.0008

20

Estimated Zyx for each segment and each equation:

Table 6.7: Zyx using equation (4.15)

Zyx from segment1

Zyx from segment2

Zxy from segment3

Zyx from segment4

Zyx from segment5

Mean Zyx

0.0013 0.0003 0.0004 0.0003 0.0009 0.0006

0.0015 0.0004 0.0019 0.0019 0.0015 0.0015

0.0025 0.0019 0.0025 0.004 0.0015 0.0025

0.0022 0.0035 0.002 0.0017 0.0022 0.0023

0.0046 0.0042 0.0021 0.003 0.0019 0.0032

0.0045 0.0074 0.0068 0.0066 0.0054 0.0062

0.0037 0.009 0.0034 0.0057 0.0093 0.0062

0.0015 0.0011 0.0016 0.0011 0.0025 0.0016

0.0041 0.0016 0.0049 0.0018 0.0036 0.0032

0.0044 0.0031 0.0047 0.0056 0.004 0.0044

0.0038 0.0044 0.0033 0.0025 0.002 0.0032

0.0035 0.0028 0.0033 0.0023 0.0027 0.0029

0.0035 0.0025 0.0036 0.0032 0.003 0.0031

0.0079 0.0059 0.0024 0.0027 0.007 0.0052

0.0032 0.0024 0.0032 0.0026 0.0024 0.0028

0.0028 0.0009 0.0028 0.0019 0.0025 0.0022


Zyx from segment1

Zyx from segment2

Zxy from segment3

Zyx from segment4

Zyx from segment5

Mean Zyx

0.0013 0.0003 0.0004 0.0004 0.0008 0.0006

0.0016 0.0011 0.0028 0.0023 0.0016 0.0019

0.0025 0.0028 0.0029 0.0044 0.0021 0.0029

0.0023 0.0046 0.0023 0.002 0.0023 0.0027

0.0202 0.041 0.0122 0.0083 0.0224 0.0208

0.0059 0.0129 0.0095 0.0079 0.0078 0.0088

0.0056 0.0069 0.0074 0.0052 0.0094 0.0069

0.0029 0.004 0.004 0.0053 0.0054 0.0043

0.005 0.0063 0.0087 0.0033 0.0074 0.0061

0.0069 0.0088 0.0082 0.0081 0.0068 0.0077

0.0132 0.0127 0.0124 0.0193 0.0118 0.0138

0.0105 0.0196 0.0089 0.0072 0.0122 0.0117

0.034 0.0324 0.0308 0.0242 0.0439 0.0331

0.0257 0.0159 0.0111 0.0177 0.0424 0.0226

0.0249 0.0243 0.0207 0.0197 0.0341 0.0247

0.0123 0.015 0.0137 0.0178 0.0251 0.0168

21


Zyx from segment1

Zyx from segment2

Zxy from segment3

Zyx from segment4

Zyx from segment5

Mean Zyx

0.0013 0.0003 0.0004 0.0003 0.0007 0.0006

0.0016 0.0005 0.0009 0.0016 0.0014 0.0012

0.0024 0.0016 0.0025 0.0034 0.0018 0.0024

0.0023 0.005 0.0023 0.0019 0.0023 0.0027

0.01 0.0304 0.006 0.005 0.0066 0.0116

0.005 0.009 0.0072 0.006 0.0072 0.0069

0.0037 0.0054 0.0058 0.0043 0.0098 0.0058

0.0026 0.0035 0.0041 0.0044 0.0074 0.0044

0.0074 0.0063 0.0092 0.0055 0.0057 0.0068

0.0073 0.0106 0.0086 0.0084 0.007 0.0084

0.009 0.0091 0.0092 0.0114 0.0081 0.0093

0.0151 0.0223 0.0197 0.0084 0.0178 0.0167

0.0401 0.0352 0.0574 0.0706 0.0334 0.0473

0.0989 0.5687 0.05 0.0272 0.0281 0.1546

0.0363 0.0211 0.077 0.0264 0.0263 0.0374

0.0093 0.0083 0.0094 0.0087 0.0108 0.0093


Zyx from segment1

Zyx from segment2

Zxy from segment3

Zyx from segment4

Zyx from segment5

Mean Zyx

0.0013 0.0003 0.0004 0.0003 0.001 0.0007

0.0016 0.0006 0.0082 0.0019 0.0015 0.0028

0.0025 0.0029 0.0032 0.0048 0.0023 0.0031

0.0034 0.0166 0.0029 0.0026 0.0026 0.0056

0.0296 0.0099 0.0087 0.0076 0.0085 0.0129

0.0043 0.0081 0.0066 0.0056 0.0058 0.0061

0.0056 0.0061 0.0092 0.0043 0.0128 0.0076

0.004 0.002 0.0473 0.0031 0.0048 0.0122

0.0054 0.004 0.0079 0.0015 0.0058 0.0049

0.0055 0.01 0.0078 0.0071 0.005 0.0071

0.0064 0.0057 0.0055 0.0062 0.0061 0.006

0.0099 0.0088 0.0162 0.004 0.0252 0.0128

0.0195 0.0195 0.0267 0.0205 0.0203 0.0213

0.0254 0.0542 0.0481 0.0172 0.017 0.0324

0.0132 0.0122 0.019 0.01 0.0184 0.0146

0.0088 0.0069 0.0101 0.0076 0.0136 0.0094

22


Zyx from segment1

Zyx from segment2

Zxy from segment3

Zyx from segment4

Zyx from segment5

Mean Zyx

0.0013 0.0003 0.0004 0.0003 0.0008 0.0006

0.0015 0.0004 0.0005 0.0017 0.0014 0.0011

0.0024 0.0015 0.0025 0.003 0.0018 0.0023

0.0025 0.0052 0.0021 0.0017 0.0023 0.0028

0.014 0.0172 0.005 0.0046 0.0052 0.0092

0.0037 0.0059 0.0053 0.0043 0.0045 0.0047

0.0038 0.005 0.006 0.0038 0.014 0.0065

0.0013 0.0014 0.003 0.0018 0.011 0.0037

0.004 0.0029 0.0057 0.0011 0.0043 0.0036

0.0038 0.0082 0.0069 0.0053 0.0034 0.0055

0.0046 0.0043 0.0043 0.0043 0.0046 0.0044

0.0039 0.0042 0.0052 0.0017 0.006 0.0042

0.0082 0.0087 0.0096 0.0081 0.0093 0.0088

0.0069 0.006 0.0047 0.0049 0.0061 0.0057

0.0056 0.0061 0.0068 0.0042 0.0072 0.006

0.0036 0.0039 0.0046 0.0032 0.0061 0.0043


Zyx from segment1

Zyx from segment2

Zxy from segment3

Zyx from segment4

Zyx from segment5

Mean Zyx

0.0013 0.0003 0.0004 0.0003 0.0007 0.0006

0.0016 0.0004 0.0007 0.0017 0.0013 0.0011

0.0024 0.0013 0.0023 0.0028 0.0017 0.0021

0.0017 0.0014 0.0015 0.0013 0.0021 0.0016

0.0012 0.0003 0.0013 0.0016 0.0006 0.001

0.0018 0.0012 0.0017 0.0017 0.0016 0.0016

0.0012 0.0024 0.0015 0.0024 0.003 0.0021

0.0009 0.0008 0.0008 0.0007 0.0007 0.0008

0.0011 0.0007 0.001 0.0007 0.0011 0.0009

0.0022 0.0021 0.0027 0.0024 0.002 0.0023

0.0016 0.0016 0.0016 0.0009 0.0017 0.0015

0.0014 0.001 0.0017 0.0011 0.0015 0.0013

0.0009 0.001 0.001 0.0012 0.0008 0.001

0.0011 0.0019 0.0018 0.0011 0.0006 0.0013

0.0008 0.001 0.001 0.0008 0.0007 0.0009

0.0006 0.0005 0.0006 0.0004 0.0004 0.0005

23

Figure 6.2: Plot of mean Impedance tensor vs time

101

102

103

104

10-5

10-4

10-3

10-2

Time(sec)

Imp

eden

ce(o

hm

)

Estimated mean Zxy and Zyx

Zxy

Zyx

24

Chapter 7

CONCLUSION

Synthetic MT time series was shown in Figure5.2, which is similar to MT time

series which record in field.

Six different estimates of the magnetotelluric impedance tensor have been

computed from synthetically generated data by use of auto-power and cross-

power density of electric and magnetic field components.

Six different estimates of the magnetotelluric impedance tensor have value in

range of .01-.0001 which is same range of the input impedance tensor

(Figure5.1) for synthetic data generation.

Almost each impedance tensor show same trend with time period (Figure 6.2),

the decrement in value of impedance tensor when time period increase.

25

Chapter 8

REFERENCE

Alexander A. Kaufman and George V. Keller (1981), “The Magentotelluric

Sounding Method”, Elsevier Scientific Publishing Company, pp. 1-2, 431,457-

463.

Sokolova, E. Yu. B.S. Svetov and I.M. Varentsov, 1994, “The Study of MT

data Processing Techniques using Synthetic Time Series (the COMDAT

project)”, 12th

Workshop on EM Induction in the Earth, Brest, France,105.

Tomasz Ernst, Elena Yu. Sokolova, Ivan M. Varentsov and Nikolay G.

Golubev (2001), “Comparison of Two Techniques for Magnetotelluric Data

Processing using Synthetic Data Sets”, Vol. XLIX, No. 2, Institute of

Geophysics, Polish Academy of Sciences ul. Ksiecis Janusza 64, 01-452

Warszawa, Poland

W.E. Sims, F. X. Bostick, JR., and H. W. Smiths (1971), “The Estimation of

Magnetotelluric Impedance Tensor Elements from measured data”, Geophysics

Vol. 36, No. 5, pp.938-942.

Simpson, Fiona and Bahr, Karsten (2005), “Practical Magnetotellurics”,

Cambridge University Press, pp. 58-71.

26

APPENDIX 1: Program for Time Series Generation

%-------------------------------------------------------------------------- % Part-1: Selecting Frequency %-------------------------------------------------------------------------- clear all clc t = linspace(log(80),log(42752),80); timea = exp(t); omeg = 2*pi./t; for k=1:79 o = linspace(log(omeg(k)),log(omeg(k+1)),3); omegm(k) = exp(o(2)); end time = timea(2:79); omega = omeg(2:79); %-------------------------------------------------------------------------- % Part-2: Calulation of Amplitude Factor %-------------------------------------------------------------------------- alph = -1.4; A1 = omega.^(alph); %-------------------------------------------------------------------------- % Part-3: Signal part of x and y component of MT Field %-------------------------------------------------------------------------- epsA = 0.5; epsP = 1; epsH = 0.6; theta = 15; ji_1 = 2*rand(1,78); ji_2 = -pi+2*pi*rand(1,78); S_Hx = (1-epsA*ji_1).*exp(i*epsP*ji_2); S_Hy = epsH*S_Hx +(1-epsH)*((1-epsA)+ epsA*ji_1).*exp(i*epsP*ji_2); z1 = Imped_1D(time); z2 = Imped_1D1(time); W = [cosd(theta), sind(theta); -sind(theta), cosd(theta) ]; for k=1:78 Z0 = [0,z1(k);-z2(k),0]; Z = W*Z0*W; sgmZx(k) = sqrt(abs(Z(1,1))*abs(Z(1,1))+abs(Z(1,2))*abs(Z(1,2))); sgmZy(k) = sqrt(abs(Z(2,1))*abs(Z(2,1))+abs(Z(2,2))*abs(Z(2,2))); S_E = Z*[S_Hx(k); S_Hy(k)]; S_Ex(k) = S_E(1,1); S_Ey(k) = S_E(2,1); end %-------------------------------------------------------------------------- % Part-4: Noise part of x and y component of MT Field %-------------------------------------------------------------------------- epsc1 = .1; sgmEx = .01; sgmEy = .01; sgmHx = .01; sgmHy = .01;

NEH = 2.*rand(1,78).*exp(i*epsP*(2*pi+2*pi*rand(1,78))); NHx = 2.*rand(1,78).*exp(i*epsP*(2*pi+2*pi*rand(1,78))); NHy = 2.*rand(1,78).*exp(i*epsP*(2*pi+2*pi*rand(1,78))); NEx = 2.*rand(1,78).*exp(i*epsP*(2*pi+2*pi*rand(1,78))); NEy = 2.*rand(1,78).*exp(i*epsP*(2*pi+2*pi*rand(1,78))); for k=1:78 N_Ex = ((1-epsc1)*NEx + epsc1*NEH)*sgmEx*sgmZx(k); N_Ey = ((1-epsc1)*NEy + epsc1*NEH)*sgmEy*sgmZy(k); N_Hx = ((1-epsc1)*NHx + epsc1*NEH)*sgmHx; N_Hy = ((1-epsc1)*NHy + epsc1*NEH)*sgmHy;

27

end %-------------------------------------------------------------------------- % Part-5: Generation of synthetic MT Signal in Frequency Domain %-------------------------------------------------------------------------- UEx = (S_Ex + N_Ex).*A1; UEy = (S_Ey + N_Ey).*A1; UHx = (S_Hx + N_Hx).*A1; UHy = (S_Hy + N_Hy).*A1; %-------------------------------------------------------------------------- % Part-6: Transformation of data from frequency to time domain % ------------------------------------------------------------------------- Ext1 = ftot(UEx,A1,omegm); Eyt1 = ftot(UEy,A1,omegm); Hxt1 = ftot(UHx,A1,omegm); Hyt1 = ftot(UHy,A1,omegm); Ext = real(Ext1)'; Eyt = real(Eyt1)'; Hxt = real(Hxt1)'; Hyt = real(Hyt1)'; %-------------------------------------------------------------------------- %-------------------------------------------------------------------------- %-------------------------------------------------------------------------- function timeout=ftot(omegin,A1,omegm) del_t = 20; s=-1; for l=1:78 cal1(l) = omegin(l).*A1(l)./(i); end

for k=1:4320 cal3 = 0; for l=1:78 cal2 = (exp(i*s*k*del_t*omegm(l+1))-

exp(i*s*k*del_t*omegm(l)))/(s*del_t); cal3 = real(cal1(l)*cal2)+cal3; end timeout(k) = cal3/2*pi; end %--------------------------------------------------------------------------

28

APPENDIX 2: Program for Parzen Window

function [EHxy, p] = przn(sig_w,ft,Cr,freq) %-------------------------------------------------------------------------- % f = load Frequency.dat; Cr = Parzen radius; % ft = Evaluation Frequency; %-------------------------------------------------------------------------- n1=length(ft); n2=length(freq); f=freq; for j=1:n1 fr = Cr*ft(j); for k=1:n2 a(j,k) = abs(ft(j)-f(k)); u(j,k) = pi*a(j,k)/fr; if a(j,k) == 0 p(k,j) = 1; elseif a(j,k) < fr p(k,j) = (sin(u(j,k))/u(j,k))^4; else p(k,j) = 0; end end end

for s=1:5 q=1; for l=1:4 for m=1:4 for j=1:16 EHxy(s,q,j) = mean(sig_w(:,l,s).*conj(sig_w(:,m,s)).*... p(:,j))/(mean(p(:,j))); end q=q+1; end end end

29

APPENDIX 3: Program for estimation of Impedence

tensor %-------------------------------------------------------------------------- % Defining 5 Segments;;segment1=1:1024;segment2=819:1843;

% segment3=1438:2462;segment=2257:3281; % segment5=3076:4100;Take fft of each segment of each signal; %-------------------------------------------------------------------------- [Exf_w1,Exf_w2,Exf_w3,Exf_w4,Exf_w5,freq] = fft_seg(Ext,1024); [Eyf_w1,Eyf_w2,Eyf_w3,Eyf_w4,Eyf_w5,freq] = fft_seg(Eyt,1024); [Hxf_w1,Hxf_w2,Hxf_w3,Hxf_w4,Hxf_w5,freq] = fft_seg(Hxt,1024); [Hyf_w1,Hyf_w2,Hyf_w3,Hyf_w4,Hyf_w5,freq] = fft_seg(Hyt,1024);

sig_w(:,:,1) = [Exf_w1 Eyf_w1 Hxf_w1 Hyf_w1]; sig_w(:,:,2) = [Exf_w2 Eyf_w2 Hxf_w2 Hyf_w2]; sig_w(:,:,3) = [Exf_w3 Eyf_w3 Hxf_w3 Hyf_w3]; sig_w(:,:,4) = [Exf_w4 Eyf_w4 Hxf_w4 Hyf_w4]; sig_w(:,:,5) = [Exf_w5 Eyf_w5 Hxf_w5 Hyf_w5]; %-------------------------------------------------------------------------- % Defining Evaluation frequency % fmax=freq(N/2); % for k=1:17 % ft_approx(k)=fmax/2^((k-1)/2); % end % ft ~= ft_approx; %-------------------------------------------------------------------------- ft=[freq(3) freq(4) freq(6) freq(8) freq(11) freq(16) freq(23)... freq(32) freq(45) freq(64) freq(91) freq(128) freq(181) freq(256)... freq(362) freq(512)];

Cr=0.5; % Parzen Radius [EHxy,p] = przn(sig_w,ft,Cr,freq);

%-------------------------------------------------------------------------- %Zxy and Zyx for each equation:: %Zxy_1(j,k):Zxy_eqno.(evaluation frequency,segment) %-------------------------------------------------------------------------- for k=1:5 for j=1:16 Zxy_1(j,k) = (EHxy(k,9,j)*EHxy(k,2,j)-EHxy(k,10,j)*EHxy(k,1,j))/... (EHxy(k,9,j)*EHxy(k,14,j)-EHxy(k,10,j)*EHxy(k,13,j)); end end %-------------------------------------------------------------------------- for k=1:5 for j=1:16 Zxy_2(j,k) = (EHxy(k,9,j)*EHxy(k,3,j)-EHxy(k,11,j)*EHxy(k,1,j))/... (EHxy(k,9,j)*EHxy(k,15,j)-EHxy(k,11,j)*EHxy(k,13,j)); end end %-------------------------------------------------------------------------- for k=1:5 for j=1:16 Zxy_3(j,k) = (EHxy(k,9,j)*EHxy(k,4,j)-EHxy(k,12,j)*EHxy(k,1,j))/... (EHxy(k,9,j)*EHxy(k,16,j)-EHxy(k,12,j)*EHxy(k,13,j)); end end %-------------------------------------------------------------------------- for k=1:5 for j=1:16

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Zxy_4(j,k) = (EHxy(k,10,j)*EHxy(k,3,j)-

EHxy(k,11,j)*EHxy(k,2,j))/... (EHxy(k,10,j)*EHxy(k,15,j)-EHxy(k,11,j)*EHxy(k,14,j)); end end %-------------------------------------------------------------------------- for k=1:5 for j=1:16 Zxy_5(j,k) = (EHxy(k,10,j)*EHxy(k,4,j)-

EHxy(k,12,j)*EHxy(k,2,j))/... (EHxy(k,10,j)*EHxy(k,16,j)-EHxy(k,12,j)*EHxy(k,14,j)); end end %-------------------------------------------------------------------------- for k=1:5 for j=1:16 Zxy_6(j,k) = (EHxy(k,11,j)*EHxy(k,4,j)-

EHxy(k,12,j)*EHxy(k,3,j))/... (EHxy(k,11,j)*EHxy(k,16,j)-EHxy(k,12,j)*EHxy(k,15,j)); end end %-------------------------------------------------------------------------- %-------------------------------------------------------------------------- for k=1:5 for j=1:16 Zyx_1(j,k) = (EHxy(k,14,j)*EHxy(k,5,j)-

EHxy(k,13,j)*EHxy(k,6,j))/... (EHxy(k,14,j)*EHxy(k,9,j)-EHxy(k,13,j)*EHxy(k,10,j)); end end %-------------------------------------------------------------------------- for k=1:5 for j=1:16 Zyx_2(j,k) = (EHxy(k,14,j)*EHxy(k,8,j)-




EHxy(k,15,j)*EHxy(k,5,j))/... (EHxy(k,13,j)*EHxy(k,11,j)-EHxy(k,15,j)*EHxy(k,9,j)); end

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end %-------------------------------------------------------------------------- for k=1:5 for j=1:16 Zyx_6(j,k) = (EHxy(k,16,j)*EHxy(k,7,j)-

EHxy(k,15,j)*EHxy(k,8,j))/... (EHxy(k,16,j)*EHxy(k,11,j)-EHxy(k,15,j)*EHxy(k,12,j)); end end %-------------------------------------------------------------------------- Zxym1=abs(Zxy_1); Zxym2=abs(Zxy_2); Zxym3=abs(Zxy_3); Zxym4=abs(Zxy_4); Zxym5=abs(Zxy_5); Zxym6=abs(Zxy_6);

Zyxm1=abs(Zyx_1); Zyxm2=abs(Zyx_2); Zyxm3=abs(Zyx_3); Zyxm4=abs(Zyx_4); Zyxm5=abs(Zyx_5); Zyxm6=abs(Zyx_6); %-------------------------------------------------------------------------- t=1./ft; Zxym1m = Zxym1(:,1)+Zxym1(:,2)+Zxym1(:,3)+Zxym1(:,4)+Zxym1(:,5); Zxym2m = Zxym2(:,1)+Zxym2(:,2)+Zxym2(:,3)+Zxym2(:,4)+Zxym2(:,5); Zxym3m = Zxym3(:,1)+Zxym3(:,2)+Zxym3(:,3)+Zxym3(:,4)+Zxym3(:,5); Zxym4m = Zxym4(:,1)+Zxym4(:,2)+Zxym4(:,3)+Zxym4(:,4)+Zxym4(:,5); Zxym5m = Zxym5(:,1)+Zxym5(:,2)+Zxym5(:,3)+Zxym5(:,4)+Zxym5(:,5); Zxym6m = Zxym6(:,1)+Zxym6(:,2)+Zxym6(:,3)+Zxym6(:,4)+Zxym6(:,5);

Zyxm1m = Zyxm1(:,1)+Zyxm1(:,2)+Zyxm1(:,3)+Zyxm1(:,4)+Zyxm1(:,5); Zyxm2m = Zyxm2(:,1)+Zyxm2(:,2)+Zyxm2(:,3)+Zyxm2(:,4)+Zyxm2(:,5); Zyxm3m = Zyxm3(:,1)+Zyxm3(:,2)+Zyxm3(:,3)+Zyxm3(:,4)+Zyxm3(:,5); Zyxm4m = Zyxm4(:,1)+Zyxm4(:,2)+Zyxm4(:,3)+Zyxm4(:,4)+Zyxm4(:,5); Zyxm5m = Zyxm5(:,1)+Zyxm5(:,2)+Zyxm5(:,3)+Zyxm5(:,4)+Zyxm5(:,5); Zyxm6m = Zyxm6(:,1)+Zyxm6(:,2)+Zyxm6(:,3)+Zyxm6(:,4)+Zyxm6(:,5); %-------------------------------------------------------------------------- %-------------------------------------------------------------------------- %-------------------------------------------------------------------------- function [sigf_w1,sigf_w2,sigf_w3,sigf_w4,sigf_w5,freq] = fft_seg(sigt,N) % N=1024; sigt_w1 = sigt(1:1024); sigt_w2 = sigt(819:1843); sigt_w3 = sigt(1438:2462); sigt_w4 = sigt(2257:3281); sigt_w5 = sigt(3076:4100);

sigf_w1 = fft(sigt_w1,N/2); sigf_w2 = fft(sigt_w2,N/2); sigf_w3 = fft(sigt_w3,N/2); sigf_w4 = fft(sigt_w4,N/2); sigf_w5 = fft(sigt_w5,N/2);

del_t = 20; f_sam = 1/del_t; freq = f_sam/N:f_sam/N:f_sam/2; %--------------------------------------------------------------------------

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