Range ®nding of AlfveÂ n oscillations and direction ®nding of ion-cyclotron waves by using the ground-based ULF ®nder

A new approach to the problem of direction and distance finding of magnetospheric ULF oscillations is described. It is based on additional information about the structure of geoelectromagnetic field at the Earth’s surface which is contained in the known relations of the theory of magnetovariation and magnetotelluric sounding. This allows us to widen the range of diagnostic tools by using observations of Alfvén oscillations in the PC 3–5 frequency band and the ion-cyclotron waves in the PC 1 frequency band. Preliminary results of the remote sensing of the magnetosphere at low-latitudes using the MHD ranger technique are presented. The prospects for remote sensing of the plasmapause position are discussed.


Introduction
The study is devoted to the problem of direction and range ®nding of the ULF geoelectromagnetic oscillations. Our approach to this problem is in¯uenced by the paper of Hayakawa (1991) dedicated to the direction ®nding of VLF/ELF emissions (see also Hayakawa et al., 1991). We describe the operation of the ULF ®nder as a device which makes it possible to complete the range ®nding of Pc 3±5 oscillations (2±100 mHz) and the direction ®nding of Pc 1 waves (0.2±5 Hz). The range ®nder consists of a magnetometer and the recording system for measuring the Earth's currents, i.e. the normal set of standard observatory equipment. Moreover, we consider a large volume of the Earth's crust in the vicinity of the observation point as an essential element of the device. The dimensions of the volume are assumed to be of the order of the skin depth. The electrodynamic characteristics of the new element are taken into account when measurements and calculations are being carried out. Figure 1 depicts a small part of the Earth's surface in the neighborhood of the observatory. Here the x axis is directed northward, the y axis points eastward and the horizontal line stands for the projection of a magnetic shell on the Earth's surface. We consider that this shell undergoes resonant AlfveÂ n oscillations and our purpose is to locate this shell. We present our method which allows us to solve this problem by observations at one single point. In other words, we suggest an algorithm for measurements and calculations to determine the distance x up to the projection of the oscillating magnetic shell on the ground.
The ULF ®nder is also available for the Pc 1 wave propagation, i.e., for the measurement of the angle u (see Fig. 1). We start with this problem as it is easier in some respects than the range ®nding of Pc 3±5 oscillations. Although we cannot show any relevant Pc 1 data to verify the method at the moment, we introduce the basic elements of the direction ®nding method for the sake of completeness.

Direction ®nding of Pc 1 pulsations
It is customary to assume that Pc 1 are generated in the magnetosphere as ion-cyclotron waves, which propagate to the ionosphere in a longitudinal wave guide. Here a portion of wave energy is incident in the ionospheric wave guide and propagates horizontally along the Earth's surface at large distances from the end of the longitudinal wave guide (Manchester, 1966(Manchester, , 1968Tepley and Landsho, 1966;Grei®nger, 1968, 1973). Measurement at one observation point thus tells us nothing about the location of this end. The problem has been solved by several methods such as triangulation direction of arrival (Frazer and Wawrzyniak, 1978), directional analysis (Chetaev, 1978), dispersion and group delay (see Feygin et al. 1979 and references therein) and by using the measurements from a dense network of magnetometers (Hayashi et al., 1982(Hayashi et al., , 1988. In addition to these methods we present here an independent technique for the measurement of azimuth u of the Pc 1 propagation (see Fig. 1). The position of the Pc 1 source region may be established by the direction ®nding of Pc 1 at two or more points.
Let us locate the observation point in the center of a region with a relatively homogeneous distribution of conductivity of the rocks in horizontal directions. We shall register three components of the magnetic ®eld bt in the Pc 1 frequency band. The data analysis is based on the following graphic construction.
We can determine the plane perpendicular to bt. This plane intersects the horizontal plane (or Earth) in a straight line. The line rotates because the Pc 1 exhibits elliptic or, more precisely, quasi-elliptic polarization. The rotating line is coincident with the direction of Pc 1 propagation twice over the period of oscillations, much as a stopped clock shows the right time twice a day. This leads to the question: when does this coincidence take place? The answer runs as follows: the rotating line coincides with the propagation direction when z t 0. Let us put this in another way. Figure 2 shows the hodograph of the vector-function b s t, where index s indicates that we are dealing with the horizontal vector projection. The moments at which z t 0 are indicated by the bold dots on the hodograph. Let us draw a line which connects these two points. Then the perpendicular to this line de®nes the direction of Pc 1 propagation.
In order to prove this conclusion we make use of the relation where 1 stands for the surface impedance of the Earth, k axY is the velocity of light, x is the frequency of oscillations and b s x Y y . This relation is widely used in the study of the Earth's crust and upper mantle by means of magnetovariation sounding (Berdichevsky et al., 1969;Schmueker, 1970;Lilley and Sloane, 1976;Jones, 1980;PajunpaÈ aÈ , 1988). Setting z 0, we obtain r s Á b s 0. The operator r s may be replaced with ik s , where k s k x Y k y is the local wave vector of horizontal propagation. Hence b s Á k s 0, i.e., vector b s is parallel with respect to the wave front, and it is perpendicular to the direction of propagation when z 0.
The information on the surface impedance is not necessary when the assumption of horizontal homogeneity and isotropy of the Earth's crust in the vicinity of observation point is ful®lled. If it is necessary, we can take into account a weak dependence of 1 on xY y. All one has to do is to replace z by z À ikb s Á r1. This replacement follows from the evident generalization of Eq. (1) to the case of slightly inhomogeneous media: z ikr s 1b s . The realization of this version requires preliminary investigation of the geoelectric structure of the region in the vicinity of the observation point to calculate rf. In this case the information on geoelectric properties of the rocks is used here in the explicit form.
Assuming the medium to be horizontally homogeneous, but anisotropic we can use the procedure described, replacing b s by the auxiliary vector b i j 1 kk 1 À1 ji Y where 1 ij stands for the surface impedance tensor and the indices iY jY k take the values xY y. Here we consider that 1 ij 1 ji .

MHD ranger
We measure the east-west component of the electric ®eld i y and the vertical component of the magnetic ®eld z in the Pc 3±5 frequency range. Let us make use of spectral analysis and designate i j i y x jY j z x j. We may ®nd the distance from the observation point to the magnetic shell, resonating at the frequency x, from the expression if the electroconductivity of rocks has horizontal homogeneous distribution (Guglielmi, 1989). Here hx is the phase dierence between the spectral components i y x and z x. In the case of inhomogeneous distribution it is necessary to replace z by z À ikb s Á r1 as before.
Equation (2) was derived with the help of Eq.
(1) and the impedance boundary condition i s 1b s Â n Y 3 where n denotes the unit vector normal to the Earth's surface directed earthwards (e.g., Wait, 1982), with regard to the approximate expression for the known spatial structure of the ®eld-line resonance (e.g., Hasegawa and Chen, 1974;Southwood, 1974;Orr and Hanson, 1981;Gough and Orr, 1984).
Here D is the width of a resonance. Let us draw an analogy between the range ®nding of ®eld-line resonances based on Eq. (3) and magnetotelluric sounding (MTS). It is well known that the essence of MTS lies in the estimation of the vertical distribution of the crustal conductivity by the frequency dependence of the modulus of surface impedance (e.g., Rokityansky, 1982;Wait, 1982). The modulus of the surface impedance can be calculated from the Cagniard relation: Here r j x j, see Eq.
(3). Clearly MTS is based on the idea of compensation: since i y and x are proportional to the intensity of external sources, their ratio i y a x does not depend on this intensity and moreover it is proportional to f. (We recall that i y is proportional to f, and x mostly does not depend on f.) In other words, MTS is related to the compensation of the inde®nite conditions in the upper half-space. Similar to MTS, range ®nding in the form of Eq. (2) is achieved owing to a speci®c compensation: as both components i y and z are proportional to f, their ratio i y a z does not depend on f. In other words, our method is directed to the compensation of inde®nite conditions in the lower half-space.

Veri®cation
As a test of the techniques described the data generously supplied by Prof. D. Chetaev has been analyzed in order to calculate x . Pc 3 pulsations have been observed in Ukraine not far from the village of Rakhny Sobovy, / 0 43X9 . Altogether 21 events suitable for analysis were registered on August 30, 1974 between 09.30±12.30 LT. Each event represents a wave packet with a carrier frequency f xa2p and a smoothly varying amplitude envelope. In addition to the distance x also the impedance j f j has been calculated for selected events. The dependence of x on f is shown in Fig. 3. We can see that the connection between x and f is obvious. The correlation coecient r À0X89 has the expected sign (the period 1af increases with increasing latitude). The regression equation is where x is measured in km and f in mHz, respectively. At the same time the connection between j f j and f turned out to be rather weak (see Fig. 4). The correlation coecient r 0X24. This indicates that, at least in the case under study, the method of the magnetospheric MHD-range ®nding is more informative than that of magnetotelluric sounding.
The result obtained may be veri®ed, for instance, by an observation in two points displaced in latitude. As such data is not available, we carried out indirect veri®cation by means of analysis of pulsation polarization in the horizontal plane. Figure 5 displays the relative distributions of number of cases versus x for v and polarizations separately. One can see that v polarization is preferable if x b 0 and vice versa as would be expected, in the framework of the theory of ®eld-line resonances (e.g., Lanzerotti, 1976).
With a knowledge of x and the position of the device, it is possible to determine the magnetic shell v value. Also, knowing v and the frequency of resonant oscillations, we can use the familiar method of hydromagnetic diagnostics (HMD) for the estimation of plasma density q in the equatorial plane of the magnetosphere (e.g., Aubry, 1970;Jacobs, 1970;Orr, 1973;Orr and Matthew, 1971;Nishida, 1978). According to Eq. (6) d lg qadv 9 À0X9 at v 9 2, which is not in contradiction with direct and indirect measurements (e.g., Carpenter and Anderson, 1992).

Discussion
We now discuss mainly two questions, one related to MTS and the other concerns HMD, and in one way or another both of these questions are related to the problem of ULF ®nding. We point out also some limitations in using the direction ®nding method.
In order to understand the reason why the impedance 1 is almost independent of the frequency f (Fig. 4), let us consider the relation which is more general than the Cagniard relation Eq. (5).
Here d 2 p j 1 j k is the eective skin depth (e.g. Wait, 1982).
The strong inequality is usually valid, and that is why the second term on the right hand side of Eq. (7) may be omitted. Such a rejection leads to the violation of the correlation between 1 and f . Let us consider the question in a greater detail. As we may write it follows from Eqs. (1) and (4) that D ix Àk1 x a z at x 0. Based on these relations we conclude that the rejection of the second term in the right hand side of Eq. (7) leads to an error of the order of $ ar 2 . The analysis of data indicates that the random variation of value ar 2 . The analysis of data indicates that the random variation of value ar 2 is of the order of 5±10%.
On the other hand, the relative variation of frequency is of the order of 10% (see Fig. 3). It is known that 1 depends on frequency such as f p in the high frequency limit, and 1 is almost independent on f in the low frequency limit. Hence, the regular variation of j 1 j in our case is not greater than 5%, i.e., it is less than a random variation. Therefore, the poor correlation between j 1 j and f is the result of neglecting the second term on the right hand side of Eq. (7).
A more detailed analysis of the data available for the present study reinforces our conclusion. It has been found that the correlation coecient r tends to increase with the increase of distance j x j. For example, r 0X41 if j x j b 100 km (14 events), and r 0X07 if j x j`200 km (14 events). The reason is that the greater j x j, the smaller the second term in the right hand side of Eq. (7).
We now turn our attention to the discussion of the application of ULF range ®nding to HMD. We see great possibilities for the development of hydromagnetic diagnostics with the help of the MHD-ranger. The idea is as follows. If the range-®nder is placed in the band of latitude 50 À 60 it may be possible to detect the plasmapause position v by recording Pc 3, 4 pulsations. It is anticipated that a display of x against f will reveal the location of the plasmapause by a distinct discontinuity.
There are two weak points which have to be recognized when applying the direction ®nding method  Hayashi et al. (1982) found that the radius of the ionospheric secondary source of Pc 1 pulsations does not exceed 300 km it is obvious that the Pc 1 source region is often more extended in the E-W direction. On the other hand, according to the theoretical calculations (Grei®nger and Grei®nger, 1973;Fujita, 1987) the propagation of Pc 1 waves is limited mostly along the magnetic meridian plane i.e., at small values of u. Although this may not be the general case according to our experience (see also e.g., Fraser, 1975) it is probable that the method to locate the Pc 1 source by the direction ®nding method does not work in all cases. However, it is expected that the method always produces useful information about the¯uctuations of the local direction of the Pc 1 propagation.

Conclusion
The basic idea of the present study is that the impedance relationships between the components of electromagnetic ®eld on the Earth's surface can be used in hydromagnetic diagnostics. Two simple methods based on the impedance relation (1) have been introduced. This is not, however, an exact formula. We have avoided discussion of the applicability conditions but even so it is clear that the depth of ®eld penetration into the crust must be small as compared with the vacuum wavelength, the radius of curvature of the surface and the characteristic scales of the changes in the ®eld and medium properties along the surface of the Earth. Any complete answer to the question of applicability condition is dicult to present. The relationship (1) depends strongly on the geoelectric conditions in the vicinity of the point of observation. There is always the risk that one or another applicability condition will be violated or, alternatively, that some condition will turn out to be too rigid.
Is it possible to suggest an alternative approach which would retain the idea in general terms but which would not be based directly on Eq. (1)? We believe that the answer is positive. For this purpose we need to study the conductivity distribution in the lower half-space near the observation point by MTS methods and then jointly solve the internal problem (for the Earth's crust) and the external problem (for the magnetosphere). The solutions are to meet at the interface. Then, it becomes possible to use additional relationships between the ®eld components in order to improve the accuracy of magnetospheric diagnostics.