ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-36-337-2018On the relevance of source effects in geomagnetic pulsations for induction soundingsNeskaAnneanne@igf.edu.plhttps://orcid.org/0000-0001-6749-2052RedaJan TadeuszNeskaMariusz LeszekSumarukYuri PetrovichInstitute of Geophysics, Polish Academy of Sciences, ul. Ks.
Janusza 64, 01-452 Warsaw, PolandInstitute of Geophysics of the
National Academy of Sciences of the Ukraine, Kiev, UkraineAnne Neska (anne@igf.edu.pl)7March20183623373473March201713January201824January2018This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/36/337/2018/angeo-36-337-2018.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/36/337/2018/angeo-36-337-2018.pdf
This study is an attempt to close a gap between recent
research on geomagnetic pulsations and their usage as source signals in
electromagnetic induction soundings (i.e., magnetotellurics, geomagnetic
depth sounding, and magnetovariational sounding). The plane-wave assumption
as a precondition for the proper performance of these methods is partly
violated by the local nature of field line resonances which cause a
considerable portion of pulsations at mid latitudes. It is demonstrated that
and explained why in spite of this, the application of remote reference
stations in quasi-global distances for the suppression of local
correlated-noise effects in induction arrows is possible in the geomagnetic
pulsation range. The important role of upstream waves and of the magnetic
equatorial region for such applications is emphasized. Furthermore, the
principal difference between application of reference stations for local
transfer functions (which result in sounding curves and induction arrows) and
for inter-station transfer functions is considered. The preconditions for the
latter are much stricter than for the former. Hence a failure to estimate an
inter-station transfer function to be interpreted in terms of electromagnetic
induction, e.g., because of field line resonances, does not necessarily
prohibit use of the station pair for a remote reference estimation of the
impedance tensor.
Geomagnetism and paleomagnetism (geomagnetic induction) – ionosphere (ionosphere–atmosphere interactions) – magnetospheric physics (general or miscellaneous)Introduction
In this section we motivate our study by adducting a number of contradictions
and ambiguities that are encountered both within the literature on
electromagnetic induction soundings and between literature and empirical
findings. These are demonstrated on data which will be documented in the Data
section and which have been processed by methods introduced in the Method
section. The contradictions will be solved in the light of recent research in
the Results and discussion section.
Here we refer to the magnetotelluric (MT), magnetovariational
(MV), and geomagnetic depth sounding
(GDS) methods as induction methods. The aim of such induction soundings is to
obtain a model of the distribution of electrical resistivity in the
subsurface of the solid earth which enables a geologic–tectonic
interpretation. Transfer functions between electromagnetic field components
measured at the surface or on the sea floor constitute the input data for
such a modeling. The precondition for the proper functioning of such a
modeling is that transfer functions depend only on electromagnetic induction
in the subsurface. This means in particular that the geometry of the source
field, i.e., of the incident electromagnetic wave which drives that
induction, can be neglected. This condition is referred to as the plane-wave
assumption. By this term it is understood that the incident wave field is
homogeneous over a large area on the surface. “Large” means here large
compared to the skin depth that characterizes how deep this wave penetrates
into the solid earth. This is an elementary consequence of a basic equation
in electrodynamics – the telegrapher's equation which describes the damped
propagation of an electromagnetic wave in an electrically conducting medium.
The source field in the range of long-period magnetotellurics (LMT,
comprising periods from 10 to 10 000 s) is constituted by the geomagnetic
pulsations and variations sensu stricto, e.g., of geomagnetic storm,
substorm, and bay types. All of them take their energy from the solar wind,
and they are explained by either fluctuations in the solar wind or its
interaction with the terrestrial magnetosphere. In this study we focus on the
pulsation range that is limited to periods < 600 s and
characterized by frequently occurring sinusoidal signals. For this range
textbooks of MT and other works introducing source signals for soundings
e.g., list field line
resonances, waveguides, and cavity modes as generation mechanisms. Whereas it
is accepted that in general LMT sources meet the plane-wave assumption, two
exceptions to this rule are highlighted in the literature, which will be
considered in the following.
The first exception concerns certain regions in which all LMT source signals
are inhomogeneous according to, e.g., . These regions are
situated beneath a polar or the equatorial electrojet. We focus on the latter
here because this region is more important for our study (and because the
high latitudes are rather different in terms of geomagnetic pulsations). The
equatorial electrojet is (similar to the Sq variation at mid latitudes) a part
of the current system developing in the dayside ionosphere due to solar
radiation; i.e., it has a diurnal periodicity at a given place at the
surface. The literature warns of
using data recorded during local daytime for determining transfer functions
in equatorial regions and advises one to limit their estimation to a basis of
local nighttime data. Unfortunately, it is not explained how an ionospheric
current system with diurnal periodicity can make LMT source signals (which
have much shorter periods and are generated at a much larger distance from
the surface) inhomogeneous, and why this happens in electrojet and not in Sq
regions. In other words, the impression is created that the equatorial
electrojet is causal for anomalous properties of daytime LMT sources in the
same region, although the mechanism of cause and effect is not revealed. The
statement that the diurnal electromagnetic signals originating from that
electrojet itself are not appropriate for sounding does
not contribute to an answer to this question given the fact that the Sq
variation cannot be used in the classical induction methods either. As a
result, a large region of the earth is to some degree “distrusted” in terms
of induction methods without thorough reasoning.
The second exception to the rule that LMT sources produce plane incident
waves concerns all pulsations generated by field line resonances. Field line
resonances are an everyday phenomenon. They are standing magnetohydrodynamic
waves (Alfvenic shear waves) propagating along the main field lines through
the plasmasphere and ionosphere. Their periods match the length of the given
field line (which is characterized by the L-value, )
and therefore depend significantly on geomagnetic latitude. For geomagnetic
mid latitudes most of them are in the pc3 range (10–45 s). At the
footpoints of the field lines ionospheric currents are driven which radiate
energy in the form of electromagnetic waves to the ground where they are
recognized as pulsations . Given the situation that in
principle every field line is able to generate its own electromagnetic wave
with a latitude-dependent period, it is quite clear that the wave fronts
observed on ground will be inhomogeneous rather than plane. This will be the
case even if a set of neighboring field lines is excited synchronously
(coupled resonances, ), and inhomogeneity will be most
striking when observed at two longitude-aligned stations in a distance
greater than the one between footpoints of two field lines. This problem is
pointed out in textbooks and is the objective of
several research articles . The latter
show a synthetic apparent resistivity curve with distortions caused by
resonant features in the source field. The former address violations of the
plane-wave assumption, which have been noticed in inter-station transfer
functions and which changed with a diurnal periodicity. The identification of
field line resonances as the cause was possible due to an additional large
eigenvalue in the spectral density matrix of the measured field components.
It is pointed out in the same work that such an additional large eigenvalue
in the applied processing method is indicative of DC railway noise as well.
DC railway noise is an obviously harmful anthropogenic violation of the
plane-wave assumption that often causes bad distortions in transfer functions
. A further proof of the local,
inhomogeneous character of pulsations seems to be their coherence. Calculated
for March 2013, it is only between “neighboring” observatories (BEL–KIV
sites in Poland and Ukraine, Europe) greater than 0.5, but is close to zero
for most distant pairings and periods (Table ). Given these
findings one may honestly wonder whether induction soundings in the pulsation
range are possible at all.
However, this pessimistic view is counterbalanced by the empirical situation.
MT in the pulsation range is done routinely and as a matter of course, and
neither are hints on possibly source-related problems in this range a
ubiquitous phenomenon in the literature on regional MT nor can the authors of
this study confirm such problems from their own experience. A further
argument comes from the remote-reference (RR) technique, which utilizes data
of a second site to stabilize and correct local transfer functions of some
site in MT and GDS. This technique relies on coherence of data between both
sites, and it turns out that its successful application is possible even for
distant reference sites. report on such an application
of a German observatory to data from the Pyrenees and even from Morocco
(northern Africa). Figure shows how a heavy distortion by DC
railway noise in a GDS sounding at a Polish observatory (BEL, treated in
detail in ) is corrected by means of a number of
different reference sites. It is shown that reference sites from other
continents yield results basically equal (even if not as smooth) to a
classical reference site from the same region. This works under the condition
of a sufficient data amount, where “sufficient” means a multiple of the
data amount necessary for an everyday LMT sounding (ca. 3 months in contrast
to 2–3 weeks). One of these references (MBO) is situated in Senegal in the
region of the magnetic Equator.
Another contradiction is demonstrated in Fig. . The upper part
(a) shows an inter-station transfer function between two Polish sites which
completely confirms the objections in . Its behavior at
periods < 100 s is not a small deviation varying smoothly over
period from the unity matrix, but makes no sense at all in the paradigm of
induction and would be treated as failure. Interestingly, the magnetotelluric
transfer functions based on data from the same time (b of one of the sites, c
of both sites) are not unusual, problematic or suspicious in this period
range. Thus we finish this section by posing two questions. First, why does
RR with distant reference sites work in spite of low coherences between both
sites? And second, why does a violation of the plane-wave assumption lead to
a failure in estimating a reasonable inter-station transfer function, but is
not an obstacle to determining a magnetotelluric transfer function?
Coherences between H components of observatories in
“neighborhood” (KIV-BEL) and “inter-continental” (all others) distances.
See Sect. for their locations. Data from the entire
March 2013.
Period (s)224488176352distanceKIV-BEL0.360.520.580.640.77700 kmKIV-MBO0.010.070.160.350.315800 kmMBO-BOU0.060.040.040.030.138900 kmKIV-BOU0.110.010.020.070.029000 km
Real (a, c, e, g) and imaginary (b, d, f, h) induction arrows for
Belsk observatory. The north-pointing real arrows in the single-site
processed upper panel are an effect of distortion by DC railway noise. This
effect vanishes if data are processed with the remote reference method (three
lower panels). Very far reference sites (MBO, Senegal, and FRD, US) are
similarly helpful to a classical regional one (SUW) if the data amount is
sufficiently high (15 days for both upper panels, 101 days with MBO, 85 days
with FRD).
The inter-station transfer function between sites GRB and SUW (a)
behaves erratically (from an induction point of view) for periods below 100 s. Sounding curves for GRB from the same data (amount 5 days, (b)
single-site, (c) remote-reference with SUW, only off-diagonal elements shown)
in this period range do not show unusual features.
Data
In this study selected data from the International Real-time Magnetic
Observatory Network (INTERMAGNET, www.intermagnet.org) have been used,
which consists of ca. 100 observatories all over the globe. Some years ago
INTERMAGNET introduced the possibility of publishing and accessing
geomagnetic variation data with an increased sampling rate of 1 Hz
additionally , whereas the shortest sampling interval
that could be published prior to that was 1 min .
These 1 s data offer the chance to investigate almost the whole pulsation
range (with the exception of pc1 and pi1) practically in a world-wide manner.
We included data of five INTERMAGNET observatories in our analysis. KIV and
BEL are situated in Europe, BOU and FRD in North America, and MBO in Africa
at the geomagnetic Equator. Further data have been taken from ODE (an
observatory in the Ukraine), SUW (a permanent variometer station in Poland),
and GRB (a long-term magnetotelluric station in Poland from a running
project). Table introduces corrected geomagnetic coordinates
(CGM from omniweb.gsfc.nasa.gov for 2013) and L-values
(ibid.) of these sites. They are mapped in Fig. . For
all sites the north (X) and east (Y) components of the magnetic variation
data have been used. Because of the GDS study the vertical magnetic component
(Z) has been included for BEL. The GRB magnetotelluric
site also comprises 1 s records of
both perpendicular horizontal electric field components.
MethodCoherence analysis
The INTERMAGNET geomagnetic variation data have been rotated from the
geographic (X,Y,Z) to geomagnetic (H,D,Z) coordinate systems. Spectral
values have been produced by means of the wavelet transform, a tool that is
often applied in geomagnetic and related studies, e.g., by
, , , and
. Thereby a signal depending on time t, e.g., H(t),
is subjected to a convolution with a wavelet function Ψ (an asterisk
* denotes the complex conjugate):
WH(a,b)=1a∫-∞∞H(t)Ψ*t-badt.
As Ψ we used the Morlet wavelet (as done by all authors cited
above) which has a shape matching pulsations:
ΨMorlett=π1/4exp-t22expiω0t,ω0=π2/ln2,
where ω0 is the dimensionless frequency parameter of the Morlet
wavelet . The resulting wavelet coefficients
WH are functions of a and b, where a describes the
dilatation of the wavelet pattern. It represents the frequency about which
the given coefficient yields information. b controls the shifting of the
wavelet along the signal and ensures that the information on frequency
content of a given coefficient can be assigned to a particular time. This is
important when analyzing non-stationary processes or looking for rarely
occurring phenomena, and it constitutes the main advantage of the wavelet
transform over the Fourier transform F.
Locations, geomagnetic coordinates (CGM), and L-values of the used
sites.
Map with site locations. The red line marks the magnetic Equator
following .
Certain properties of the wavelet transform allow for a re-consideration of
Eq. () that can save much computing power. First, according to the
convolution theorem, for a constant a, WH can be regarded as
the inverse Fourier transform of the product of the Fourier transforms of
signal H and wavelet Ψ:
WH=F-1{F(H)F(Ψ)}.
Second, (F(Ψ)) has specific properties. Its Fourier
spectrum is narrow and self-similar under dilatation. This means that if we
are not interested in a very fine frequency resolution and consider only a
values differing by a constant factor (typically 2), F(Ψ)
has to be calculated only for one a and for the rest F(Ψ)
can be reconstructed, basically by shifting it through frequency space.
Furthermore, the multiplication in Eq. () has to be carried out
only where (F(Ψ)≠0). These considerations lead to
the fast wavelet transform cf., e.g.,. Both versions,
the latter and the one described in Eq. (), have been applied in
this study.
By means of the wavelet coefficients of synchronous signals of two
observatories, e.g., H1 and H2, their coherence can be calculated
according to
γ2(ω)=|〈H1(ω)H2*(ω)〉|2〈H1(ω)H1*(ω)〉〈H2(ω)H2*(ω)〉.
The relation between a, period T, and angular frequency ω is
T=a2ln(2)=2πω.
Brackets 〈〉 indicate a stacking over a certain number of
coefficients at subsequent b values.
Transfer functions
When carrying out induction soundings, the area for which a model of electric
resistivity shall be derived is covered with a profile or an array of
stations which measure the variations of magnetic (X, Y, Z) and, in the
case of MT, electric (Ex, Ey) field components. For an LMT campaign
each measurement lasts usually 2–3 weeks and the sampling interval is 1 s
or longer. During the data processing transfer functions between these
components are derived which are introduced in the following. All of them are
complex functions of frequency ω. Usually they vary only smoothly over
frequency (or period), and their real and imaginary parts (or absolute values
and phases) are connected with each other in a relationship that is typical
of induction processes.
GDS is of special interest here because geomagnetic observation data can be
utilized. Here a two-component transfer function (A(ω),B(ω))
between a station's Z component on the one hand and the X and Y
components on the other hand is estimated which satisfies the equation
Z(ω)=A(ω)X(ω)+B(ω)Y(ω).(A(ω),B(ω)) is usually displayed as induction arrows (a real and
an imaginary one) with A as north and B as the east component as in
Fig. . The solution of Eq. () for
(A(ω),B(ω)) in terms of least squares is
(A,B)={(X→,Y→)†(X→,Y→)}-1{Z→†(X→,Y→)},
where the vectors consist of a large number N of coefficients at the same
frequency, e.g., X→(ω)=(X1(ω),X2(ω),…,XN(ω))⊤. A⊤ denotes the
transpose and a† the Hermitian transpose.
In MT the magnetotelluric transfer function between the horizontal electric
and horizontal magnetic components of a station is calculated. It has the
shape of a 2 × 2 tensor Z(ω)=((Zxx(ω),Zyx(ω))⊤,(Zxy(ω),Zyy(ω))⊤)
and satisfies the equation
(Ex(ω),Ey(ω))⊤=Z(X(ω),Y(ω))⊤.
The solution for Z(ω) is
Z⊤={(X→,Y→)†(X→,Y→)}-1{(Ex→,Ey→)†(X→,Y→)}.Z is displayed as so-called sounding curves consisting of apparent
resistivy ρa (e.g., ρaxy=μ0/(2π)|Zxy|2T) and phases
ϕxy=arctan(ImZxy/ReZxy). In
case of not too complicated resistivity structures the off-diagonal elements
Zxy, Zyx are much larger than the others; thus, often only they are
presented; cf. Fig. b and c.
The solutions for (A,B) and Z can become biased and unusable in the
presence of certain types of anthropogenic noise. A crass example is the
electromagnetic signals transmitted from DC railways, since they spread over
distances of ca. 100 km from the source and are contradictory to the
plane-wave assumption. A remedy for this is the remote-reference method. This
means that data XR→,
YR→ are included in the processing which have
been measured at another variometer station that is sufficiently remote for
the noise to be damped away, whereas the natural signal remains correlated.
In many cases it is practical to use observatories as remote-reference
stations. Thereby, e.g., Eq. () changes to
ZRR⊤={(X→,Y→)†(XR→,YR→)}-1{(Ex→,Ey→)†(XR→,YR→)}.
This leads to a stacking over cross spectra (instead of auto spectra in
Eq. ) which suppresses noise features that are incoherent between
both sites. The usage of remote data in both enumerator and denominator
warrants that the information content remains unchanged compared to
Eq. (). The approach in Eq. () is referred to as the
single-site solution to distinguish it from the remote-reference approach
(Eq. ).
If the plane-wave assumption is taken seriously, the following argumentation
is possible: the incident magnetic field at two stations of an array is
exactly equal. Then differences between stations come only from the much
smaller secondary magnetic fields which accompany currents locally induced in
the conducting solid earth by the incident field. Therefore the inter-station
transfer function HMT=((HMTxx,HMTyx)⊤,(HMTxy,HMTyy)⊤)
between the horizontal magnetic components of two stations
(X,Y)⊤=HMT(XRR,YRR)⊤
only gently deviates from the unity matrix and is indicative of induction and
conductivity distribution. This is the idea of magnetovariational sounding,
where inter-station transfer functions serve as input data for a modeling of
the resistivity distribution in the subsurface. It has already been mentioned
that the HMT displayed in Fig. a does not meet this expectation
for part of the period range.
It can be shown
that the remote-reference
solution for MT (Eq. ) and GDS transfer functions can be formulated
in a way that makes use of the inter-station transfer function. Thereby a
quasi-magnetotelluric transfer function Zq is calculated between
horizontal electric components of one station and horizontal magnetic
components of another (reference) station, which is then multiplied by the
inverse of the inter-station transfer function between the same stations:
ZRR⊤=(HMT⊤)-1Zq⊤.
For the estimation of most transfer functions within this study, Egbert's
code has been used. It applies, like all modern
processing codes in this domain, robust instead of least-square statistics to
stabilize results against outliers in the data. To finish this section we
provide the formula for the skin depth δ. For the simplest model of
resistivity distribution in the solid earth, a homogeneous halfspace of
resistivity ρ (in Ωm), the skin depth in km amounts to
δ≈12ρT.
Results and discussion
Figure shows dynamic amplitude spectra and mutual coherences
for H components of the KIV, MBO, and BOU observatories during 1 day in
March 2013. The amplitude spectra have been obtained via the “slow” wavelet
transform (Eq. ); for this reason the resolution over period is
better. The time resolution, i.e., the interval from which coefficients were
taken for one stacked value, is about 10 min, with small differences from
period to period. The coherences come from the fast wavelet transform
(Eq. ) and the time resolution is 15 min. Thereby one coherence
value for 100 s is stacked over 85 coefficients, for 50 s over 170, and for
25 s over 341 ones. Although maximum activity takes place around local noon
and is therefore shifted between observatories, a number of striking events
in the spectra happen synchronously in all observatories and lead not
throughout but often to medium or high coherence values. That the signals
behind such values are really pulsations is frequently very obvious in time
series, too; cf. Fig. . From both pictures it is evident that
there exist pulsations that cannot originate from field line resonances since
they are present at the Equator (far from footpoints of field lines) and have
a quasi-global coherence length. If one looks up the literature for a
possible origin of pulsations with such properties, one gets a clear answer,
at least for the pc3 and pc4 range. These are most probably upstream waves.
The following description of their generation follows mainly but not only
.
Dynamic amplitude spectra of (a) and coherences between
(b)H components of KIV, MBO, and BOU observatories for
4 March 2013, for a number of pc3–4 periods. Although the main activity
takes place around local noon (marked with vertical dashed lines in
a), many events take place synchronously in these distant places,
and often short-term coherence (time resolution: 15 min) is not low. Such
events are marked with grey vertical lines.
Two examples of highly coherent synchronous time series between KIV,
BOU, and MBO H components. The upper part shows pulsations in the pc3 range
which originate most probably from upstream waves. The example belows
contains pc5 pulsations to the left, probably from another global source
mechanism. The time series are high-pass filtered with a cut-off period of
300 s.
In the terrestrial foreshock region ions originating from solar wind and
having been deflected from the terrestrial magnetosphere are subjected to
cyclotronic resonance in the interplanetary magnetic field. Thereby they
produce a signal at the cyclotron resonant frequency that propagates upstream
from the solar wind. But since the velocity of the latter is greater than the
propagation velocity of the former, these upstream waves are swept back
against the magnetopause and, under convenient conditions, couple to the
magnetosphere. Then they propagate in the form of compressional waves (fast
Alfvénic mode) partly directly through the magnetosphere. Where they
encounter a magnetic field line with a component parallel to their
oscillation direction, they couple to it, and if its resonant frequency
matches their own frequency, they excite field line resonances which have a
different oscillation direction. In the ionosphere all ULF waves experience a
rotation of their polarization e.g.,, such that
field line resonances and direct upstream waves cannot be distinguished
easily in the geomagnetic pulsations as which they are observable on the
ground. Data from the interplanetary magnetic field enable identification of
upstream waves since there is a (in first approximation) simple linear
relationship between its strength and their frequency
e.g.,.
Multi-spacecraft data led also to the result that the coherence length of
upstream waves in the foreshock region is on the order of magnitude of 1
earth radius . Comparison of spacecraft data mapping the inner
magnetosphere with a mid latitude ground station revealed a practical
dayside-wide coherence of upstream waves in geomagnetic low and mid latitudes
. The same study confirms a maximum of upstream wave power
in the equatorial pre-noon region. Upstream waves can also be refracted into
the nightside of the magnetosphere and then have smaller amplitudes
; this effect is visible for BOU in the upper part of
Fig. .
Pulsations in the region of the geomagnetic Equator are special. They possess
practically no D component and the amplitudes of the H component are
larger than at the adjacent low latitudes. This phenomenon is referred to as
the equatorial enhancement. It is not clear whether this is the reason for
difficulties in equatorial LMT. Similarly, the reason for the enhancement
itself is not clear. Proposed explanations comprise both interactions with
the electrojet and the idea that the geometry of the magnetosphere causes the
equatorial plane to be less dissipative for waves traveling through it
and citations therein.
xx element of the inter-station transfer function between SUW and
GRB (red) and KIV and ODE (blue) for 1 h of data. Both show the typical
pattern for field line resonances that allows for determination of the
resonant frequency (indicated by arrows). It is 18 s (55 mHz) for the
KIV-ODE pairing (midpoint latitude 44.5∘) and 26 s (37 mHz) for
SUW-GRB (48.2∘). The green curve representing KIV-ODE for another
hour of data demonstrates that this transfer function is not stable in time,
and such features may not reliably appear because of a temporal lack of such
resonances (almost zero absolute value below 20 s).
Sounding curves from the quasi-magnetotelluric transfer function
between electric components of GRB and horizontal magnetic ones of SUW. Data
are the same as in Fig. . Note the similarity to the
inter-station transfer function (Fig. a) in the “erratic”
short-period behavior.
Ground-based research beyond low latitudes confirms that field line
resonances are excited by upstream waves and that the latter are not less
common than the former . They also report
pulsation events coherent over thousands of kilometers
. Analysis tools in this domain are
extraordinarily fine compared to MT ones; frequency resolution can reach a
few millihertz and in time structures can be caught that exist only for some
minutes as shown in . These authors also report that
structures typical for upstream waves and structures typical for field line
resonances alternate rapidly and in a way that suggests that upstream waves
initiate and finish field line resonances. Identification of field line
resonances is, beside other techniques, possible by means of two
longitudinally arranged stations spaced by at least 80–100 km in distance,
which is the estimated surface width of resonant shells
. The technique makes use of the fact that a
signal resulting from the superposition of two (monochromatic) signals with
nearly but not completely the same frequency exhibits a minimum–maximum
transition in amplitude and a maximum in phase at the midpoint between both
frequencies . It is called the cross-phase technique and
applied not only for verification of field line resonances, but also for
determination of their resonant frequency
e.g.,. Interestingly, its
implementation is formally almost equal to an inter-station transfer function
. Figure shows xx components of such a
transfer function between two appropriate station pairs (SUW-GRB and KIV-ODE;
cf. Fig. ) obtained from 1 h of data and based on wavelet
coefficients. The expected features are visible in the blue and red curves;
according to the phase maxima the resonant frequencies amount to ca. 55 mHz
(18 s) for KIV-ODE and ca. 37 mHz (26 s) for SUW-GRB. It meets
expectations that the pairing at a higher latitude has a lower frequency.
Comparison with the correspondent time in Fig. hints at
broadband activity in both and coherence between KIV and MBO. Since this
cannot be explained by field line resonances alone, it is very probable that
upstream waves stand behind the blue curve in Fig. as well.
The midpoint latitude of SUW-GRB is 48.2∘, which is almost equal to
the geomagnetic latitude of Niemegk observatory in 1985 (48.0∘). Data
from this place and time were analyzed by , who found
that the resonant frequency amounts to 30–35 mHz. Given the poor frequency
resolution in our method, we regard this as a confirmation of our result. The
green curve in Fig. has been obtained for another hour of
KIV-ODE data. We interpret the scatter in phase and the missing maximum in
amplitude as a sign of weak or absent field line resonances at this time.
Comparison with Fig. reveals that activity at the resonant
frequency was absent in both KIV and MBO at this time. There is, however,
some coherent activity at longer periods that could be possibly ascribed to
upstream waves.
Inter-station transfer function between BEL (Poland) and MBO
(Senegal) for 101 days of data.
The result in Fig. confronts us with two insights that may
appear somewhat heretical from the point of view of induction soundings.
Inter-station transfer functions can change in time (depending on the
presence/absence of field line resonances), and they can have a physical
meaning beyond electromagnetic induction (exhibiting a resonant frequency).
In this light the inter-station transfer function in Fig. a does
not appear meaningless. The putative distortions are simply a (even if
coarsely resolved) signature of field line resonances. Obviously the
violation of the postulate that the incident field at both stations must be
equal prevents the inter-station transfer function from taking a shape
interpretable for MV. The question why MT works for this example can be
answered in two ways. The first one is formal. Figure shows
Zq for these data which exhibits analogous resonance signatures.
Application of Eq. () means, roughly speaking, that it is divided
by the HMT in Fig. a. Thereby resonance features are cancelled
out and a flawless RR result remains (Fig. c). The second way
consists in taking the postulates seriously. The plane-wave assumption for
local (not inter-station) transfer functions does not mean that the incident
waves must be truly plane. They have to be plane on a scale greater than the
skin depth. If we take the width of a resonant shell as the scale of
inhomogeneity and assume that field line resonances at mid latitudes are
limited to the pc3 range (10–45 s), application of Eq.
reveals that up to a halfspace resistivity ∼500Ω m field line
resonances are not an obstacle for MT and GDS. This is confirmed by
, who point out that only low-conductive structures can
be affected by this problem. In cases where MT or GDS results in
high-resistive regions are affected by field line resonances, it might be a
good suggestion to attempt a remote-reference application of an equatorial
station (which is free of such resonances). In this way at least resonances
that do not occur at the same time as upstream waves would be prevented from
entering the transfer functions.
An inter-station transfer function between mid latitudes and the Equator
(corresponding to the third panel in Fig. ) is shown in
Fig. . As discussed above, an induction-based interpretation
seems highly improbable. However, some parts of it are not too far from the
unity matrix, e.g., the pc5 range (150–600 s) of the xx element. This
fits to the finding that long-term coherences for a similar station pairing
in this range are not really low (>0.3; see Table ).
There remains the apparent contradiction that for shorter periods, long-term
coherences are very low (ibid.) but short-term ones are not
(Fig. b). This seems to be a systematic tendency. We have
additionally calculated coherences for the whole of March 2013 during 4 h
intervals. In general such pc3–4 coherences are low on an inter-continental
scale, but for a few 4 h intervals during that month they are medium to high
(not shown). Analogous traces of such a trend can be found in the literature.
reports that 20 min coherence in airborne pulsations
⩽ 25 s drops at a distance of 150 km, whereas
find dayside-wide 1 min coherence for upstream waves.
These findings could be a hint at a possible time dependency in far-distant
inter-station transfer functions in the pulsation range.
Conclusions
It has to be admitted that currently, it is difficult to understand what is
state of the art in fundamental research on geomagnetic pulsations. That is
because “The bibliography […even on a section of this domain –
insertion by the authors] includes several hundred publications”
and because currently there is a large number of new
insights in this area, but also a series of open questions
and therefore no clear, comprehensive, and consistent picture. Nevertheless
it raises doubts that 20 years after their importance for geomagnetic
pulsations was noticed, upstream waves are not mentioned in the signal source
sections of magnetotelluric textbooks. Given the fact that due to their large
coherence lengths these waves have very convenient properties for induction
soundings, it seems inappropriate that many MT workers appear unaware of
them, whereas the much more problematic field line resonances are relatively
well represented in the MT literature.
The peculiarities of LMT sources in the magnetic equatorial region are
currently not well understood, but are still under debate. This should be
clearly pointed out instead of making vague references to the electrojet.
Such a clarification would clear the way for unprejudiced research in this
area. Research of this type appears promising, particularly with regard to
equatorial stations as distant reference sites. This idea is justified by the
findings that (a) equatorial pulsations are more directly ascribable to
upstream waves and not “disturbed” by field line resonances, and (b) the
equatorial region is privileged in terms of high large-distance coherence of
upstream waves with (at least) mid latitude regions.
When considering the question whether the plane-wave assumption is violated
by field line resonances, a clear discrimination between estimation of
inter-station transfer functions (for MV sounding) and MT/GDS transfer
functions has to be made. This is because the MV precondition that the
incident field at both stations has to be equal is very strict. It is readily
violated in everyday situations. A north–south spacing of 80–100 km is
sufficient to produce effects around the resonant frequency in the
inter-station transfer function that make it unusable for a sounding
interpretation unless special MV processing schemes which remove data
fragments leading to non-stationary results far from the unity
matrix, are applied. In contrast, the precondition for usable
local transfer functions is weaker. It only says that the skin depth of the
incident wave has to be significantly smaller than the mentioned distance.
This is the case at mid latitudes for low to moderate resistivities of the
subsurface and only massive high-resistive structures give reason for
caution. In other words, situations may be encountered where MT/GDS is well
possible in the pulsation range although MV (with the same data) is not. RR
can be applied with such data, and it is not a contradiction that RR can be
formulated in a way that relies on the inter-station transfer function. This
is because RR does not require that the inter-station transfer function makes
sense in the MV paradigm; it seems to be sufficient that it exists. The
latter sentence is – together with the insight that many LMT sources have a
global rather than local origin – also the key to the understanding that
reference sites may be situated at much larger distances than assumed so far.
Data from BEL, BOU, FRD, and MBO
are available at the INTERMAGNET database. Data from SUW used for Fig. and from KIV and
ODE have been added to this publication as supplement. Data from SUW used
in Figs. , , and and from GRB
are part of a running project and therefore are not available before the
project ends in summer 2018. They may become available at a later time on
request to Vladimir Semenov (sem@igf.edu.pl) or Mariusz Neska
(nemar@igf.edu.pl). The MT processing code by
Gary Egbert is available, see reference list (Egbert and Booker, 1986). All other codes used in this study are available on
request to Anne Neska (anne@igf.edu.pl).
The Supplement related to this article is available online at https://doi.org/10.5194/angeo-36-337-2018-supplement.
The authors declare that they have no conflict of
interest.
This article is part of the special issue “The Earth's magnetic
field: measurements, data, and applications from ground observations
(ANGEO/GI inter-journal SI)”. It is a result of the XVIIth IAGA Workshop on
Geomagnetic Observatory Instruments, Data Acquisition and Processing,
Dourbes, Belgium, 4–10 September 2016.
Acknowledgements
The results presented in this paper rely on data collected at magnetic
observatories. We thank the national institutes that support them and
INTERMAGNET for promoting high standards of magnetic observatory practice
(www.intermagnet.org). This work was supported within statutory
activities no. 3841/E-41/S/2017 and relies on data from grant no. NCN
2014/15/B/ST10/00789 of the Ministry of Science and Higher Education of
Poland. We thank the topical editor Arnaud Chulliat for handling the
manuscript and the reviewers Frederick Menk and Andreas Junge for many
invaluable hints that helped to improve it. The topical editor, Arnaud Chulliat, thanks Frederick
Menk and one anonymous referee for help in evaluating this paper.
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