The observations presented above can be explained
by the field line resonance theory [Southwood, 1974] and the
ionospheric effect on ground magnetic pulsations [Hughes and Southwood, 1976].
In the magnetosphere, the phase change of 
across the
resonant point is a significant feature of the field line resonance theory
[Southwood, 1974; Chen and Hasegawa, 1974].
This feature has been clearly observed by radar experiments
[Walker et al., 1979] that measure the electric field oscillations
on the ionosphere.
Applying the calculation of the magnetic field for field line
resonances by Southwood [1975];
Hughes and Southwood [1976]
included the ionospheric effect in their model and numerically
solved the magnetic field oscillations on the ground.
Based on the same principle, an approximation that is valid in the
vicinity of resonance has also been used to interpret the gradient
method [Best et al., 1986; Guglielmi, 1989; Green et al., 1993].
When the Earth is considered to have a finite conductivity,
ground pulsations can exist in the Z component and
their results show that the phase change
for the Z component is generally greater than that for the H component.
The phase change for the D component is comparatively small.
Although the two-oscillator model proposed by Waters et al. [1991] can interpret some of the features of the cross-phase spectrum, we find that its physics differs from the field line resonance theory and it cannot explain aspects of our observations. According to their model, the two simple harmonic oscillators represent the two oscillating field lines whose footpoints are located at the two ground stations of concern. The oscillations of the two field lines are assumed to be independent, except that both of them are driven by a common wave source. In this model large phase differences take place at the frequencies between the eigenfrequencies of the two field lines.
The alternative and more traditional paradigm is that the magnetosphere
is excited by a broad band of fast mode waves that cross magnetic field
lines. In the magnetosphere the Alfvén velocity is a function of L,
and theory [Southwood, 1974; Chen and Hasegawa, 1974] indicates that the field line
resonances take place at the L-value where the driving frequency
matches the resonant frequency of the field line. The broad spectrum
will excite many field lines oscillating independently. Across
any resonating L-shell, the phase changes .
Understanding the effects of the ionosphere is essential for a proper interpretation of ground magnetic signals. The Hall currents in the ionosphere corresponding to the transverse oscillations of magnetospheric field lines are most directly responsible for the magnetic pulsations observed on the ground. For two closely separated ground stations, the magnetic induction of ionospheric currents can be seen by both stations even if only a single resonant L-shell is present. The phase difference between two ground stations can be calculated by considering the scale size of the phase change in the ionosphere. This mechanism is quite different from the model by Waters et al. [1991], which does not include the ionospheric effects and assumes that the two stations see the oscillations of two different field lines. The results for the Z component can also be interpreted easily by the ionospheric currents, if a finite conductivity is considered for the Earth.
Figure 4 illustrates how the phase difference arises between
stations on the same magnetic meridian.
Consider several resonances with resonant frequencies
, 
, and 
.
The top diagram plots the phase as a function of latitude,
that is predicted by
field line resonance theory with inclusion
of an ionospheric model as calculated by
Hughes and Southwood [1976].
The phase difference, 
, of the signals
measured by the two stations 
and 
may be read off the left hand scale and is
plotted in the bottom diagram.
If the resonant field line is to the left or the right of the center of the
two stations, its phase difference
must be smaller than the one for
resonance halfway between the stations.
If we move the two stations closer together all the phase
differences will be smaller.
If we move them apart they will be larger.
This is consistent with the observation in Figure 3.
The size of the phase difference of course depends on
the station separation relative to the scale size for the phase
change in the ionosphere.
Our results can all be interpreted in terms of this simple picture.
In summary we have presented observations that are consistent with
the interpretation of the ground based signature of magnetic
pulsations as being due to field line resonances modified by
ionospheric effects. This simple picture enables the properties
of cross-phase spectrum of geomagnetic pulsations to be explained.
In particular the phase difference between two stations increases
as the station separation in the meridian plane increases and the
observed phase difference in the Z component is explained.
The two-oscillator model differs from the conventional
field line resonance model, which has a phase change of
across any resonating L-shell.
Care must be taken in the analysis. For instance, Green et al. [1991] noted that the nonsteady nature of the signal may be important for short Pc3-4 wave packets. Also the latitudinal profile of phase differences on the ground may be different from that expected for just the resonating field lines if the energy of nonresonant waves is significant. Finally we emphasize that our reinterpretation of the origin of the phase differences seen at adjacent sites does not alter the fact that the cross phase technique is a powerful technique for probing both the ionosphere and magnetosphere.