2.1 Overview
It is well known that the eastwest component of the magneticfield 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 northsouth directed.
As shown by Pilipenko (1990), Pilipenko and Fedorov (1994), and Pilipenko et al. (2000) (see also references therein), the solution to the above equation can be asymptotically decomposed, and its leading term is expressed as follows (its valid range will be discussed later in the main text and in Appendix A):
with
where x is the ground position (latitude), f is the frequency, H_{
m
}(x, f) is the Fourier component of the ground H component (the subscript m indicates that this is a theoretical model), x_{
R
}(f) is the position (latitude) of the resonance point, δ is the resonance width, h_{
R
}(f) is the amplitude at the resonance point, and ζ is the dimensionless distance of x from x_{
R
}.
Since x refers to the ground latitude here, when it is fieldaligned 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.
Now we return to the hodograph method itself. It is based on Eq. (1). In the method, we divide the model Eq. (1) for a station by the same model but for another station, as follows:
where
is the latitude of the midpoint between the two stations,
is the distance between the two stations (x_{1} > x_{2} is assumed here),
is the dimensionless resonance latitude (offset to zero at x_{0}), and
is the dimensionless resonance width.
X(f) can be regarded as a value of the dimensionless coordinate
at f, where x (referring to the latitude in general) is equal to x_{
R
}(f). It is to be noted that X = 0 at x = x_{0}, X = 1 at x = x_{1}, and X = −1 at x = x_{2}.
If δ is a constant (which means that D is a constant), and if we regard Eq. (9) as a special case of complex transformation from complex variables X to R in which we only use the real axis in the X complex plane (because X(f) is a real number), then Eq. (9) has the form of a Mebius transformation (fractionallinear transformation); in this transformation, it is known that a line is transformed into a circle. With Eq. (9), the real axis in the complex X plane is transformed into a circle having the radius a which satisfies the equation
and having the center at
(a straightforward verification of this is given in Appendix B). This hodograph is illustrated in Fig. 2.
2.2 Key features of the hodograph
Here we summarize the important features of the aboveexplained (model) hodograph, as follows.

The hodograph is tangent to the real axis at 1 in the asymptotic limit of X(f) → ±∞. That is,

From the radius of the hodograph a, one can obtain the resonance width δ with the following equation:
2.3 How to fit the hodograph to observed data
As stated above, the model hodograph (Eq. (9)) is a circle. On the other hand, one can also draw a data hodograph in the complex plane, as follows. One first calculates H_{o,1}(f) and H_{o,2}(f), Fourier Transform (FT) of the observed ground magnetic field H component at station #1 and #2. The Fast Fourier Transform (FFT) algorithm is widely used for its swiftness and convenience. (See Appendix C for another method using the crosscovariance function.) Thus obtained H_{o,1}(f) and H_{o,2}(f) are complex functions and include the amplitude and phase as follows:
where j = 1 or 2, A_{
j
}(f) is the amplitude, and ψ_{
j
} ( f ) is the phase. Then, for all f values of the FT’ed data, one can calculate the observationbased R, R_{
o
}( f ), with
One can plot thus obtained R_{
o
}( f ) in the complex plane for all the f’s to make a data hodograph.
It is to be noted here that, since
R_{
o
}( f ) includes both the amplitude ratio
and the cross phase
and merges the two in a natural way.
Having plotted R_{
o
}(f) on the complex plane, the next step is to bestfit 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 modelfitting process means the signalnoise 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 bestfitted 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 nonFLR signal (whitenoise 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 “signalnoise separation” is widely used and intuitive, and because we are interested in the FLR signal in this paper.
A question regarding the abovestated 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
}(x_{0})) (f_{
R
}(x_{0}) is estimated by using the standard amplituderatio or crossphase methods). A point to note here is that, as f goes far from f_{
R
}(x_{0}), 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
}(x_{0}) (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
}(x_{0}). 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
}(x_{0}). 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
}(x_{0}) makes the abovestated signalnoise 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 moreorless casebycase.
We note that Pilipenko and Fedorov (1994) suggested the inequity
to roughly identify the range of x for which Eq. (1) is close enough to the FLR field of Southwood and Chen and Hasegawa (see Appendix A for more details). In the following we will use this inequity to indicate the valid, or differently put, preciseenough frequency range of the hodographmethod results.
Another point to note in doing this modelcircle 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 bestfitted 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 modelhodograph 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 hodograph method is improved over the standard amplituderatio method and the crossphase method in the following aspects.

The amplituderatio method and the crossphase method provide single frequencies, the obtained frequency includes the effects of both the FLR signal and nonFLR signals (which cannot be perfectly cancelled out by these methods) in the amplituderatio and crossphase data, and there is no mathematically established procedure to estimate the error (i.e., the effect of the nonFLR signal) in the obtained frequency. On the other hand, the hodograph method bestfits a model hodograph to several data points (corresponding to several frequencies), thus can separate the FLR signal (i.e., the fitted hodograph) from nonFLR 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.)