IRREGULAR
pi2 P U L S A T I O N S OF
RESOLVED
SINUSOIDAL
INTO
A SUPERPOSITION
OSCILLATIONS
JAROSLAV ST~E§TiK.
Geophysical Institute, Czechosl., Acad. Sci., Prague*) Zusammenfassung: Als Grundlage der Arbeit dient das Verfahren, dem zufolge man eine unregelmiissige Sehwingung auf die Summe der Sinussehwingungen iiberfiihren kann. Man 9eht dabei vom Komplexspektrum der gegebenen Sehwingung aus, yon dem Spektren der Sinussehwingungen mR entspreehender Frequenz, Amplitude und Phase sukzessiv abgezogen werden. Die beschriebene Methode wurde in der vorliegenden Arbeit fiir magnetische und tellurisehe Pi2-Pulsationen angewendet. Man hat gefunden, dass die Pi2-Pulsationen aus 3--6 Sehwingungen bestehen. Auf solehe Weise entstand das Material flit die magnetotellurisehe Sondierun#. Die Ergebnisse schliessen gut an die Pc3-Pulsationen an. Der Hauptbeitrag des besehriebenen Verfahrens liegt darin, dass die Anzahl der Punkte zum Finden des spezifischen Widerstandes, und zwar in verschiedenen Perioden, erh6ht wird. 1. I N T R O D U C T I O N One of the important sources of information o n the fundaments and structure of irregular pulsations are their spectra. How to obtain them, their properties and other data resulting from t h e m was described in [1]. It was also found that other oscillations with different frequencies are superimposed on the basic pi2 oscillation. On the whole a pi2 p h e n o m e n a is composed of 2-- 3 oscillations. The spectra, however, are very complicated, especially due to fictitious subsidiary maxima. F o r this reason the analysis of pi2's, carried out in [1], can be said to be qualitative. The determination of the amplitude, the phase and even the period of other oscillations than the basic is subject to a considerable error. The conditions given in [1] for identifying a real oscillation are quite stringent and, therefore, it is not probable that a non-existing oscillation would be considered as real. On the other hand, however, it is possible that there are more associated oscillations, which, however, have a smaller amplitude and which cannot be identified reliably. In this paper, the quantitative method for resolving irregular pulsations into a superposition of sinus~idal oscillations is used. The method is based on the successive subtraction of spectra of sinusoidal oscillations with all fictitious subsidiary maxima from the pulsation spectra, until a spectrum of a non-periodical function is obtained. It will be shown that there are more associated frequencies. After the analysis of the results, obtained by this method, an example of their use f ) r deep magnetotelluric sounding is given.
2. D A T A The pulsations were recorded at the Budkov Observatory ((p = 49004 , N, 2 = 14°01 ' E, • = = 49°01 ', A = 96002 ') by a LaCour magnetometer, a n IVJ induction variometer and an ET-30 telluric instrument in 1963. The LaCour instrument records the relative changes of the intensity of the components AH, AD, AZ. Its scale value is 1"2 7 / m m in A H and AD and it is independent of the period in the considered range of pariods. The recording speed is 6 mm/min. The selected pulsation records were magnified four times. *) Address: Bo6nf II, Praha 4 - Spofilov.
64
Studia geoph, et geod. 15 (1971)
Irregular pi2 Pulsations Resolved into a Superposition of Sinusoidal Oscillations The induction variometer IVJ records the current which is induced in coils with a permalloy core due to the variations of the intensity of the magnetic, field. The components recorded are X(NS), Y (EW) and Z (vertical). The scale value of the variometer depends on the pulsation period and it is in the range between 0.1 and 0.3 7/ram in all components. The phase distortion also depends on the period. The records of this instrument have been described in greater detail in [2]. The recording speed is 15 ram/rain.
I
os2
~ _ E W ~~--__
1 0
NS
I 0.2
I
I 0.4
_ lO
o
0o
Fig. 1. Calibration of the ET-30 telluric instrument, valid for .1962--63. Vertical scale in mV/km/mm.The phase is the same for both components.
co
The ET-30 tellurie instrument has four electrodes, connected crosswise. Their distances are 1026m in the NS direction and 1119 m in t h e E W direction. The potential difference is measured by a galvanometer and recorded on a common record. The'scale value and the phase distortion depend on the pulsation period. The calibration curves have not yet been published and they are, therefore, shown in Fig. 1. It is necessary to add that the curves only correspond to the period 1962--1963. The recording speed of the instrument is 30 mm/min.
3. P U L S A T I O N S R E S O L V E D I N T O A S U P E R P O S I T I O N OF S I N U S O I D A L OSCILLATIONS
3.1. Fourier Spectrum and Its Properties Consider the fundamental formulae of the Fourier transformation: (1)
S(co) =
(t)exp(-icot)dt,
f(t) = (2n) -1
(co) exp (loot) dco.
S(co) is a complex function called the spectral density. It will be considered in the form S(co) = A(co) + iB(co). Thus A(co) = f~o~f(t) cos cot dt, and B(co) = - J'~o~f(t) sin cot dt. We shall also introduce the real spectral density F(co) -- IS(co)i, S(co) = F(co) exp [@(co)], where ~(co) is the phase spectrum. f(t) will be used to denote the pulsation record. The time of the first onset will be considered as t = 0, and the time when the oscillations vanish completely will be t = z. The function f(t) is defined in the'interval (0, z), and it is zero outside this interval. S(co) is then the instantaneous spectrum of the function f(t) in the interval (0, z), The spectra of pi2 in [1] were computed in this way. Studia geoph, et geod. 15 (1971)
65
J. StPegtik Some of the properties of function S(e~) will now be mentioned. Let S(co) corresponds to function f(t) according to Eq. (1), SI(o)) to fs(t), and Sz(m)to f2(t). If f (t)= = f l ( t ) + f2(0, then S(o))= St(o))+ $2(o9). If f ( t ) = k f~(t), where k is a real constant, then S(co) = k S1(o)). Both can be expressed by the formula (2)
as $1(°9) + a2 $2(c°)= fo~ [alfl(t) + a2f2(t)]exp(-i°)t) d - oo
dr"
Appart from the main maximum in o)0, there is also a number of subsidiary maxima in the pi2 spectra. However, not all of them have a corresponding oscillation which really exists. Subsidiary maxima also occur in the spectrum of function sin o)t, if integrated in the interval (0, ~) [3]. Their position and magnitude is precisely determined and they depend on z (they are, therefore, not harmonics). If in Eq. (2), for example, fl(t) = sin (colt + 01) and f2(0 sin (OJzt -t- I//2), S(o)) and also F(09) have two maxima in 09s and c02. However, the value of F(o)s) is distorted by the presence of subsidiary maxima corresponding to f2(t) and this also applies to F(c02). Also the phase values qh = q~(oh), and ~02 = q~(092), are distorted, so that ~0s ~e 0a, q~2 • 02. The distortion is not always the same. If, e.g., in Eq. (2) as >> a2, F(o)I) and q~(oh) are subject to a much smaller error than F(o~2) and q~(c02). As F(co) depends on the amplitude, the amplitude of this oscillation, a~, can be determined more accurately than a2. This practically means that the amplitude and the phase of the main oscillation can be determined from the spectrum comparatively accurately, but for the remaining oscillations the determination is the more inaccurate, the smaller their amplitude. In [1] the magnitude of the subsidiary maxima was estimated. The spectral density in the subsidiary maximum only amounts to 0.3 of the spectral density in the main maximum at the most and it is greater on the side of lower 09. In the example mentioned above, the amplitudes were determined directly from the spectrum with an accuracy of a~ _ 0"3a2, a2 -t- 0.3a~. The value 0"3 appears when the subsidiary maximum of one frequency falls in with the main maximur~ of another frequency. Otherwise the error is much smaller. The spectral density F(coo) in the main maximum is related to the amplitude A by the relation A = 2z -s F(o)o) or A = ~-1 S~: F(o)) de). For sake of simplicity the first relation will be used, although the latter formula would, according to [2], serve to determine the amplitude a little more accurately (col and o9~ are the positions of the minima closest to coO). -----
3.2. M e t h o d o f S u b t r a c t i o n o f S p e c t r a o f S i n u s o i d a l O s c i l l a t i o n s Let us again consider the foregoing example; the following form will now be used: (3)
S(o~) = S,(co) + Sz(m)= f ~ [gl(t) + #2(t)] exp ( - i c o t ) d t , d - ao
9(t) = ga(t) + gz(t), 66
gl(t) = al sin(ogtt + ~kl), gz(t) = a2 sin(cozt + 02). Studia geoph, et geod. 15 (1971)
Irregular pi2 Pulsations Resolved into a Superposition of Sinusoidal Oscillations
Let it be assumed that g(t) and S(o)) are known (and thus also F(co) and 9(co)). The main maximum o)t, the phase 9t = q~(o)t) and the value F(cot) can be found from F(co). As the amplitude As = 2z -1 F(cot), the first (main) oscillation can be written in the form A1 sin (cost + qh), where As - al, opt - St. By computing the spectrum St(co) of this oscillation and subtracting it from S(o)), one arrives at S2(co). These spectra are complex and the substraction must be vectorial, This means that one 'F
502 .
0
0
.
.
.
0
O.I
0,2
I
d
e
I
f
Fig. 2. Spectrum of a pi2 pulsation recorded on 12.3. 1963 at 20"43 LMT, A H component after successive substraction of spectra of sinusoidal oscillations, a) full line Is(co) I, dashed line [S0(co)[, b) full line [Z'l(co)] , dashed line [St(co)I, c) full line I222(co)], dashed line [S2(o)l , d) full line [X3(co) ], dashed line 1S3(00)I, e) full line I224(00)], dashed line Is4(co)l, f) Ixs(co)[. All the curves are on the same scale.
has to compute A2(o) ) = A ( c o ) - As(o), B2(o)) = B ( o ) ) - Bl(o) ) and from them F2(co ) and ~02(o)). This spectrum is no longer distorted by the presence of subsidiary maxima St(o)) and it is approximately identical with the spectrum of the function g2(t) itself(the error is due to the fact that the following did not hold exactly: A s = al, (01 = ~1). The mare maximum is in o)2, the phase 92 = 9(<92) and A2 = 2z -1 • F2(o)2) - a2. The original F(o)2) can differ considerably from F2(o)2). In this way it is possible to determine more accurately the amplitude and the phase of other oscillations than the fundamental. This will be applied to pi2 pulsations. As an example thepi2 of 12. 3. 1963, 20.43 LMT, component AH, will be considered. Its spectrum S(co) is shown in Fig. 2a. We shall find the main maximum Oo, the phase (ao and the amplitude A o = 2 r - 1 F(coo)" The fundamental oscillation is A o sin (coot + (ao) and its spectrum is So(e)) (dashed line in Fig. 2a). The difference of the spectra 221(o)) ~ S(co) -- So(O ) is shown in Fig. 2b. Now cot, (as, A1 are found and the spectrum Sl(co) is computed (dashed line in Fig. 2b). Then 272(00) = 271(00) -- St(o) ) (Fig. 2c), from which e)2, ~2, A2 follow, etc. Final one arrives at a spectrum which has no distinct maximum (in our case 225(o)) and this is then the spectrum of a non-periodic function. We shall call this the residual Studia geoph, et geod. 15 (1971)
67
J. St~egtik function z(t). The pulsation has thus been resolved into (4)
y(t) = y' A i sin (colt -t- qh) + z(t). i=0
In terms of numbers: y(t) = 1.15 sin (0-078t -- 170°) -[- 0.81 sin (0.051t -- 120°) + 0.39 (0.108t + -[- 110°) + 0-32 sin (0.140t + 60°) -t- 0.32 sin (0.015t ÷ 140°) + z(t), where the amplitude are in 7 and the phases in degrees. As already mentioned, the amplitudes At are subject to an error, which differs with the various Ai and reaches 0.3 of the amplitude of the neighbouring oscillation at the most. In the substraction of the spectra it may thus happen that one is liable to substraction of the spectrum of a sinusoidal oscillation with the incorrect amplitude. Let the latter amplitude be Ai and the relevant spectrum St(co). Thus the oscillations with frequency cot has not been completely removed from the spectrum St + 1(co)After a few steps, when the spectra of the oscillations with larger amplitudes have been substracted, some 2~j(co) will have a maximum in coj "__ cot. Also in this case the amplitude Aj a n d t h e phase pj are found, cos is put equal to cot, and the following are added: At sin (cott + ~0t) + A s sin (co~.t + ~0j). In this way the corect amplitude is obtained, as well as the phase, for the oscillation with frequency cot. In the example given in Fig. 2 this did not occur. At the most, on could consider one of the maxima in spectrum 2;5(co) as a supplement of the oscillation with frequency co~. However, its amplitude is so small that it can be neglected. In evaluating the pulsation on the record, it is necessary to devote attention to the selection of the base. The ideal base should go through the centre of the pulsation so that ttie deviations from it up and down are approximately the same. If the selected base is at a distance k from the ideal the pulsation is resolved as follows:
(5)
y(t) =
At sin (cott +
+ 4 0 + k.
t=0
The pulsation spectrum, will then also contain the spectrum of constant k. We shall now investigate the spectrum of function k(t), k(t) = k for t ~ (0, z), k(t) = 0 for the remaining t's. The spectral density is Fk(co)= 2kco-llsin½cozl • This function has a maximum in co = 0, where Fk(O) = kz. The zero points are in 2~/z, 4~/z,..., the subsidiary maxima in 3~/z, 5~/z, ..., and the value of Fk(m) in them is 2kz/3~, 2kz]5~, etc. Therefore, the spectrum of the pulsation is most affected in the lowfrequency range. The subsidiary maxima will distort the spectrum only little. In order that the effect of the constant k in Eq. (5) on the spectrum should be negligible, it must hold that k < 0"5 7 when the double pulsation amplitude is 3 7. 3.3. C o m p a r i s o n
of Results Obtained
from Different Instruments
Using one special example, it was shown that the pi2 oscillation could be resolved into a superposition of 5 sinusoidal oscillations. Also other p i t s can be resolved in this way and the number of sinusoidal oscillations usually varies between 3 and 6. {J8
studia geoph, et geod. 15 (1971)
lrre#ular pi2 Pulsations Resolved into a Superposition of Sinusoidal Oscillations Let us now investigate how the subsidiary frequencies depend on the principal frequency. This relation is illustrated in Fig. 3. The ogt-values are accumulated i n two regions, approximately about the straight lines o9, = o9o-t-~o90 and partly also about o9t = o90 --- ~o9o. N o r is this then a case of some higher harmonics or subharmonics. ,d •
// //
¢.~
,/
i~lo jl
/ •
O. l
i
/
J ,"
Fig. 3. Dependence of subsidiary frequencies cot on the fundamental frequency coo (AH component). Straight line a) cot ~ coo, b) coi= coo-t-£2, c) col= coo--~, d) c o i = C ° o + 2 0 , e) Cot= = coo -- 2.Q. o
//b
/
,/ i /
/ " '/
~
//j
a
• /
te / i, / p" ,,,/ /z l•
~o
/g ¢/•
~"
e
, .
o •-" -'" "
i
\ '0
.~¢
I 04
~ ~"
I
,~ 11!
However, this does not mean that all frequencies, found in this way, are actflally real. In [1] the effect of frequency modulation on the spectrum was mentioned. A frequency modulated oscillation (and this applies to pi2 as shown in [1]) can be resolved into a superposition of sinusoidal oscillations [3]. I f the fundamental carrier frequency is o9o and the frequency of~the modulating function is O, o9 = o9o + + flY2 cos t?t, where fl is the modulation index, the modulated oscillation is (6)
y = A Jo(fl) sin ogot + A ~ Jk(fl)[sin(o90 + kO) t + (--1)ksin(o90 -- kf2) t] , k=l
where Jk(fl) are Bessel functions. The frequency modulated oscillation thus has a line spectrum. A number of oscillations in the pi2 resolution can really be interpreted as the effect of modulation. The modulating long-periodic function would then have a frequency around f2 = ½co. Hoivever, there are also oscillations, which are o f another origin. They usually have a larger amplitude than would follow from the ratio of Bessel functions Jk/Jo, and their position does not fall in with the system COo, o9o -- f2, o9o --- 2f2, etc. Unfortunately, the limited data only provides the basic information about them. I f one only keeps these cases in Fig. 3, one will not obtain the dependence of these frequencies on o9o. Their generation, therefore, is at least partly independent of the fundamental frequency pi2. Also the following conclusions; made in [1], are proved here: there are two real oscillations, one with a period of 2 . 5 - 3 mins, the second with a period of about 40 s. The former has an amplitude of about 1 7, in components AH and AD, the latter 1 , / i n AH and less than 0.5 7 in AD. S t u d i a geoph, et geod. 15 (1971)
69
•L Stfegtlk
Whether the found oscillations are real, or whether they are due to frequency modulation, the oscillations are not fictitious. They will, therefore, appear on the records o f various instruments in the same way. In [2] the data on pulsations recorded , by a LaCour variometer and an induction variometer were ~¢ ¢. / • o.~o x // compared.This can be adopted Ca ," / here and applied to the pi2 re,/ solution. In working with the ,," records of the IVJ induction / variometer, the curve on the re./ •/ / cord is used for computing the / / /, .p/
. •
/ °
/ •
/
/ /
// -f' /
/
•/
,
~
,
,
,
4/4
,
O.l
Fig. 4. Correlation between indivi° dual cos in AHand X. The straight , .. line illustrates coi( A H ) = o~i(X ) .
0.16
3T
l
,p • /
A
X
I
o
// / °/
[
/ / / / / /
2:
/ )/ /" / /,, / / / / / / /
Fig. 5. Correlation between individual A l in AH and AT.The straight line illustrates A(AH) = A(X).
/ /,
A
4
,~H
e•
2
spectrum and only at the end the result provided by Eq. (4) is adjusted by using the calibration curves. In the X and Y components the same frequencies cos are found as in the AH and A D components. The correlation between cot in the A H and X c o m p o nents is illustrated in Fig. 4. The correlation coefficient is 0.977 and for the AD and Y components 0.991. The coo itself has a correlation coefficient o f 0.938 in A H and X
70
Studia geoph,
et g e o d . 15 (1971)
Irregular pi2 Pulsations Resolved into a Superposition of Sinusoidal Oscillations and 0.910 in AD and Y [2]. The correlation between the amplitudes in AH and X is shown in Fig. 5. The correlation coefficient in this case is 0"795, and for AD and Y 0.665. For the amplitudes A o these values are 0.753 and 0.661, respectively. The correlation coefficient between co~in the AH and A D components is 0.981, and between the amplitudes 0.699. On the whole, it may be said that for sinusoidal oscillations the values of the correlation coefficients in this comparison are much higher than those found for directly observed periods and amplitudes of pi2. The values of the correlation coefficients in this case are much closer to the values of the coefficients for pc3 in [2]. 3.4. T h e F i e l d a f t e r S u b t r a c t i n g the S i n u s o i d a l O s c i l l a t i o n s Composition of the Original Pulsation
a n d the
The formulae of the Fourier transformation (1) are, but for the factor 1/2~, symmetric with respect to t and co. It is, therefore, possible to compute f(t) if S(co) is known using the same formula and also the same computer programme as that applied to S(co). This reverse computation was described in [4]. The function f(t) is given by (7)
2n f(t) =
(co) cos cot dco -
(co) sin cot de).
d
o
¢
Fig. 6. Functions fi(t) corresponding to spectra in Fig. 2. a) pulsation record, b) function fl(t), corresponding to 2;1(~o), c) fe(t), d)f3(t), e)f4(t), f) residual function z(t). We shall now investigate what functions f~(t) correspond to spectra S,(co) in Fig. 2. In other words, we are going to investigate the shape of the pulsations after the successive subtraction of the sinusoidal oscillations. We must know A'i(co) and B'~(co) for the computation, where A'i2(co) + B'i2(co) = ]Si(o)[ 2. However, these functions are known from computing ri(co ) in Section 3.2. They only need substitution into Eq. (7) and then f/(t) can be determined. The results are given in Figs. 6 b - e . The residual function fs(t) -- z(t) (Fig. 6f) then corresponds to spectrum X,(co). Studia geoph, et geod. 15 (1971)
71
Y. St[egtik If we have the pulsation resolved according to Eq, (4), we can compose it again and find out h o w accurate the resolution was. z(t) in Fig. 6f is added to the sum of the sinusoidal functions now. The, pulsation composed in t h i s manner, is compared with the original record in Fig. 7. T h e
correlation coefficient between these curves is 0-980. The small differences between both curves are compatible with z(t). This means that the residual function can no longer affect the pulsation and that the other frequencies, contained in 2"5(o)), -are negligible.
Fig. 7. Comparison of the shape of the
pulsation composed of sinusoidal oscillations with the original shape on the record. a) composed pulsation, b) record.
4. A P P L I C A T I O N TO M A G N E T O - T E L L U R I C D E E P S O U N D I N G
I f a subsurface which is homogeneous and has an electric conductivity a is assumed, this value can be computed from the amplitudes of the magnetic and telluric pulsations
Es]: (8)
a = #o
°~tH,/Exl 2 (MKS units).
I f E is expressed in m V / k m and H in 7 (1 m V / k m -- 10 -6 V/m, 17 -- (4n) -1 x x 10 -2 A/m, #0 = 4n x 10 -7 Vs/Am, co = 2n/T), Eq. (8) becomes (9)
a
=
(SIT)I
;ExI 2 ,
=
T]E
m2 .
If the subsurface is inhomogeneous, the quantity 0 is an apparent resistance of a kind, relevant to the sub surNce. 4.1. C o m p a r i s o n
of Magnetic
Results with Telluric
Telluric pulsations are recorded by galvanometers. Therefore, due to their frequency characteristics, the pulsation record does not correspond precisely to the real field variation and the record must be corrected by means of the calibration curve. The situation is similar to that of the induction coils. Therefore, the pi2 pulsations have a similar shape on telluric records to that on induction coil records. The period of the pulsations is extended a t first, and then it decreases again. The amplitude increases at first, to decrease later on. There is no monotonous increase in period and decrease in amplitude. We shall now resolve the telluric pi2 pulsations into a sum of sinusoidal oscillations. We shall proceed in the same way as with the records of the induction coils, i.e. the calibration curves will be used at the end. It is possible to find the same frequencies in the resolution as these of the corresponding magnetic pulsations. 72
Studia geoph, et geod. 15 (1971)
Irregular pi2 Pulsations Resolved into a Superposition of Sinusoidal Oscillations
The correlation between the frequencies of magnetic and telluric pulsations is shown in Fig. 8. The correlation coefficient is 0.987 for AH and EW, 0.981 for AD and NS, 0.971 for X and EW, and 10.964 for Y and NS. The correlation is slightly better, if the pulsations are recorded by a °/ LaCour variometer. The correlation / / / coefficient between the periods mea/ EW / sured directly is 0.874 for AH and / • / I o/ EW and 0.924 for AD and NS. /
/ /
0.14 /
~L / / / •
0.1
z.*
zy ;62~.
I
•//
2
y °
°o
/ o/
o o~ o
/
l
/
°•
I
/•
m
o•
./ 0.5
/
/
/ aH I
I
o'. I
'
o )~
'
Fig. 8. Correlation between individual coi in A H a n d EW. The straight line illustrates coi(AH) = c o i (EW).
T%
0.2
i
!
5
c
7
n
I
]
t
10
Fig. 9. Dependence of impedance Z r on T(s1/2). Empty circles represent pc3 data.
A similar relation was not found between the amplitudes of the magnetic and telluric pulsations. The ratio [E/HI = f(~/T)is called the impedance and its dimension is fL Let us put Zy = AEw/AB and Z~ = ANs/AD. The dependence of Zy on x/Tis shown in Fig. 9. For this purpose Zy, expressed in units of mV/km/y, were transformed to f~ by multiplying by 4n x 10 -4. A certain degree of asymmetry can be observed here between the NS and EW components. The Z~ values are smaller in comparison with Zr. Inspire of the large scatter of the points, one may conclude that the impedance decreases slightly with increasing x/T. The average value of Zr is about 0.01 f~, of Z x about 0.007 f~, and the latter also decreases with increasing ~/T. The data in this case was supplemented by a few values of pc3 pulsations recorded between January and April, 1963. They fit in well with the series of pi2 data. The average of the periods of the magnetic and telluric components was always taken as the period, the two differing slightly. 4.2. A p p a r e n t R e s i s t a n c e o f the S u b s u r f a c e When the impedances are in D, it is sufficient to use Eq. (8) to compute the resistance. However, it is simpler to compute in mV/km and 3' according to Eq. (9). Also in this case the period T considered was the average period of the magnetic and S t u d i a g e o p h , et g e o d . 15
(1971)
73
3. Stfegtik telluric components. Let us denote by 0x the apparent resistance for the pair D-NS, by 0y for the pair H-EW. The relation between Qy and x/Tis shown in Fig. 10. For purposes of comparison and as additional data pc3 pulsations were used in this case as well, together with data obtained from direct observations of pi2 amplitudes. Both fit the data series well. The relation between Ox and x/T has the same form as for Or. Only as regards the absolute lO~ ~m 1 Sy values, Q:, is smaller. The same asymmetry as with the impedance can be observed.
:°t :t 0
4
o o
oo o o
o
T~ I
5
I
I
7
I
I
[
I
'"
Fig. 10. Dependence of specificresistance of subsurface on x/T (s 1/2). Empty circlesrepresent pc3 data, crosses pi2 amplitudes observed directly.
I0
Figures 10, as well as 9, only indicate a weak dependence on x/T. However, one must realize that usually a much wider range of T is used, from 1 to 104 s. Z and Q change considerably in this interval and the dependence on x/Tis clear. The apparent resistance of the subsurface was investigated in greater detail in Budkov, using more numerous data [6]. The curve ~ = O(x/T) was very undulated and as regards the period range, considered here ( x / T = 4 - 10s 1/2) practically horizontal, only slightly increasing towards larger x/T. The results fully substantiate it. However, there is a small difference in the absolute values. Whereas in [6] the average value of Qr is about 1800 Q and of Ox about 600 ~ in the interval T = 4 - 10 s x/2, the values in this case were 1200 f], and 400 ~, respectively. This difference is not very large on a logarithmic scale. At other stations the values of the specific resistance differ considerably more. For example, at ~rob~trov~i [7] the apparent resistance is considerably smaller and in the interval considered x/T decreases from 50 fl to 10 £'1 in both directions, the differences between 0x and 0r being negligible. The situation is similar at other stations given in [7]. The value of ~ was computed under the assumption of a homogeneous subsurface and, therefore, represents a kind of average specific resistance of the subsurface down to a certain depth. According to the equation of the skin-effect the amplitude of the oscillations decreases with depth the faster, the higher the frequency. The slow oscillations penetrate deeper and thus the value of Q for larger x/T indicates and average specific resistance of the subsurface at greater depth. Q, of course, includes also the uppermost layer. In the latter the resistance varies with the season, precipitation, etc., and this affects the average value of 0. This is apparently the origin of the large scatter of points in Figs. 9 and 10. Another property of the 0x and 0y values was found. If the value of 0x is larger than the average, also the value of 0y, computed from the same pulsation, is higher than the average, and vice versa. This phenomenon can also be explained by the 74
Studia geoph, et geod. 15 (1971)
Irregular pi2 Pulsations Resolved into a Superposition of Sinusoidal Oscillations
fluctuation of the resistance in the uppermost layer, because this resistance fluctuates independently of the direction. Therefore, the frequently used value, 0 = x/(0xOy), does not decrease the scatter appreciably. 5. CONCLUSION
T h e main results are as follows: 1) A method was derived for transforming an irregular oscillations to a sum of sinusoidal oscillations. The method consists in substracting successively from the spectrum of the oscillation considered the spectra of sinusoidal oscillations with appropriate frequencies, amplitudes and phases, until the residue is a spectrum of a non-periodic function. 2) This method was applied to pi2 pulsations. These pulsations are usually composed of 3 - 6 oscillations, which are mostly due to the frequency modulation of pi2's. However, there are also other oscillations which are of another origin, independent of pi2. 3) There is a good correlation between the frequencies of the oscillations, of which the pi2's are composed, even if these frequencies were obtained from records of different instruments (LaCour and induction variometer). The correlation between the amplitudes is also good. 4) The resolution of pi2 oscillations into a sum of sinusoidal oscillations can be carried out also for telluric pulsations, which can be used together with magnetic pulsations, for magneto-telluric sounding. The results correspond to the results of the magneto-telluric sounding carried out in Budkov earlier. The scatter among the points is not decreased substantially by applying the method described, but the main advantage is the increase of the number of points on the graph, moreover, with different periods. This is important if other irregular pulsations, besides pi2, are used. Received 29.8. 1969
Reviewer: M. HvoSdara References
[I] J. St~'e~tik: Properties of Spectra of Geomagnetic Pi2 Pulsations Recorded at the Budkov Observatory. Studia geoph, et geod., 13 (1969), 42. [2] J. St[egtik: Comparison of Data on Geomagnetic Pulsations Recorded by Various Instruments. Studia geoph, et geod., 13 (1969), 293. [3] A. A. Xap/ce~r[~t: Cne~TpbI~ a~a.rm3. Foc. ~t3~. rex.-Teop, nrIr., Moc~ma 1957. [4] J. St~egtik: Determination of the Original Shape of Pulsations from Their Spectra. Studia geoph, et geod., 14 (1970), 344. [5] L. C a g n i a r d : Principe de la m6thode magndtotellurique, nouvelle m6thode de la prospection g6ophysique. Ann. de gdoph., 9 (1953), 95. [6] J. P6~ovfi, O. Praus, M. TobyRgov~: A Study of the Electric Conductivity of the Earth's Mantle from Magnetotelluric Measurements of the Budkov (Czechoslovakia) Station. Studia geoph, et geod., 10 (1966), 184. [7] V. Petr, J. P~6ovA, O. P r a u s : A Study of the Electric Conductivity of the Earth's Mantle by Magnetotelluric Measurements at ~robfirov~ (Czechoslovakia). Travaux Inst. G6ophys. Acad. Tch6cosl. Sci. No 208, Geofysik/tlnf sbornik 1964, N(~SAV, Praha 1965. Studia geoph, et geod.15 (1971)
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