generation of synthetic wide-band electromagnetic time series · 2019. 11. 4. · generation of...

ANNALS OF GEOPHYSICS, VOL. 45, N. 2, April 2002

289

Generation of synthetic wide-bandelectromagnetic time series

Mariano Loddo, Domenico Schiavone and Agata SiniscalchiDipartimento di Geologia e Geofisica, Università di Bari, Italy

AbstractThe estimation of the earth transfer functions in MT prospecting method poses the greatest difficulty. As in theseismic prospecting method, this task requires the development of advanced processing techniques. In order toassess the performance of each technique, controlled synthetic data and different noise types, which simulate theobserved signals, are required. This paper presents a procedure to generate a wide-band noise-free electromagneticfield to be used both for magnetotelluric and audio-magnetotelluric studies. Furthermore, an effort was made toextend the simulation procedures to generally stratified and simple inhomogeneous earth structures. The discrete-time magnetic field values are generated through the inverse Fourier transform of a continuous amplitude spectrumand a sampling procedure. The electric field time series are obtained by the convolution of the magnetic field timeseries, calculated in the interested frequency band, with a non-causal impedance impulse response. Polarizedfields, which are important when inhomogeneous media are considered, are also generated.

1. Introduction

Computer simulation of electromagnetic fieldcomponents has frequently been used for testingestimation techniques of the magnetotelluricimpedance tensor (Goubau et al., 1978; Mc-Mechan and Barrodale, 1985; Yee et al., 1988;Larsen et al., 1996) and for assessing pre-processing methods of extraction of the stationaryand coherent part of signals corrupted by noise(San Filipo and Hohmann, 1983; Lamarque,

1999). The testing and assessing tasks aregenerally accomplished in the frequency domainthrough the power spectra of the measuredsignals. Hence, the simulation is carried out inthe frequency domain.

Some authors have pointed out the numerousproblems arising in the frequency-domainapproach and have proposed estimation methodsin the time-domain (Kunetz, 1972; Ernst, 1981;McMechan and Barrodale, 1985; Yee et al., 1988;Spagnolini, 1994). Moreover, for some purposes(e.g., noise identification and removal, fillingmissing data, search for outliers) the time-domainmethods are also useful in a frequency-domaincontext (Egbert, 1992; Egbert et al., 1992).Therefore, these considerations justify theinterest in generating synthetic time series forsimulation purposes.

In the above-mentioned papers, the syntheticdata are constructed by means of differenttechniques. Larsen et al. (1996) used measuredmagnetic time series and an estimated 1D transfer

Mailing address: Prof. Mariano Loddo, Dipartimentodi Geologia e Geofisica, Università di Bari, CampusUniversitario, Via E. Orabona 4, 70125 Bari, Italy; e-mail:[email protected]

Key words magnetotelluric – electromagnetic fieldsimulation

290

Mariano Loddo, Domenico Schiavone and Agata Siniscalchi

function, modified by a distortion function, togenerate the electric time series in a test area.Simple pseudo-random numbers were used torepresent the noise-free random components of thetime series (Lamarque, 1999) or the real andimaginary parts of the incident magnetic fieldcomponents (Goubau et al., 1978). The latterauthors calculated the electric field spectra using abi-dimensional impedance tensor made by simplecomplex relationships. San Filipo and Hohmann(1983) and Yee et al. (1988) assume a functionwhich tends to zero exponentially with increasingvalues of frequency to represent the main spectralcharacteristics of the natural magnetic field,simulating this behaviour through recursive digitalfilters. These same authors generated the magnetictime series by convolving a pseudo-random numbersequence with the filter coefficients.

Yee et al. (1988) computed the noise-freeelectric field components by convolving in time themagnetic time series with the impedance impulseresponses modelled as rational forms.

Uncorrelated noise is simulated by addingrandom numbers either to the spectra (Goubauet al., 1978) or to the time series (Yee et al., 1988;Larsen et al., 1996). Some authors introduce thenoise by adding sinusoids to the signal. These arecharacterized by frequencies and amplitudes whichare typical of industrial and atmospheric noise(Lamarque, 1999).

In recent years, the English version of a sys-tematic study on the MT synthetic data genera-tion, carried out by Russian researchers, waspublished (Varentsov and Sokolova, 1995). Thestudy was conducted within the framework of theproject to compare MT data processing tech-niques, using synthetic data sets (COMDATProject). The discrete values of the spatialcomponents of the magnetic field spectrum,considered in the MT frequency band, wereconstructed by opportunely multiplying selectedrandom functions with a simple realistic fieldstructure (the form was used with < 0). Theelectric field spectra were calculated via arbitraryone-dimensional scalar impedances. Finally, thetime series were obtained by a discrete inverseFourier transform of the spectral data approx-imated by a piecewise constant function. Noisecomponents were added both in the frequencyand in the time domain.

The present paper describes an approachto generate wide-band electromagnetic timeseries. The procedure adopted is representedin a flow chart (fig. 1). The aim was tosimulate real acquisition techniques to be usedboth in magnetotelluric and audio-magne-totelluric studies. In this first work only noise-free signals have been considered. The resultswere then applied to 1D and simple non 1Dstructures.

2. EM field simulation

The noise-free horizontal electromagneticfield components are related by means of theearth transfer functions in a two-input/two-output linear system (Cantwell, 1960). In thetime-domain, the relation between the time-varying input magnetic field h(t) and the time-varying output electric field e(t) can be writtenin terms of convolution equations among thefield components

ex(t) = zxx(t) hx(t) + zxy(t) hy(t) (2.1)

ey(t) = zyx(t) hx(t) + zyy(t) hy(t)

where zxx, zxy, zyx, zyy are the elements of theimpedance impulse response tensor. Theinput and output channels in the system areperturbed by added noise (Swift, 1967; Simset al., 1971) giving the observed field com-ponents

ex

ob(t) = ex(t) + nex(t)

ey

ob(t) = ey(t) + ney(t)

(2.2)hx

ob(t) = hx(t) + nhx(t)

hy

ob(t) = hy(t) + nhy(t).

For application purposes, the continuous-timeconvolutions of eq. (2.1) must be replaced bydiscrete forms. This task can be accomplishedby applying the well-known classicalsampling theorem (Freeman, 1965) to low-pass (anti-alias) filtered continuous signals.In the following, the procedure adopted to

291

Generation of synthetic wide-band electromagnetic time series

Fig. 1. Flow chart outlining the procedure for generating the electromagnetic field time series.

Construction of a

continuous function

representing the magnetic

field spectrum H(ω ) in the

10-4-104 Hz frequency

range (eq. (2.5) and Fig. 2)

Construction of the

analytical expression for

the frequency domain

earth impedance Z (ω) for

a stratified model (eq. (2.9))

Inverse Fourier

Transform of H(ω )

(eq. (2.7))

Inverse Fourier

Transform of Z(ω )

(eq. (2.12))

Input the sub-

band to use in

simulation and

the resulting

sampling rate

Sampled Magnetic

Field Impulse

Response (eq. (2.8)

and Fig. 3a)

Sampled Impedance

Impulse Response

(eq. (2.13) and Fig. 5)

Sampled Electric

Field Impulse

Response (Fig. 6)

Input a sequence

of Random

Numbers

Noise-Free Magnetic

Field Time Series

Noise-Free Electric

Field Time SeriesDFT DFT

H(∆ω )

Fig. 4

E(∆ω )

Fig. 7

292


generate the noise-free electromagnetic fieldcomponents and the impedance impulseresponse is illustrated.

2.1. H-field generation

The natural EM field is due to a variety ofcauses. At frequencies above 6 Hz, roughly, itis primarily due to spherics, which are EMtransients generated by lightning strokes. Below6 Hz, the field is of geomagnetic and ionosphericorigin. The general features of the natural fieldin different frequency bands have been illustratedby many authors (Campbell, 1967; Filloux, 1973;Serson, 1973; Mcnae et al., 1984; Labson et al.,1985; Malergue et al., 1986).

The horizontal components of the fieldpresent average amplitudes, generally increasingwith longer periods and reaching a value of about10 nT at 10 4-10 5 Hz. Moreover, two distinctminima in the 1 Hz to 6 Hz and 1 kHz to 2 kHzfrequency ranges are observed. These minimarepresent the separation between ionosphericpulsations and spherics and between low-frequency and high-frequency spherics, respec-tively.

A smoothed, mean spectrum of the magneticfield is shown in fig. 2. To obtain a relationshipwhich describes the field, the plotted behaviourcan be modelled through a linear time invariantsystem.

Depending on the goal of the study (geo-magnetic,magnetotelluric, audiomagnetotelluricsprospecting), the wide-band field is used in alimited frequency range. This operatingcharacteristic influences the procedure for thegeneration of the magnetic field time series. Infact, in accordance with some authors (San Filipoand Hohmann, 1983; Yee et al., 1988), theapproximation of the frequency characteristicsof the field could be performed directly in thediscrete domain by writing the transfer functionof the system in terms of poles and zeros in the zdomain. An alternative is to transform the transferfunction from the continuous to the discretedomain through a bilinear transformation(Oppenheim and Schafer, 1975). In the presentstudy, these methods cannot be adopted for thefollowing reasons:

1) For each sub-band, the power of the Hfield at frequencies greater than the Nyquistone is not negligible, thus resulting in thealiasing problem when the normal z transformis used.

2) With regards to the bilinear z, thismethod maps the complex continuous s planeinto the complex sampled z plane withoutaliasing, but with a frequency distortion(warping) (Oppenheim and Schafer, 1975).This distortion, resulting from the non-linearcharacter of the transformation, is com-pensated when the frequency-magnitudecurve of the system can be approximated bya piecewise constant function in which onlytwo or three characteristic frequencies needto be transformed without distortion (pre-warping procedure) (Golden and Kaiser,1964). This requirement is not fulfilled in thesimulation of a physical field which ischaracterized by a complex morphology spec-trum.

Considering the previous arguments againstthe use of a discrete-time relationship, acontinuous function for the H field wasconstructed. The sampled impulse response was

10-4

10-3

10-2

10-1

100

101

102

103

104

FREQUENCY (Hz)

10-3

10-2

10-1

100

101

HO

RIZ

ON

TA

L M

AG

NE

TIC

(n

T)

Fig. 2. Smoothed, mean amplitude of the horizontalelectromagnetic field component variations simulatedthrough a linear time invariant system.

293


obtained by a windowed integral Fouriertransform, computed with the time intervaldictated by the Nyquist frequency of thestudied band.

In particular, by using the poles and zerosform in the continuous (Laplace) domain, asystem function can be expressed (Seshu andBalabanian, 1959) as

(2.3)

where, s = σ + iω is the Laplace complexvariable with σ the damping factor of thesystem and ω the frequency. The complexnumbers s0k are the m zeros while the sph arethe n poles of the function, respectively; K isa scale factor.

The analytic properties of the function aredetermined by its poles and zeros. In order forthe system to be stable, the poles must be in theleft half of the complex s-plane and any pole onthe imaginary iω axis must be simple. Moreover,since the network function must be real for areal variable s, it follows that the poles and zerosmust either be real or in complex conjugatepairs.

The simple zeros and poles and the complexconjugate pairs in the numerator and denom-inator of (2.3) can be factored in the followingform (Gatti et al., 1966)

(2.4)where the indexes i, j, l, k indicate the genericsimple zero, simple pole, complex conjugatezero and complex conjugate pole, respectively.The scale factor K1 and the correction factors ζl

and χk are defined by the following equations:

If s = iω is inserted into eq. (2.4), the magneticfield relationship in the frequency domain can beobtained. The frequency and characteristics of thepoles and zeros are selected by considering thatsimple poles and zeros introduce a 20 dB/decadeslope change in the function curve on a cartesiandecibel versus the log (ω) representation (Bode’splot), while the complex conjugate pairs doublethese changes. Moreover, the ζl and χk quantitiesact as damping factors at each critical frequency.

Considering the magnetic field spectrumshown in fig. 2, the downward change of the slope(energy decrease) can be modelled by poles, whilethe upward changes (energy increase) by zeros.In particular, four poles and four zeros, with valuesas shown in table I, were sufficient to representthe changes in the spectrum. The simple poles andzeros were chosen to be real, while for thecomplex conjugate pairs, correction factors ζl = 1and χk = 1 were adopted. These conditions areequivalent in considering real double poles anddouble zeros. For the sake of system stability, allthe poles have negative values. Finally, the scalefactor K1 was purposely selected in order to obtaina low frequency asymptote of 10 nT for the fieldspectrum.

With the above conditions eq. (2.4) becomes

(2.5)

H s Ks s s s s s

s s s s s s

m

p p pn

( )−( ) −( ) ⋅ ⋅ ⋅ −( )−( ) −( ) ⋅ ⋅ ⋅ −( )

= 01 02 0

1 2

K Ks s

s s

ii ll

pjj pkk

1

0 0

2

2=

[ ] [ ][ ] [ ]∏ ∏∏ ∏

ζ l

l

l

s

s= −

ℜ( )0

0

.

( )s

s

s

s

s

sii

l

l

ll

0

2

0

2

0

1 2 1−

+ +

∏ ∏ ( )ζ

( )s

s

s

s

s

spjj

pk

k

pkk

−

+ +

∏ ∏1 2 1

2

2( )χ

χk

pk

pk

s

s= −

ℜ( )

H s( ) =

( )( )( )( )ω ω ω ω ω ω ω2

12

1

2

22

2 3

2

42

4

2 1 2 1 1 2 1p

ip p

ip

i

p pi

p+ + − + + − − + +

K= 1

( )( )( )( )i

z zi

z

i

z zi

z

ω ω ω ω ω ω

1

2

22

2 3

2

42

4

1 2 1 1 2 1− − + + − − + +

H ω( ) =

K= 1

294


The continuous-time impulse response of the EMfield can be obtained from the inverse Fouriertransform of eq. (2.5)

(2.6)

If ωl and ωh are the low-frequency and high-frequency cut-off of the band, and Hr(ω) andHi(ω) the real and imaginary parts of thecomplex field H(ω), respectively, then eq. (2.6)becomes

(2.7)

If ∆t is the sampling period and T the maximumacquisition time, being Hr(ω) and Hi(ω) evenand odd functions, respectively, the sampledimpulse response can be written as

(2.8)

Both for the numerical testing of the procedure,discussed at the end of Section 2.3, and for thegeneration of polarized fields in Section 3, themagnetic field in the frequency range 0.0005-5 Hzhas been computed. The latter range is adequatefor investigating the earth models considered. Theresulting sampling period and maximum periodwere 0.1 s and 2000 s, respectively.

The sampled magnetic field impulse response,calculated from eq. (2.8), is shown in fig. 3a. TheDiscrete Fourier Transform (DFT) of the impulseresponse (fig. 3b) is in perfect agreement with thecontinuous magnetic field amplitude spectrum,within the frequency range of interest.

In order to generate a magnetic field time series,the impulse response of fig. 3a was convolved witha sequence of pseudo-random numbers with a zeromean and unity-variance. The DFT of the resultingtime series is shown in fig. 4.

The generation of the electric field could beperformed in the frequency domain as the productof H(ω) and Z(ω). However, if eq. (2.5) for H(ω)and, for example, eq. (2.9) for Z(ω) are considered,their product is a very complex function; thus, itsinverse Fourier transform is performed withdifficulty. Hence, it was decided to inverse transformthe two functions separately, thereby obtaining theelectric field by a convolution procedure in the timedomain.

The determination of the discrete-time impe-dance impulse response which produces the sameoutputs as the continuous-time counterpartremains a problem which is thoroughly andelegantly discussed by Egbert (1992). As thecontinuous-time impedance impulse response is

Table I. Characteristics of the poles and zeros of the system function used in the generation of the magnetic fieldvariations. All poles and zeros are real and shown as absolute values.

Critical frequencies Frequency value (Hz) Multiplicityp1 0.002 Doublez1 0.006 Simplez 2 0.8 Doublep2 7.0 Doublez3 10.0 Simplep3 100.0 Simplez4 1500.0 Double

p4 20000.0 Double

h t H e di t( ) = ( )−∞

∞

∫1

2πω ωω .

= ( ) + ( )[ ] ( ) + ( )[ ] +−

−

∫1

2πω ω ω ω ω

ω

ω

cos sin H iH t i t dr i

h

l

+ ( ) + ( )[ ] ( ) + ( )[ ]∫12π

ω ω ω ω ωω

ω

H iH t i t dr i

l

h

cos sin .

h t( ) =

h n t∆( ) =

= ( ) ( ) − ( ) ( )[ ]∫1

2π

ω ω ω ω ωπ

π

cos sin H n t H n t dr i

T

t

∆ ∆∆

.

295


unbounded, the non-causality of thecorresponding discrete form generally follows.Hence, the discrete-time impulse response can beestimated from the frequency-domain impedance byband limiting and sampling it, as well as, by takinginto consideration the negative time values. Thisprocedure, applied to a stratified model, is presentedin the following paragraph.

2.2. Impedance impulse response

The analytical expression of the frequencydomain MT impedance, relating magnetic andelectric field components, can be written for astratified earth as (Kunetz, 1972)

(2.9)

a b

Fig. 3a,b. a) Impulse response of the electromagnetic field in the frequency range 0.0005-5 Hz, sampled every100 ms; b) theoretical amplitude spectrum of the magnetic field in the frequency range considered (continuousline) and DFT of the impulse response in a) (dashed line).

Fig. 4. Amplitude spectrum of the magnetic time se-ries generated by the convolution of the impulseresponse of fig. 3a with a sequence of pseudo-randomnumbers.

Z ii

q em

m

am i( ) = + ==

1

1

2 4

41 2

e q ei

mam i

m

= +=

41 2 4

141 2

296


where the first term on the right hand siderepresents the impedance over a half space ofconstant resistivity 1, the second one resultsfrom the reflections of the qm images on theboundaries between the layers and ,h1 being the thickness of the first layer (seeKunetz, 1972, for the determination of theimages from the reflection coefficients).

The corresponding continuous time impulseresponse can be written as

(2.10)

If the low-frequency and high-frequencylimits of the band are indicated, as previously,with l and h , respectively, then eq. (2.10)becomes

(2.11)

By integrating by parts, eq. (2.11) yields

(2.12)

By placing l = 2 /T and h = / t, with tand T the sampling period and the longest

period, respectively, the solution of the integralson the right side of (2.12) leads to the followingalgorithm for the discrete-time impedanceimpulse response for a stratified earth:

(2.13)

The above expression can be easily calculatednumerically for a given sequence of resistivitiesand thicknesses. Figure 5 shows an example ofthe impulse response for an H-type three-layer

Fig. 5. Non-causal discrete-time impedance impulseresponse for a three-layer earth model (shown in theupper part of the figure).

e q e e di

mam i

m

i t

h

l

= + +=

12 4

1 24 1 2 4

1

e q e e di

mam i

m

i t

l

h

.+ +=

12 4

1 24 1 2 4

1

z n t n t d

T

t

( ) =( )

+13

222 4

cos( )

z t( ) =

z t t d

h

l

( ) =( )

+13

22 4cos( )

q em

m

am

h

l

cos+=1

2 224

t am d2 2 .

q e n tm

mT

tam+

=1 2

24

2 2 cos

am d2 2 .

z t Z e di t( ) .= ( )1

2

a h= 1 110

297


earth (the model is shown in the same figure).The impulse response shows a structure whichis mainly localized around the origin, withoscillating decreasing values. Moreover, adifferent behaviour between positive and negativetime lags is observed.

The procedure previously adopted for layeredearth models could be extended and applied tomore complex situations, the analytical solutionsof which are available, e.g., vertical discontinuity(d’Erceville and Kunetz, 1962), vertical dyke(Rankin, 1962), anisotropic stratified sequence(Douglas et al., 1967).

2.3. E-field generation

The electric field time series is calculated byconvolving in time the magnetic field time serieswith the earth impulse response (eq. (2.13)). Theprocedure adopted for the convolution operationrequires further discussion.

If M values for h( t) and N values for z( t)are considered, with N M, the e( t) resultingfrom the convolution will contain M + N – 1values. However, there are (N – 1) values both atthe beginning and at the end of the convolvedseries which are not fully constrained by thedata. A solution to this problem is to strip offthese edge regions from the data, thus obtainingM – N + 1 e( t) values. Moreover, in order togenerate magnetic and electric field time seriesof the same length as well as a causal electricfield, the (N – 1)/2 values at the beginning and atthe end of h( t) must be discarded. Thus, thevalue e(N) will be generated by h[(N + 1)/2].

Figures 6 and 7 show the impulse responseand the spectral amplitudes of the noise-freeelectric field for the 1D structure of fig. 5. Forthe generation of the electric field impulseresponse 20 000 samples of the magnetic fieldimpulse response, together with 4001 samplesof the impedance impulse response were used.

For the purpose of numerically testing theprocedure, the impedance transfer functionZ( ), obtained as E( )/H( ) for a singletime series, has been computed. The computeddiscrete impedance transfer function (fig. 8) fitsthe theoretical curve well in all but low frequencyvalues, due to the number of samples used.

Fig. 6. Electric field impulse response resulting fromthe convolution in time of the magnetic field impulseresponse (fig. 3a) with the non-causal impedanceimpulse response of fig. 5.

Fig. 7. Amplitude spectrum of the electric field gen-erated by the magnetic field time series.

297


earth (the model is shown in the same figure).The impulse response shows a structure whichis mainly localized around the origin, withoscillating decreasing values. Moreover, adifferent behaviour between positive and negativetime lags is observed.

The procedure previously adopted for layeredearth models could be extended and applied tomore complex situations, the analytical solutionsof which are available, e.g., vertical discontinuity(d’Erceville and Kunetz, 1962), vertical dyke(Rankin, 1962), anisotropic stratified sequence(Douglas et al., 1967).

2.3. E-field generation

The electric field time series is calculated byconvolving in time the magnetic field time serieswith the earth impulse response (eq. (2.13)). Theprocedure adopted for the convolution operationrequires further discussion.

If M values for h( t) and N values for z( t)are considered, with N M, the e( t) resultingfrom the convolution will contain M + N – 1values. However, there are (N – 1) values both atthe beginning and at the end of the convolvedseries which are not fully constrained by thedata. A solution to this problem is to strip offthese edge regions from the data, thus obtainingM – N + 1 e( t) values. Moreover, in order togenerate magnetic and electric field time seriesof the same length as well as a causal electricfield, the (N – 1)/2 values at the beginning and atthe end of h( t) must be discarded. Thus, thevalue e(N) will be generated by h[(N + 1)/2].

Figures 6 and 7 show the impulse responseand the spectral amplitudes of the noise-freeelectric field for the 1D structure of fig. 5. Forthe generation of the electric field impulseresponse 20 000 samples of the magnetic fieldimpulse response, together with 4001 samplesof the impedance impulse response were used.

For the purpose of numerically testing theprocedure, the impedance transfer functionZ( ), obtained as E( )/H( ) for a singletime series, has been computed. The computeddiscrete impedance transfer function (fig. 8) fitsthe theoretical curve well in all but low frequencyvalues, due to the number of samples used.

Fig. 6. Electric field impulse response resulting fromthe convolution in time of the magnetic field impulseresponse (fig. 3a) with the non-causal impedanceimpulse response of fig. 5.

Fig. 7. Amplitude spectrum of the electric field gene-rated by the magnetic field time series.

298


3. Field polarization

The fields simulated through the previouslydescribed procedure were generated byconsidering a horizontally stratified isotropichalf-space. In this case, it is well-known that thepolarization of the waves is not important(Porstendorfer, 1975). On the contrary, in an earthhaving a complicated electrical structure, themeasured electric and magnetic fields dependstrongly on the orientation of the primary electricand magnetic fields. The effect of an electrical

inhomogeneity varies as the orientation of theprimary field varies. Thus, the total fieldpolarization is a function of the current sourcesin the ionosphere. Since, in general, the locationof the sources changes with time, or severalsources may contribute simultaneously to thefield, the total electromagnetic field observedalong arbitrary directions often changes with timeat a given frequency.

The representation of the impedance as atensor instead of as a scalar quantity was de-veloped to take into account the orientation ofthe primary field and the structural complexities.

In considering the most general polarizationof the magnetic field, that is the elliptical one,an electric field, itself elliptically polarized(Porstendorfer, 1975), results.

A 2D structure is now considered andmeasuring axes which coincide with the principalaxes of the anisotropy tensor, are assumed.Remembering that any elliptically polarizedmagnetic wave can be synthesized from twolinearly polarized waves, the field may be dividedinto two parts, parallel and perpendicular tothe strike of the structure, respectively. In thisparticular case, each of the electric field com-ponents relates to the perpendicular magneticfield component alone. Thus, the magnetotelluriceqs. (2.1) become

ex(t) = zxy(t) hy(t)

ey(t) = zyx(t) hx(t)

where zxy and zyx are the impedance impulseresponses of two different 1D models.

Fig. 8. Fit of the impedance transfer function gen-erated by synthetic data (solid line) to the theoreticalcurve (dashed line) for the three-layer model in fig. 5.

Table II. Two 1D layered earth sections representing a weakly anisotropic stratified structure, used to generatethe impedance impulse response along the anisotropy directions.

zxy zyx

Resistivity ( m) Thickness (km) Resistivity ( m) Thickness (km) 16 1 19.36 1

1 0.75 1.21 0.75 16 19.36

299


Fig. 9. Horizontal components of the time series filtered at 5 s and total field polarization of short sections. Theellipses are centered at each of the sections and are in the same scale as the corresponding field.

300


In summary, to preserve the real fieldbehaviour it must be reminded that it presents(Grillot, 1975):

1) An impulsive nature, occurring in shortbursts.

2) A distinct, randomly variable, polarizationin each burst.

3) More generally, an elliptic polarization.Together with the previous characteristics

of the fields, the field components must becorrelated to each other.

In order to satisfy the previous requirements,the magnetic time series have been modulatedwith a pseudo-random sequence, uniformlydistributed in the interval (– 1,1). This modifiedtime series was assumed to represent onecomponent of the field. The other component wasobtained by modulating the same initial field witha sequence of numbers, complementary to thepseudo-random ones.

As earth model, a stratified anisotropicstructure, characterized by the two 1D modelsshown in table II has been selected. The electricfield components were generated by the pro-cedure discussed in the previous sections.

In accordance with Grillot (1975), theoccurrence of the field characteristics was testedthrough a band-pass filtering of the time series.Figure 9 shows a 1600 s specimen of the hori-zontal components of the field time series filteredwith a band-pass centered at 5 s and a bandwidth-centre frequency ratio of 0.3. In the figure thepolarization ellipses of the total horizontal fieldvectors for short sections of the data have alsobeen plotted.

The figure shows ellipses well defined over2-3 cycles with polarizations changing fromsection to section. As expected, the magnetic andelectric ellipses are perpendicular.

Considering independent sets of fieldcomponents at a given frequency, which occurswhen the sections have different polarizations,is the basis for any statistical estimation of theearth impedances.

4. Conclusions

A procedure to generate synthetic wide-bandelectromagnetic time series has been developed

and described in detail. It attempts to take intoaccount and to reproduce the main characteristicsof the actual electromagnetic field. The mainintent was to carry out a simple yet accurate MTexperiment simulation, consisting in the inter-action of a synthetically generated magnetic fieldwith assigned resistivity models of the under-ground. These models were complex with regardsto their analytical representation. The electricfield time series were calculated in the time-domain through a non-causal impedance impulseresponse. The procedure was extended to simpleinhomogeneous structures considering the fieldpolarizations.

Further studies will include:1) More complex earth structures.2) The effects of different noise types.3) The relative performances of different

statistical approaches for a reliable impedanceestimation.

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CANTWELL, T. (1960): Detection and analysis of lowfrequency magnetotelluric signals, Ph.D. Thesis, MIT,Boston.

D’ERCEVILLE, I. and G. KUNETZ (1962): The effect of a faulton the earth’s natural electromagnetic field, Geophysics,27, 651-665.

DOUGLAS, P., D.P. O’BRIEN and H.F. MORRISON (1967):Electromagnetic fields in an n-layer anisotropic half-space, Geophysics, 32, 668-677.

EGBERT, G.D. (1992): Noncausality of the discrete-timemagnetotelluric impulse response, Geophysics, 57, 1354-1358.

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