- Open Access
Hodograph method to estimate the latitudinal profile of the field-line resonance frequency using the data from two ground magnetometers
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012
- Received: 4 July 2012
- Accepted: 27 February 2013
- Published: 10 June 2013
The hodograph method enables estimating the latitudinal profile of the field-line resonance (FLR) frequency (f R ) using the data from two ground magnetometers. This paper provides the full details of this method for the first time, and uses a latitudinal chain of ground magnetometers to examine its validity and usefulness. The hodograph method merges the widely-used amplitude-ratio and cross-phase methods in a sense that the hodograph method uses both the amplitude ratio and the phase difference in a unified manner; further than that, the hodograph method provides f R at any latitude near those of the two ground magnetometers. It is accomplished by (1) making a complex number by using the amplitude ratio (phase difference) as its real (imaginary) part; (2) drawing thus obtained complex numbers (one number for one frequency) in the complex plane to make a hodograph; and (3) fitting to thus obtained hodograph a model satisfying the FLR condition, which is a circle with the assumption that the resonance width is independent of the latitude. To examine the validity and usefulness of the hodograph method, we apply it to a Pc 4 event observed by the Scandinavian BEAR array. We also apply the amplitude-phase gradient method (Pilipenko and Fedorov, 1994; Kawano et al., 2002) to the same event, and compare the results; this is the first article applying the both methods to the same dataset.
- Hodograph method
- field-line resonance
- ground magnetometers
- amplitude-phase gradient method
The field-line resonance (FLR) is the mechanism in which fast-mode (compressional) magnetosonic waves, propagating from the magnetopause into the inhomogeneous magnetosphere, excite the Alfven-mode waves via the resonant mode conversion. This is one of the key physics of ULF waves in the terrestrial magnetosphere. According to the theoretical concept of the FLR by Tamao (1965), the mode conversion is the most effective where the frequency of the incoming fast-mode waves matches the local frequency of the Alfvenic field-line oscillations (called the FLR frequency (fR) below). The growth of thus generated field-line oscillations is terminated at a certain level depending on the dominant dissipation mechanism, from which we can obtain the effective Q-factor of the FLR.
Southwood (1974) and Chen and Hasegawa (1974) formulated the FLR within the framework of a simplified model: a “plasma box” with 1D inhomogeneity across straight field lines. Despite the apparent simplicity of such a model, the spectral properties of the relevant system of MHD equations turned out to be useful in explaining observations. Following their work, the formulation was generalized for more complicated systems: multi-dimensional inhomogeneity, curvilinear magnetic field, finite plasma pressure and finite conductivity plasma boundaries; the results of these studies confirmed that the basic predictions of the 1D theory remain valid in a more realistic situation.
Ground magnetometer data include FLR signatures (in the ULF range), characterized by key parameters such as the FLR frequency fR and the resonance width δ which is an indicator of the damping scale of the resonance. Since fR is related to the magnetic field strength and the plasma density along the field line, we can use fR to diagnose the magnetospheric plasma density from the ground.
However, in actually observed ground magnetometer data, FLR-generated ULF waves are often difficult to identify due to superposed non-FLR signals (mainly compressional-type). To overcome this problem, the amplitude-ratio method (Baransky et al., 1985; Kurchashov et al., 1987; Gugliel’mi, 1989) and the cross-phase method (Waters et al., 1991), using ground magnetometer data from two stations separated by 50~250 km along a geomagnetic meridian, have turned out to be the most effective. These methods utilize the characteristic feature of FLR-generated ULF waves: Since the amplitude and phase of FLR-generated ULF waves sharply changes across the resonance latitude while the superposed signals are more global and almost the same at the two stations, taking the difference of the data from the two adjacent stations can cancel out the most of the non-FLR signals and extract even weak FLR signals. These methods have been applied to Pc3–4 pulsation data from two ground stations in middle to low latitudes, and the FLR parameters at a point (usually regarded as the midpoint) between the two stations have been successfully obtained (e.g., Baransky et al., 1985, 1989; Kurchashov et al., 1987; Green et al., 1993; Waters et al., 1994; Russell et al., 1999; Chi et al., 2000; Menk et al., 2000).
As stated above, the standard amplitude-ratio (cross-phase) method uses only the amplitude ratio (phase difference) between two stations to determine the FLR frequency. A problem here is that the two methods can provide different values of the FLR frequency.
As a countermeasure against this problem, the hodograph method, originally suggested by Kurchashov and Pilipenko (1996) and used in this paper, employs both the amplitude and phase information at the same time. Furthermore, the hodograph method allows to determine fR not only at the midpoint between the two stations but at (theoretically) any latitude; differently put, fR is obtained as a function of latitude or L (fR(L)). The details of the method will be presented in the next section.
It is well known that the east-west component of the magnetic-field perturbation in space (associated with the radial electric field perturbation) caused by the FLR follows the equation of Southwood (1974) and Chen and Hasegawa (1974). On the ground, it is observed as the northward (H) component magnetic field perturbation, because the magnetic field perturbation caused by the ionospheric Hall (Pedersen) current is observed (neglibly small) on the ground (e.g., Hughes and Southwood, 1976) and because the electric field mapped from space to the ionosphere is north-south directed.
Since x refers to the ground latitude here, when it is field-aligned mapped to the geomagnetic equator in space, it increases with increasing distance from the Earth. We note here that Pilipenko and Kurchashov (2000) used the opposite increasing direction of x.
Figure 1 shows the amplitude (top) and phase (bottom) parts of Eq. (3) as a function of ζ for the case of h R = 1. The typical FLR pattern (amplitude peak and sharp phase change at the resonance point, i.e., at ζ = 0) is seen in this figure. The top panel also shows that the amplitude at ζ = ±1 (i.e., at x = x R ± δ) is times the peak amplitude, so that the wave power is half.
As a detail of the resonance width δ, or differently put the damping scale of the resonance, it is related to the damping rate in a system χ, which is the imaginary part of the complex frequency ω + iχ, as follows: δ = −(χ/2π)(∂f R (x)/∂x)−1 (e.g., Menk et al., 1994; Pilipenko and Fedorov, 1994; Waters et al., 1994). The sign of the gradient of f R (x), and correspondingly δ, determines the direction of an apparent phase velocity in the resonant region. Throughout the magnetosphere, except for the narrow region near the plasmapause (Dent et al., 2003) and the region very close to the Earth where the mass loading effect is significant (Menk et al., 2000; Kawano et al., 2002), f R (x) decreases with increasing x, and thus the meridional component of the apparent phase velocity is directed toward higher latitudes.
2.2 Key features of the hodograph
2.3 How to fit the hodograph to observed data
Having plotted R o (f) on the complex plane, the next step is to best-fit the model hodograph R m (f) of Eq. (9), i.e., a circle, to the data hodograph of R o (f). It is to be noted here that this model-fitting process means the signal-noise separation in the data (R o (f)): Here the signal is estimated by R m (f), and the difference from R m (f) to R o (f) refers to the noise. Then, from thus best-fitted model hodograph, one can obtain the resonance width δ using Eq. (18).
Note here that, throughout this methodology description, we use the term “signal” to refer to the FLR signal, and “noise” to mainly refer to the non-FLR signal (white-noise component is expected to be small because we usually smooth the FFT results in the frequency domain). Why we use this terminology is because the term “signal-noise separation” is widely used and intuitive, and because we are interested in the FLR signal in this paper.
A question regarding the above-stated procedure is if the circle should be fitted to all the frequency range of R o (f), or to data points near R o (f R (x0)) (f R (x0) is estimated by using the standard amplitude-ratio or cross-phase methods). A point to note here is that, as f goes far from f R (x0), both the bias and the noise increase in R o (f); the bias increases because the model of Eq. (1) is an asymptotic approximation around f R (x0) (see Appendix A for its comparison with the exact solution to Southwood’s equation), so that it deviates from the true FLR solution far from f R (x0). The noise increases because the FLR amplitude decreases with increasing distance from the resonance point, which means the decrease in the signal/noise ratio as f goes away from f R (x0). Thus, it is not a good idea to fit the model hodograph R m (f) to all the frequency range of R o (f). On the other hand, selection of a too narrow frequency range near f R (x0) makes the above-stated signal-noise separation inefficient (as an extreme example, if we choose only three datapoints, a circle is perfectly fitted to these three datapoints, yielding zero noise term). A guideline for the selection is to first draw all the R o (f) data in the complex plane, and then to select the frequency range for which R o (f) looks circular. This selection has to be made by visual inspection, and thus the selection process is more-or-less case-by-case.
Another point to note in doing this model-circle fitting is that the model assumes a constant resonance width; thus, the data points can also shift from the circle where the resonance width systematically changes near the utilized ground station pair. One has to remember this point when using the hodograph method.
The next and important step is to estimate the FLR frequency f R as a function of latitude x. This is achieved by an inverse procedure, as follows. For any given f (denoted as f g here), R o (f g ) is already plotted on the complex plane, and the best-fitted model hodograph (circle) is also plotted; then, one can find R m (f g ), i.e., the corresponding point on the model hodograph closest to the R o (f g ), by simply drawing a line running through both the center of the model-hodograph circle and R o (f g ) and finding the point where this line intersects the model hodograph. Then, by inserting thus obtained R m (f g ) into Eq. (9), one obtains X(f g ), and by inputting thus obtained X(f g ) into Eq. (12), one obtains x R (f g ). After having finished obtaining x R (f) for all f’s, one can then inverse the x R (f) function to obtain f(x R ), which is the same as f R (x) in its meaning.
2.4 Advantages of the hodograph method
The amplitude-ratio method and the cross-phase method provide single frequencies, the obtained frequency includes the effects of both the FLR signal and non-FLR signals (which cannot be perfectly cancelled out by these methods) in the amplitude-ratio and cross-phase data, and there is no mathematically established procedure to estimate the error (i.e., the effect of the non-FLR signal) in the obtained frequency. On the other hand, the hodograph method best-fits a model hodograph to several data points (corresponding to several frequencies), thus can separate the FLR signal (i.e., the fitted hodograph) from non-FLR signals. (A caution here is that the model hodograph assumes a constant resonance width; thus, the data points can also shift from the model hodograph where the resonance width systematically changes near the utilized ground station pair. One has to remember this point when using the hodograph method.)
The obtained f R (x) (explained above) is a continuous function of latitude; that is, with the hodograph method, one can obtain the FLR frequency at (theoretically) any latitude by using the data from the two ground magnetometers. (In actuality, the accuracy of the solution decreases with increasing distance from the two stations, as stated above.)
The above-stated model hodograph reaches R m (f) = 1 at f → ±∞, as stated above. However, in fact, it is possible that the observed R o does not approach R = 1 at f → ±∞ because of the difference between the two stations in the underground conductivity, which modifies the original signal and distorts the hodograph. (We note here that, for the standard amplitude-ratio and cross-phase methods, neglecting this effect can result in biases in the obtained f R at the midpoint between the two stations. We also note that the difference between the ionospheric conductivities above the two stations could also modify the original signal, but in this paper we are interested in the FLR signal, thus we simply call these non-FLR effects by the term “influence of the underground conductivity.”)
To correct for this effect, we use a procedure which is a generalization of the procedure of Green et al. (1993) (applied to the amplitude-ratio and cross-phase methods). That is, we assume that the underground effect changes the amplitude ratio by a constant factor, and changes the cross phase by a constant number, in the vicinity of the FLR frequency at x = x2 ~ x1. (In general, frequency dependence is likely to exist, but the current methodology of the hodograph method does not include it; it is a topic of future research to include it in the methodology. At least Pilipenko and Kurchashov (2000) and Vellante et al. (2002) applied the hodograph method (with the above assumption) to their observations and obtained reasonable results.)
As a result of this modification, the model hodograph R m (Eq. (9)) is ǀMǀ-times expanded and rotated by arg(M) around the coordinate origin. However, the modified hodograph (referred to as R mm below) is still a circle.
Best-fit a circle R mm to the data hodograph R o . Let us denote the center of the best-fitted circle as α + iβ and its radius as γ.
Calculate the modulus ξ and argument θ of the circle center by using the equations and .
Draw a line (Line (3) below) from the coordinate origin to the circle center.
Draw a line (Line (4) below) which starts from the coordinate origin and is tangent to the upper part of the circle. The modulus, η, of the tangent point satisfies the equation .
Calculate the angle φ from the Line (3) to Line (4), by using the equation (φ is always positive.)
- 6)Conversion from the fitted circle R mm (including the underground effect) to the model circle R m (from which the underground effect is removed) is accomplished by multiplying M−1 to R mm (see Eq. (25)). This conversion refers to the rotation (around the coordinate origin) of the fitted circle R mm by −(θ + φ) (note here that θ can be positive or negative according to its definition (2)), and dilation/contraction which moves η + i0 (the circle’s tangent point at the real axis after the rotation) to 1 + i0 (tangent point of R m ); see Eq. (17) and Fig. 2. Thus, M satisfies the following equation:
Shows the locations of the BEAR stations used in this paper. Longitudinal distances of the stations from the 105° magnetic meridian are less than 0.7°. See text for more details.
Geogr. lat. [deg]
CGM lat. x [deg]
Relative CGM lat. [deg]
Table 1 also shows the dimensionless coordinate (latitude) X of the stations and the midpoints of adjacent stations, calculated by using Eq. (14) with x1 (x2) set to the CGM latitudes of B38 (B37). That is, we apply the hodograph method to the B38–B37 pair (more details are described just below).
In this section we apply the hodograph method to Pc 4 magnetic pulsations observed at 0750–0830 UT on Day 177 (June 26), 1998 and study the latitudinal profile of the FLR frequency. The H-component amplitude was significantly enhanced during this interval with the peak-to-peak amplitude of ~20 nT (not shown). We apply the hodograph method to the B38-B37 pair, while the standard two-station methods (amplitude-ratio and cross-phase methods) are applied to the other two pairs (B28-B38 and B37-B36), and the results are compared.
The two vertical dashed lines in Fig. 4 surround the frequency range in which the amplitude ratio shows a smooth bipolar pattern and the cross phase shows a smooth unipolar pattern: At the leftside dashed line the amplitude ratio shows a sudden change in its gradient, and at the rightside dashed line the cross phase starts to decrease again. That is, the frequency range between the two dashed lines are where an FLR signature is dominant. Thus, we apply the hodograph method to the data points between the two vertical dashed lines (there are 24 data points).
In Fig. 8, the solid-line part of the f R (x) curve corresponds to the valid range calculated by using Eq. (24). We can see that the valid range is centered around the midpoint of B38 and B37 (their data were used for the calculation), and that the valid range is about three times wider than the distance between the two stations (B38 and B37) used for the calculation.
Figure 8 also shows the results of applying the amplitude-ratio method (squares) and the cross-phase method (open circles) to each adjacent two stations. In the amplitude ratio method, we have determined the FLR frequency as the average of the frequencies where the amplitude ratio becomes maximum and minimum. In the cross-phase method, we have determined the FLR frequency as that where the cross phase has the peak. As the error bar of the cross-phase FLR frequency, we have used the half width of the cross-phase profile; this could be an overestimate, but there is no mathematically established procedure to estimate the confidence interval of the cross-phase FLR frequency. As the error bar of the amplitude-ratio FLR frequency, we have used the half of the frequency difference between the two peak frequencies of the amplitude ratio. Figure 8 shows the above-stated error bars, and we can see that the lengths of the two error bars are not very different, and that the error bars for the three station pairs touches the f R (x) curve (the B37-B36 pair’s error bars barely miss the curve.)
The hodograph method merges the amplitude-ratio method and cross-phase method in a natural manner and provides a single value of the FLR frequency (f R ) at the midpoint between the two stations (Fig. 8). (The amplitude-ratio or the cross-phase methods by themselves give respectively just a single value of f R , but it often happens that the two f R ’s are fairly different.)
While the cross-phase method and the amplitude-ratio method provide f R ’s only at the midpoint between the two stations, the hodograph method provides f R as a continuous function of latitude x, i.e., f R (x), by using the same data from the two stations only (Fig. 8).
The hodograph method can make a signal-noise separation of the data in a systematic manner, in terms of model fitting and residual (Fig. 7). (To be more specific, the term “signal” here refers to the FLR signal, to which we are interested, while the term “noise” here mainly refers to non-FLR signals. A point to note here is that the model hodograph assumes a constant resonance width; thus, the data points can also shift from the model hodograph where the resonance width systematically changes near the utilized ground station pair.)
The hodograph method includes the procedure to correct the effects of the inhomogeneous underground crust conductivity (Section 3).
The essence of the hodograph method (leading to the above features) is, for each frequency f of the FFT’ed H-component data, to make a complex-number ratio R o (f) between the two stations’ data (including both the amplitude ratio and the phase difference; see Eq. (20)), to plot thus obtained R o (f)’s on the complex plane, and to best-fit to them a model hodograph R m (f) (Eq. (5)) which conforms to the FLR theory (Eq. (1)).
As an illustration of the hodograph method application, we have used the ground-magnetometer data from a BEAR station chain along the 105° meridian (from higher latitude, B28, B38, B37, and B36 (see Table 1)). That is, we (a) applied the standard amplitude-ratio method and cross-phase method to the B28-B38, B38-B37, and B37-B36 pairs, (b) applied the hodograph method to the B38-B37 pair, and compared the results of (a) and (b). We note that this kind of comparison has been made for the first time in literature.
As a result of the comparison of (a) and (b), Fig. 8 shows that, taking into account the estimation errors in the amplitude-ratio method and the cross-phase method, the f R (x) profile obtained from the B38-B37 pair provides fairly good estimate of f R at the midpoints of B28-B38 and B37-B36, even though the both midpoints are located outside the latitudes covered by the B38-B37 pair.
Still, if one draws a best-fit line to the cross-phase FLR and amplitude-ratio FLR data (six in total) shown in Fig. 8, the six points are fairly aligned to that line, and that line makes a finite angle to the above-obtained curve of f R (x). This is of some concern, so here we apply the hodograph method to another pair (i.e., B28-B38) and see the result, as follows.
The top panel of Fig. 12 shows f R (x)’s from the two station pairs; the slopes of the two curves are now steeper, and the two curves crosses with each other, which is a good sign. On the other hand, the resonance widths calculated with APGM, shown in the bottom panel of Fig. 12, show sharp enhancement near the edges, which is probably unrealistic.
What we can deduce from the comparison between Fig. 11 and Fig. 12 is that, for the event analyzed in Section 4, the most realistic values of the underground-origin component exist somewhere between zero (resulting in Fig. 12) and those obtained by the hodograph method (resulting in Fig. 11); the hodograph method subtracted too much for the event of Section 4.
A possible cause of this too much subtraction is that the amplitude ratio and the cross phase of the event in Section 4 did not have an asymptotic flat part: For example for the event of Kawano et al. (2002), their figure 4 shows flat parts of the amplitude ratio (cross phase) at the edges of its bipolar (unipolar) part. With the asymptotic flat part, we can obtain an idea on the order of the underground effect before applying the hodograph method.
On the other hand for the event in Section 4 of this paper, Fig. 9 shows that the bipolar (unipolar) part of the amplitude ratio (cross phase) is not surrounded by such asymptotic flat parts. It could mean that the adjacent parts, having different frequency dependence, comes from different sources, and the different sources are in fact affecting the bipolar/unipolar part and deforming its waveform. (It could also bias the results of the amplitude-ratio method and the cross-phase method.) A lesson from the data-analysis part of this paper would be: “Avoid events whose amplitude ratio and cross phase do not show a gradual shift toward asymptotic flat parts.”
As long as we avoid such events, we believe the hodograph method is a useful method, as summarized as four items at the beginning of this section; it is a topic of future research to do the same analysis as this paper for another event having the amplitude ratio and the cross phase whose edges are flat. Another topic of future research is to improve the hodograph method so that it enables a flexible latitude dependence of δ(x).
The authors would like to thank the SVEKALAPKO BEAR Working Group, Toivo Korja from the University of Oulu, Finland, and David Milling from the University of Alberta, Canada for the availability of the BEAR data. The research of V.A.P. is partly supported by the Program 22 of Russian Academy of Sciences. This research was also made possible in part by the fellowship to V.A.P. from Nagoya University. The research of H.K. is partly supported by a research program of the International Center for Space Weather Science and Education, Kyushu University. Calculations of the geomagnetic coordinates were made at http://nssdc.gsfc.nasa.gov/space/cgm/cgm.html. A matlab function to realize the Taubin method, CircleFitBy-Taubin.m (written by N. Chernov) was obtained through the “matlab central” website http://www.mathworks.co.jp/matlabcentral.
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