Introduction
The radio occultation (RO) remote sensing has been known for the last 50
years as a powerful tool for investigation of the atmospheres, ionospheres
and planetary surfaces (Fjeldbo, 1964; Marouf and Tyler, 1986;
Lindal et al., 1983, 1987; Hinson et al., 1997, 1999; Yunck et al., 2000;
Yakovlev, 2002, and references therein). With regard to the study of
near-Earth space, the RO method should be competitive with other means of
remote
sensing (Gurvich and Krasilnikova, 1987; Yunck et al., 1988; Melbourne et al., 1994; Steiner et al., 1999; Beyerle and Hocke, 2001; Beyerle et al., 2001, 2002; Yakovlev, 2002; Liou et al., 2002, 2003, 2005a, 2010; Manzini and Bengtsson, 2008; Anthes, 2011). Assumption of spherical
symmetry – cornerstone of RO method – should be carefully analyzed when
the RO technology is applied to global monitoring of the Earth's ionosphere
and atmosphere at different altitudes (Vorob'ev and Krasilnikova, 1994;
Melbourne et al., 1994; Syndergaard, 1998, 1999; Yunck et al., 2000).
In particular, effectiveness of the RO method applied for investigation of
the Earth's ionosphere can be compared with radio tomographic approach
(Kunitsyn and Tereshchenko, 2003). The tomographic method allows obtaining
2-D distributions of electron density in the ionosphere using a chain of
ground-based receivers, which capture signals of low Earth orbital (LEO) or
navigational satellites along a set of intersecting radio rays (Kunitsyn et
al., 2011, 2013). Unlike the radio tomographic approach, the RO method uses
a set of nearly
parallel radio-ray trajectories. The RO method applied for processing the
assumption of spherical symmetry of the Earth's ionosphere and atmosphere
with known location of the centre of symmetry (Melbourne et al., 1994;
Yakovlev, 2002; Melbourne, 2004). In accordance with this assumption, all
resulting altitude profiles of atmospheric and ionospheric parameters are
attached to vertical and horizontal coordinates of the radio ray perigee
relative to a spherical symmetry centre, which is close to or coincident
with the centre of the Earth or a selected planet.
Highly stable signals synchronized by atomic frequency standards and
radiated by GPS satellites at frequencies F1 = 1575.42 MHz and
F2 = 1227.60 MHz create at the altitudes from 0 to 20 000 km radio fields that can be
used for the development of the RO method as a new tool for
global monitoring of the ionosphere and neutral atmosphere (Gurvich and
Krasilnikova, 1987; Yunck, 1988). Several LEO missions were launched during
1995–2014 for the study of the atmosphere and ionosphere: GPS/MET
(Melbourne et al., 1994; Ware et al., 1996; Gorbunov et al., 1996; Kursinski
et al., 1997; Vorob'ev et al., 1997; Hajj and Romans, 1998), SAC-C (Schmidt et al., 2005), CHAMP
(Wickert et al., 2001, 2009), FORMOSAT-3 (Liou et al., 2007; Fong et al., 2008),
GRACE (Hajj et al., 2004; Wickert et al., 2005), METOP (Von Engeln et al., 2011;
Joo et al., 2012), TERRA-SAR, TANDEM-X (Zus et al., 2014), and FY-3 CNOS
(Bai et al., 2014). The success of these missions demonstrated that the RO
technique is a powerful remote sensing tool for obtaining key vertical
profiles of bending angle, refractivity, temperature, pressure and water
vapour in the atmosphere and electron density in ionosphere with global
coverage, high spatial and temporal resolution (Zhang et al., 2013).
Explicit analysis of experimental data of LEO missions introduced important
contributions in the following areas: (i) the theory of radio wave propagation (Gorbunov and
Gurvich, 1998a; Gorbunov et al., 2002; Benzon et al., 2003; Gorbunov and
Lauritsen, 2004; Gorbunov and Kirchengast, 2005; Pavelyev et al., 2004,
2010a); (ii) climate changes detection (Kirchengast et al., 2000;
Steiner et al., 2001; Foelsche et al., 2008); (iii) space weather
effects and ionosphere monitoring (Rius et al., 1998; Jakowski et al., 2004;
Wickert et al., 2004; Arras et al., 2008, 2010); (iv) deriving new
radio-holographic methods of the RO remote sensing (Karayel and Hinson,
1997; Mortensen and Hoeg, 1998; Pavelyev, 1998; Gorbunov and Gurvich, 1998b;
Mortensen et al., 1999; Hocke et al., 1999; Sokolovskiy, 2000; Sokolovskiy et al., 2002; Gorbunov, 2002a, b;
Gorbunov et al., 1996, 2002, 2010; Igarashi et al., 2000; Jensen et al.,
2003, 2004; Pavelyev et al., 2002, 2004, 2010a, b, 2012; Liou et al.,
2010; Yakovlev et al., 2010).
Recently, an important connection between the intensity and time derivatives of the phase, eikonal, Doppler frequency of radio waves propagating
through the ionosphere and atmosphere has been discovered by theoretical
analysis and confirmed by processing of the RO radio-holograms (Liou and
Pavelyev, 2006; Liou et al., 2007, 2010; Pavelyev et al., 2008a,
2012). This connection is a key regularity of the RO method. Now this
relationship gets a possibility to recognize that the phase (eikonal)
acceleration (proportional to the time derivative of the Doppler shift) has
the same importance for the theory of radio waves propagation in a layered
medium and solution of the RO inverse problem as the Doppler frequency,
phase path excess, and refractive attenuation of the RO signal (Liou and
Pavelyev, 2006; Liou et al., 2007; Pavelyev, 2008, 2013; Pavelyev et al.,
2009, 2010a, b, 2012, 2013). It follows from this connection that the
phase acceleration technique allows one to convert the phase and Doppler
frequency changes into refractive attenuation variations at a single
frequency. Note that this is similar to classical dynamics when the
derivations of the
path and velocity on time and acceleration are connected by the Newton's
laws. From such derived refractive attenuation and intensity data, one can
estimate the integral absorption of radio waves. This is important for
future RO missions for measuring water vapour and minor atmospheric gas
constituents, because the difficulty of removing the refractive attenuation
effect from the intensity data can be avoided. The phase
acceleration/intensity technique can be applied also for determining the
location and inclination of sharp layered plasma structures (including
sporadic Es layers) in the ionosphere. Advantages of the phase
acceleration/intensity technique are validated by analyzing the RO data from
the Challenging Minisatellite Payload (CHAMP) and the FORMOSA Satellite
Constellation Observing Systems for Meteorology, Ionosphere, and Climate
missions (FORMOSAT-3/COSMIC).
The locality principle generalizes the phase path excess
acceleration/intensity technique to the practically important case in which
the location of the symmetry centre of layered medium is unknown
(Pavelyev, 2013; Pavelyev et al., 2012). New
relationships have been revealed to expand the scope and applicable
domain of the RO method. These relationships allow, in particular, measuring
the real height, inclination, and displacement of atmospheric and
ionospheric layers from the RO ray perigee relative to the Earth's (or other
planetary) surface. This implies the possibility of determining the position
and orientation of the fronts of internal waves, which opens a new RO area
in geophysical applications for remote sensing of the internal waves in the
atmospheres and ionospheres of Earth and other planets (Steiner and Kirchengast, 2000; Liou et al., 2003, 2005b, 2007; Pavelyev et al., 2007; Gubenko et al., 2008a, b, 2011).
The goals of this paper are the following: (i) to formulate a principle of locality; (ii) to
present several important findings arising from the locality principle; and
(iii) to introduce a new index of ionospheric activity. The paper is
structured as follows. In Sect. 2 the formulation of the locality principle is
presented. Section 3 describes three important findings following from the
locality principle: (i) a possibility to determine the total absorption at a
single frequency; (ii) a possibility to evaluate the height, displacement
from the radio ray perigee, and tilt of the inclined ionospheric
(atmospheric) layers; (iii) method for separation of the contributions of
the layered and irregular structures in the RO signal, and technique for
measurement of parameters of layers and turbulence at a single frequency
using joint analysis of the amplitude
and phase variations. In Sect. 4 a new scintillation index based on the
refractive attenuation found from the phase variations of the RO signal is
introduced and its correlation with the S4
index is established. Conclusions are given in Sect. 5.
Principle of locality
The principle of locality is based on a previously established connection
(Liou and Pavelyev, 2006; Liou et al., 2007; Pavelyev et al., 2008a, b,
2009, 2010a), which relates the eikonal acceleration a and
refractive attenuation Xp′(t) of the RO signal emitted by a
transmitter G and received by satellite L after passing through a
spherically symmetric medium with a centre of symmetry at point O′ (Fig. 1):
1-Xp′(t)=m′a,a=d2Φ(t)dt2=λdFd(t)dt,m′=d2′d1′(d1′+d2′)(dps′/dt)-2;ps′=O′D′;Φ(t)=∫GLn(l)dl-R0,
where λ is the length of radio waves; d1′,d2′, and R0 are the distances along the straight lines GD′, D′L and
GL, respectively; point D′ is the projection of the centre of symmetry
O′ onto the straight line GDL; point D is the projection of the
centre O on the straight line GL; ps′ is the impact parameter
of the straight line GDL relative to the centre O′; Φ(t) is the
difference between the eikonal of radio waves propagating along the radio
ray trajectory GTL and the length GDL as a function of time t; n(l) is
the refractive index; dl is the differential length of the radio ray GTL;
and point T is the radio ray perigee having the altitude h relative to
the Earth's surface (or that of another planet) (Fig. 1). Another important geometric
parameter is the height H of the line of sight GDL above the surface.
During a RO event the magnitude H changes from positive to negative
values. The eikonal acceleration a (Eq. 1) is proportional to the time
derivative of the Doppler frequency of radio waves Fd(t). Equality
(Eq. 1) is fulfilled under the following conditions (Liou et al., 2007; Pavelyev
et al., 2008a, b):
p′-ps′dR1,2′dt≪ps′dps′dt;d1′≫d2′;p′-ps′ddt∂θ′∂ps′dps′dt≪dp′dt-dps′dt∂θ′∂ps′dps′dt.
where p′ is the impact parameter of the ray GTL relative to the centre
O′. In the case of the RO ionospheric research the centre of spherical
symmetry can be shifted relative to the centre O, and the first and third
inequalities are satisfied only if the distance OO′ is significantly less
than the Earth's radius ρe (Fig. 1). The second inequality is
necessary for excluding uncertainty because of symmetry in the dependence of
the coefficient m′ with respect to variables d2′,d1′. In the case of GPS RO atmospheric sounding, these
inequalities are satisfied if the orbits of satellites G and L are
circular, the centre of spherical symmetry O′ is almost identical to the
centre of the Earth (or another planet) O, and point T′ coincides with perigee
T of the ray GTL (Fig. 1). Location of the radio ray perigee T in
accordance with solution of the RO inverse problem determines temporal
dependencies of the height h above the Earth's surface and horizontal
coordinates of atmospheric layers (Melbourne et al, 1994; Yakovlev, 2002;
Melbourne, 2004).
Scheme of radio occultation measurements.
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Comparison of the refractive attenuations Xa(h) and
Xp(h) and their polynomial approximations, corresponding to the
FORMOSAT-3 RO measurements carried out on 18 October and 11 April 2008
(left and right plots, respectively). Thick and thin rough curves (marked by
indices “p” and “a”) describe the vertical profiles of Xp(h) and Xa(h), respectively. Smooth curves describing polynomial
approximations of the altitudes dependences of Xp(h) and Xa(h)
are also highlighted by indices “p” and “a”.
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When absorption is absent, the refractive attenuation Xp′(t)
found from (Eq. 1) should be equal to the refractive attenuation Xa(t)
determined using the RO intensity data (Liou and Pavelyev, 2006):
Xp′(t)≡Xa(t);Xa(t)=I/I0,
where I0 and I are intensities of the RO signal before and after the
moment when radio ray enters the medium, respectively. Identity (Eq. 3) is
fulfilled if the coefficient m′ in Eq. (1) is evaluated in accordance
with location of the tangential point T′, which is the perigee of radio
ray GTLrelative to the centre of symmetry O′ (Pavelyev et al., 2013).
The refractive attenuation Xa(t) measured from the RO signal intensity
data does not depend on position of the point T′ on the ray GTL and,
naturally, on coefficient m′. However, calculated value of the refractive
attenuation Xp′(t) does depend on the coefficient m′
(Eq. 1) and location of the tangent point T′ on the ray GTL. This permits to
formulate principle of locality under the conditions of single-ray radio
wave propagation and absence of absorption (Pavelyev et al., 2012, 2013;
Pavelyev, 2013): the refractive attenuations Xp′(t) and Xa(t) are equal if evaluation of Xp′(t) is provided with the
coefficient m′ corresponding to locations of the spherical symmetry
centre O′ and ray perigee T′. In accordance with the locality
principle, the rapid amplitude and phase variations of the radio waves
registered at point L may be considered as connected with influence of a
small neighbourhood of the ray perigee T′ corresponding to the local
spherical symmetry centre O′.
Left plot – Comparison of the refractive attenuations Xa(h) and Xp(h) (groups of curves I–IY). Each group consists of four
curves. In each group thick and thin rough curves (marked by indices “p”
and “a”) describe the experimental vertical profiles of Xp(h) and Xa(h), respectively. Smooth curves in each group describe
polynomial approximations of the altitudes dependences of Xp(h) and
Xa(h) and are also highlighted by indices “p” and “a”, respectively.
For convenience groups of curves II–IY are displaced by 0.6; 1.2; 1.8 units.
Right – The total absorption Γ corresponding to the refractive
attenuations Xa(h), Xp(h) measured at frequency F1 has been
calculated from the smooth curves I–IY (marked by indices a,p in left panel).
Curves II, III, and IY are shifted for comparison by 1, 2, and 3 db,
respectively.
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Left plot – Comparison of the polynomial approximations of
refractive attenuations Xa(h), Xp(h) (curves 1–4, indexes “p” and
“a”, respectively). Curves 2–4 are displaced for convenience by 0.6; 0.4;
0.2, respectively. Right – The total absorption Γ corresponding to the
refractive attenuations Xa(h), Xp(h) has been calculated from the
curves 1–4 (left panel). Curves 2, 3, and 4 are shifted for convenience by
2, 4, and 6 db, respectively.
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Consequences of the locality principle
Next important findings that follow the locality principle are addressed
below.
Possibility of determination of the total absorption
If location of the symmetry centre is known (for example, when the point
O′ coincides with the Earth's centre O), the total absorption Γ
in the atmosphere (ionosphere) can be defined by eliminating from value
Xa(t) the refractive attenuation Xp(h) found from eikonal
variations.
Γ=10lgXa(t)Xp(t)
Some results of determination of the refractive attenuations Xa(h) and Xp(h), and total absorption Γ are considered
below. Figure 2 (left and right plots) shows two vertical profiles of the
refractive attenuations Xa(h) and Xa(h) highlighted
by indices “a” and “p” (rough curves), respectively, and their polynomial
approximation (smooth curves) measured at the first GPS frequency F1. The
measurements were provided in 2008 above two regions located in central
Africa (RO events 18 October 17:52 UT (left), and 11 April 03:14 UT
(right), with geographical coordinates 5.0∘ N 332.5∘ W, and
6.7∘ N 328.1∘ W, respectively) using the FORMOSAT-3
satellites. The altitude dependences of Xa(h) and Xp(h) and their polynomial approximations are described by corresponding pairs
of rough and smooth curves indicated by indices “a” and “p”,
respectively. Values of the altitudes of the line of sight GDLH and the
height of the ray perigee h are plotted on the horizontal axis. The
Xa(h) and Xp(h) profiles and their polynomial
approximations are almost coincident at the heights between 12 and 40 km,
and significantly different below 8–9 km (Fig. 2, right and left plots).
The correlation between variations of Xa(h) and Xp(h) gradually decreases with height H and magnitude of Xa(h)
is obviously well below the corresponding values of Xp(h) in the
5–9 km altitude interval h. This indicates a possible influence of total
absorption of the radio waves in the atmosphere.
Other results obtained from the RO experiments carried out during four
events on 5 June 2008, are shown in Fig. 3 (left part, groups of curves
I-IY) and correspond to measurements of the refractive attenuations Xp(h) and Xa(h) at GPS frequency F1. These RO experiments were conducted
in the following areas: Norwegian Sea (I), the eastern Siberian Plateau (II), and
Alaska (III, IY). The time and geographical coordinates of these RO events
are the following: 16:18 UT; 70.8∘ N 1.7∘ W (curves I); 13:28 UT; 60.6∘ N 252.5∘ W
(curves II); 02:42 UT; 64.9∘ N 139.7∘ W (curves III); and 10:44 UT;
60.0∘ N 155.8∘ W (curves IY). A weak but perceptible absorption in
1–2 db interval at frequency F1 is observed below 8 km altitude. However, the
altitude dependence of absorption Γ is different in the considered
regions. Two cases revealed significant decrease of absorption Γ in
the 5–6 km (IY) and 7–8 km (II) height interval. In the remaining areas
the total absorption changes from 1.5 db (I) up to 3.5 db (III) below 7 km
altitude h.
Polynomial approximations of the refractive attenuations Xa(h) (measured at frequency F1); Xp1(h),Xp2(h)
(evaluated from the eikonal data at frequencies F1, F2); and Xp0(h) (estimated from the combined eikonal
F0=(F12Φ1-F22Φ2)/(F12-F22))
are shown in Fig. 4.
These results are relevant to four RO events and correspond to four equatorial regions located in central Africa. Figure 4 (left plot) shows vertical profiles of refractive attenuations Xa(h), Xp1(h) found from the FORMOSAT-3 RO data corresponding to four
events carried out on 11 April 2008 03:14 UT, 6.7∘ N 328.1∘ W; 29 May 21:41 UT 3.1∘ N
329.5∘ W; 11 October 00:56 UT, 5.7∘ N 333.4∘ W; and 18 October
17:52 UT, 5.0∘ N 332.5∘ W; (curves 1–4, respectively). Values of the
altitudes h of the ray perigee T and the height H of the straight line
GDL relative to the Earth's surface are marked on the horizontal axis.
The Xa(h) and Xp(h) profiles are marked by indices “a” and “p”,
respectively. All three curves Xp(h)=Xp1(h),Xp2(h),Xp0(h) are coincident in Fig. 4 (left plot). However, some distinction is seen
in the right part of Fig. 4, where the altitude dependence of the total
absorption is shown. This may be connected with influence of the ionosphere.
Measured values of the total absorption coincide with a mean value of
Γ equal to 0.0096 ± 0.0024 db km-1 and correspond to the
MIR–geostationary satellites RO data at the 32 cm wavelength obtained in
monochromatic regime (Pavelyev et al., 1996).
Determination of tilt, height, and displacement of inclined layers
If centre of symmetry does not coincide with the expected location – point
O, principle of locality states:
Xa(t)≡Xp′(t);1-Xa(t)=m′a=m′m1-Xp(t),
where magnitudes of the refractive attenuation Xp(t) and coefficient
m correspond to position of the point O. Using relationship (Eq. 1)
connecting coefficient m′ with the distances d1′,d2′, and impact parameter ps′, one can obtain
m′m-1=d2′d1′(dps/dt)2d2d1(dps′/dt)2-1;d1+d2=d1′+d2′=R0.
If the displacement of spherical symmetry centre satisfies the following
conditions:
dpsdt≈dps′dt;d2R0≪1;d2′R0≪1,
one can find from Eqs. (6), (7):
Xp(t)-Xa(t)=d2′-d2d21-Xp(t)=dd21-Xp(t),
where d is the distance DD′ (see Fig. 1). It follows from Eqs. (5),
(8)
that amplitudes Aa,Ap of variations of the refractive
attenuations – 1-Xa(h), 1-Xp(h) and distance d are connected
by the following equations:
m′=Aa(t)Ap(t)m;d=Aa(t)Ap(t)-1d2.
The coefficients m,m′ are slowly changing as functions of
time. Therefore the coefficient m′ and the layer's displacement d can be
estimated in the time instant when the amplitude Ap(t) achieves
maximal magnitude. This allows finding, if absorption is absent, the
displacement d of the tangential point T′ with respect to the ray
perigee T (Fig. 1) as well as the layer's height h′ and inclination
δ from following equations:
d=d2′-d2=d2α-1;α=AaAp;d2=RL2-ps2,h′=h+Δh,Δh=dδ2,δ=dre,re=TO.
The amplitudes Aa,Ap of variations of the refractive
attenuations 1-Xa(h) and 1-Xp(h), can be evaluated, for example,
using the Hilbert numerical transform. The amplitude Ap(t) of
refractive attenuation Xp(t) is evaluated using the coefficient m
corresponding to the centre of Earth (or another planet) O. Depending on the sign
of the difference Aa-Ap, the value of d is positive (or negative),
and point T′ is located on the GT or TL lines. Note that
relationship (Eq. 9) is fulfilled if one of the satellites is much farther
away from the centre of symmetry than the other. This condition is usually
satisfied during the Earth or planetary RO missions (Fjeldbo, 1964;
Yakovlev, 2002).
(a) Comparison of the refractive attenuations Xa(h), Xp(h) found from the RO intensity and eikonal data at GPS
frequency F1 (curves 1 and 2, respectively). (b) The amplitudes Aa,Ap of analytical signals corresponding to the variations of
the refractive attenuations 1-Xa(h) and 1-Xp(h) (curves 1 and 2),
respectively. (c) Location of the first layer using amplitudes Aa,Ap. (d) Vertical profiles of the gradients of electron
density in the layers.
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The spherical symmetry of a medium with new centre O′ justifies
application of the Abel transformation for solution of the inverse problem
(Pavelyev et al., 2008a, b). The time derivative of the phase path excess
Φ(t) is used to obtain the temporal dependence of the impact
parameter p′:
p′-ps′=-m′dΦdtdpsdt=-mαdΦdtdpsdt=αp-ps.
To solve the inverse problem, the following formulas are used for the Abel
transform (Hocke, 1997; Pavelyev et al., 2012) (for simplicity, the bar
in designations of the impact parameters p′,ps′ is deleted):
N(p)=-1π∫p∞lnxp+x2p2-1dξ(x)dxdx,
where p is the impact parameter corresponding to ray GTL at the instant
of time t and N(p) is the refractivity. The vertical gradient of
refractivity dN(p)/dh can be found from Eq. (12) using a relationship:
dN(p)dh=1+N(p)1-dN(p)dprTdN(p)dp,
where rT is the distance T′O′ (Fig. 1). Derivative of the bending
angle ξ(p) on impact parameter p can be found from the RO signal
intensity or the RO phase path excess data (Pavelyev and Kucheryavenkov, 1978; Pavelyev et al., 2012, 2013),
i.e.,
X(p)=pps1-R22-p2R12-p2dξ(p)dpR0-1;dξ(p)dp=1-psp-1X-1R0R22-p2R12-p2≈1-X-1R0R12-p2R22-p2.
The last equation Eq. (14) for dξ(p)/dp is valid under
condition: p≈ps. Substitution Eq. (14) in Eq. (12) gives with
accounting for relation dpdt=X(p)dpsdt (Kalashnikov
et al., 1986):
N(p)=1π∫-∞t(p)lnxp+x2p2-1X(t)-1R0R12-x(t)2R22-x(t)2dps(t)dtdt;x(t)=p(t);-∞<t≤t(p).
From Eqs. (5), (14), and (15), one can obtain a modernized formula for the
Abel inversion, i.e.,
N(p)=-1π∫-∞t(p)lnxp+x2p2-1m′aR0R12-x(t)2R22-x(t)2dps(t)dtdt;x(t)=p(t);-∞<t≤t(p),
where m′ can be determined from the first equation Eq. (9).
When the position of the spherical symmetry centre is known (for example, a
centre of symmetry coincides with the centre of the Earth), Eqs. (15) and
(16) are new relationships for solution of the RO inverse problem. Unlike
the previous solution (Eq. 12), the Eqs. (15) and (16) do not contain the angle
of refraction, and include only temporal dependences of the refractive
attenuation X(t), eikonal acceleration, and impact parameter. Note that
Eqs. (15) and (16) provide the Abel transform in the time domain where a
layer contribution does exist. The linear part of the regular trend due to
the influence of the upper ionosphere is removed because the eikonal
acceleration a in Eq. (16) contains the second derivative of time. However,
the influence of the upper ionosphere exists because it may contribute in
the impact parameter p(t). Also, nonlinear contribution of the upper
ionosphere remains in the eikonal acceleration a. Therefore, Eq. (16)
approximately gives that part of the refractivity altitude distribution,
which is connected with the influence of a sharp plasma layer. The electron
density vertical distribution in the Earth's ionosphere Ne(h) is
connected at GPS frequencies with the refractivity N(h) via the following
relationship:
Ne(h)=-N(h)40.3f2,
where f is the carrier frequency [Hz], and Ne(h) is the electron
content [el m-3].
Examples of application of Eqs. (12)–(16) for estimation of the location,
inclination, and real height of ionospheric layers are given in Fig. 5. To
consider a possibility to locate the plasma layers, a CHAMP RO event 026
(04 July, 2003, 02:27 UT; geographic coordinates 68.5∘ N 82.8∘ W) with
strong quasi-regular intensity and phase variations is used. The refractive
attenuations of the CHAMP RO signals Xa(h), Xp(h) found from the RO
signal intensity and eikonal data are shown in Fig. 5a as functions of the
RO ray perigee altitude h. The eikonal acceleration a has been estimated
by double differentiation of a second-power least square polynomial over a
sliding time interval Δt=0.5 s. This time interval approximately
corresponds to the vertical size of Fresnel zone of about 1 km since the
vertical component of the RO radio ray was about 2.1 km s-1. The refractive
attenuation Xp(h) is derived from the evaluated magnitude a using
Eq. (1), and m value is obtained from the orbital data. The refractive
attenuation Xa(h) is derived from the RO signal intensity data by a
sliding least-square polynomial having the same power with averaging in the
time interval of 0.5 s. In the altitude ranges of 50–60 and 75–85 km, the
refractive attenuations variations Xa(h) and Xp(h) are strongly
connected and may be considered as coherent oscillations caused by sporadic
layers (Fig. 5a, curves 1 and 2). Using a Hilbert numerical transform, the
amplitudes Aa,Ap of analytical signals related to
1-Xa(h) and 1-Xp(h) have been computed and are shown in Fig. 5b,
curves 1 and 2, respectively. In the altitude range of 50–60 km, amplitudes
Aa,Ap are nearly identical, but the magnitude of Aa is about 1.7 times below than that of Ap. Accordingly, the
displacement d found from Eq. (9) is negative, and a plasma layer is
displaced from the RO ray perigee T in the direction to satellite L (see
Fig. 1). A similar form of variations of the refractive attenuations
1-Xa(h) and 1-Xp(h) allows locating the detected ionospheric
layer. Displacement d corresponding to a plasma layer recorded at the 51 km
altitude of the RO ray perigee is shown in Fig. 5c. Curves 1 and 2 in Fig. 5c correspond to amplitudes Aa and Ap, respectively. Curve 3 describes the
displacement d found using Eq. (9) from amplitudes Aa,Ap in the 50.7–51.4 km altitude interval. Changes in d are concentrated
in the altitude range of -900 to -950 km when the functions Aa,Ap vary near their maximal values of 0.75 and 1.36 in the
ranges of 0.7≤Aa≤0.75; 1.29≤Ap≤1.36, respectively.
The statistical error in the determination of ratio Aa(t)Ap(t) in Eq. (9) is minimal when Ap(t) is maximal. If the relative error
in the measurements of Ap is 5 %, according to Fig. 5c, the accuracy
in the estimation of d is about ±120 km. The inclination of a plasma
layer to a local horizontal direction δ calculated using Eq. (10) is
approximately equal to δ=8.2∘±0.2∘. The vertical gradient
dNe(h)dh of electron density distribution Ne(h) for the
given RO event is shown in Fig. 5d. Curves 1 and 2 correspond to the
vertical gradient dNe(h)dh retrieved using Eqs. (12) and (16),
respectively. Curve 3 is related to the vertical gradient dNe(h)dh retrieved using the refractive attenuation Xa(h) and formula
(Eq. 15). The real altitude of the ionospheric layers is indicated on the
horizontal axis in Fig. 5d. Two ionospheric layers are seen (curves 1, 2,
and 3 in Fig. 5d). The first layer is located on line TL at the 110 to
120 km altitudes at a distance of 900 km from point T (Fig. 5d, left). The
second layer is located near the RO perigee at the 95 to 105 km altitudes
(Fig. 5d, right). From comparison of the refractive variations Xa(h) and
Xp(h) (Fig. 5a, curves 1 and 2) and vertical gradients of the electron
content (Fig. 5d), the width of sporadic E-layers is nearly equal to the
altitude interval of intensity variations of the RO signals. From Fig. 5d,
variations of the vertical gradient of electron density are concentrated in
the interval -0.4×106 [elcm-3 km-1] < dNe(h)dh < 0.5×106 [elcm-3 km-1]. These magnitudes of Ne(h) are typical for sporadic E-layers (Kelley and Heelis, 2009). The height
interval of the RO signal intensity variations is nearly equal to the height
interval of variations in the electron density and its gradient. It follows
that the standard definition of perigee of radio ray in the RO method as a
minimal value of the distance of the ray path to the surface leads to an
underestimation (bias) of layers altitude in the atmosphere (ionosphere) of
Earth (or another planet). This error is zero for horizontal layers and increases
with their inclination.
Left, top – Comparison of the refractive attenuations Xa(h), Xp(h) found from the RO intensity and eikonal data at GPS
frequency F1 (curves XA and XP, respectively, displaced for
comparison). Dotted curves show the polynomial approximations of the
refractive attenuations Xa(h), Xp(h). Left, bottom – Contribution
of the atmospheric layers (coherent component of the RO signal) and small
scale irregularities (incoherent component) (curves 1 and 2, respectively).
Right – Spatial spectra of the coherent and incoherent components of the RO
signal (top and bottom panels, respectively).
![]()
Separation of layers and small-scale irregularities contributions
in the RO signal
The principle of locality allows one to separate the contributions of layers
and irregular inhomogeneities in the RO signal. According to identity (5),
the coherent and incoherent components of the RO atmospheric signal
C(h),I(h) due to the layers and irregularities influence can be estimated
as follows:
C(h)=Xa(h)+Xp(h)/2-P(h);I(h)=Xa(h)-Xp(h)/2,
where P(h) is the polynomial approximation describing main atmospheric
contribution in the RO signal. If location of the spherical symmetry centre
is known, P(h) should be the same for the refractive attenuations Xa(h), Xp(h). Example of separation of the layers and inhomogeneities
contributions in the RO signal is presented below. Figure 6 shows altitude
dependences of the refractive attenuations Xa(h), Xp(h) for the
CHAMP RO event 7 April 2003 carried out at 02:34 UT in the area with
geographical coordinates 2.5∘ S 291.7∘ W. Vertical profiles of the refractive
attenuations Xa(h), Xp(h) reveal the coherent variations of Xa(h), Xp(h) indicating contribution of the atmospheric layers (Fig. 6,
left, top, curves XA, XP are displaced for comparison). Dotted
curves in Fig. 6, left (displaced for comparison), describe polynomial
approximations P(h) of the slow changing altitude dependences of Xa(h), Xp(h). The coherent and incoherent parts of the RO signal
C(h),I(h) are obtained using formulas (Eq. 18) and are shown in the Fig. 6,
left, bottom plot (curves 1 and 2, respectively). Intensity of the coherent
component C(h) by an order of magnitude prevails the corresponding value
of I(h), thus, indicating importance of the layers contribution in the RO
signal. Usually this contribution may be assigned to effects of small-scale
irregularities (Cornman et al., 2012). An analysis of spatial spectra of
coherent and incoherent components is presented in the Fig. 6, right, top
and bottom plots, respectively. The forms of spectra C(h),I(h) are similar
in the interval of the vertical periods greater than 1 km. In the interval
below 1 km, the power degrees of spectra inclination are different and equal
to 3.7 ± 0.2 and 2.1 ± 0.2 for components C(h) and I(h),
respectively. This indicates a different origin of the
components C(h) and I(h). The inclination of the I(h) spectrum is nearly
8/3 which corresponds to contribution of the turbulent irregularities in
variations of the RO signal intensity (Gurvich and Yakushkin, 2004). The
value 3.7 corresponding to the coherent component C(h) is near the
inclination of spatial spectrum of anisotropic internal gravity waves
in accordance with the theory developed by Gurvich and Chunchuzov (2008).
Parameters of coherent and incoherent components.
Time (UT)
Location
σA
σP
σc
σin
hb/hu, km
rc
2003 04 17 21:38
58.9∘ N 117.5∘ E
0.084
0.062
0.071
0.018
12/30
0.88
2003 04 17 22:05
45.8∘ S 146.9∘ E
0.07
0.058
0.062
0.015
14/25
0.88
2003 04 17 23:13
48.7∘ N 124.4∘ E
0.11
0.097
0.104
0.022
12/27
0.91
2003 04 26 00:06
55.8∘ N 74.2∘ E
0.065
0.052
0.056
0.016
11/26
0.85
2003 04 26 00:13
31.1∘ N 74.6∘ E
0.051
0.036
0.042
0.011
18/32
0.86
2003 04 26 01:21
51.8∘ N 125.2∘ W
0.096
0.076
0.085
0.015
12/26
0.94
2003 04 26 01:23
58.1∘ N 92.8∘ W
0.055
0.039
0.046
0.011
13/25
0.88
2003 04 26 04:42
65.3∘ N 35.3∘ E
0.056
0.043
0.048
0.014
10/22
0.85
2003 05 03 00:34
0.4∘ S 63.0∘ E
0.11
0.086
0.096
0.027
17.5/26
0.86
2003 05 03 01:27
22.2∘ N 121.6∘ W
0.14
0.11
0.123
0.023
12/27
0.94
2003 05 03 03:35
12.6∘ N 38.6∘ E
0.079
0.065
0.071
0.012
20/30
0.95
2003 05 03 04:56
59.4∘ N 17.0∘ E
0.05
0.039
0.043
0.011
12.5/26.5
0.87
2003 04 12 07:48
45.6∘ S 7.6∘ E
0.066
0.047
0.056
0.012
14/25
0.915
2003 04 12 08:25
7.4∘ N 153.9∘ E
0.103
0.076
0.088
0.013
18/30
0.96
2003 04 12 08:53
57.5∘ N 42.2∘ W
0.06
0.052
0.055
0.011
12/24
0.918
2003 04 12 09:37
70.8∘ S 165.0∘ E
0.069
0.054
0.06
0.015
12/22
0.88
2003 04 1210:08
52.2∘ N 150.9∘ E
0.085
0.069
0.075
0.016
12.5/27
0.92
Correlation of index S4(I) measured from the intensity
variations of the GPS RO signal at frequency F1 and parameters S4(F1) and
S4(F2) found from the eikonal variations at GPS frequencies F1 and F2.
![]()
Correlation of indices S4(I) and [S4(F1)+S4(I)]/2.
![]()
Parameters of coherent and incoherent components introduced in the Table 1
illustrate a possibility to separate the contributions of atmospheric layers
and turbulent structures in the RO signal. The time and geographic
coordinates are shown in the first two columns of Table 1. The rms
deviations σA,σP of the refractive
attenuations Xa(h), Xp(h) and dispersion of the components
C(h) and I(h)σc,σin are presented in the
next four columns. The altitude interval hb/hu of measurements of
σA,σP; σc,σin; and correlation coefficient rc between the refractive attenuations
variations XA(h), XP(h) are indicated in the last two columns of
Table 1. As a rule the rms deviation σc of the coherent component
C(h) is greater than that one of incoherent part I(h)σin by a
factor of 4–5. This indicates on the prevailing contribution of the
atmospheric layers as compared with influence of the irregularities in the
RO signal. High level of correlation rc (in the interval 0.84–0.96)
between the variations of the refractive attenuations Xa(h), Xp(h)
indicates the practical and theoretical importance of the locality principle
for separation of the atmospheric layers and irregularities in the RO signal
at a single frequency.
Conclusions
The principle of locality is a key regularity that extends applicable domain
of the RO method, widens possibilities and opens new directions of the RO
geophysical applications to remote radio sensing the atmosphere and
ionosphere of the Earth and other planets. These directions include: (i)
innovative estimating the altitude dependence of the total absorption of
radio waves using the RO amplitude and phase variations at a single
frequency; (ii) evaluation of the slope, altitude, and horizontal
displacement of the atmospheric and ionospheric layers from the RO signal
intensity and phase data using the eikonal acceleration/intensity technique;
(iii) separation of layers and irregularities contributions in the RO
signal, determination of vertical profiles of the turbulent and small-scale
structures by joint analysis of the RO signal eikonal and intensity
variations; and (iv) introduction of the new combined phase-intensity index
for the RO study of multilayered structures and wave processes. This
regularity is valid for every RO ray trajectory in geometrical optics
approximation including reflections from the surface.
As follow from Sect. 3.1 the total absorption is stronger in the equatorial
region than at high latitudes, pointing to the role of water vapour and,
possible, the clouds of liquid water, ice and snow. The contribution of the
clouds water (fog, rain, and hydrometeors) in the RO signal should be
analyzed separately in future investigations. First of all, the theory of
radio waves propagation should be reconsidered for the case when the radio
waves are propagating along the clouds under different temperature
conditions.
Mass-scale measurements of the total absorption at the altitude below 15 km
depend on the quality of the GPS receivers onboard of the RO missions. The
total absorption measurements are possible only in the case when the low and
high frequency noise are small enough for coinciding of the polynomial approximation of the RO intensity and phase data at the altitudes between 15–60 km. Also the stability of the RO signal phase data and accuracy of the
total absorption measurements are determined by precision of the open-loop
regime of the GPS RO receivers below 8 km altitude. Analysis of these
technological aspects of the RO measurements is the task of future works.
It follows from Sect. 3.2 that RO definition of the vertical location of
layers as coinciding with the altitude of the radio ray perigee can lead to
an underestimation of their height in the atmosphere (ionosphere) of Earth
and other planets. This systematic bias is zero for horizontal layers and strongly
increases with their inclination in the range 1–10∘ from 1 km up to
about of several dozen kilometres. The measured inclination of layers can be
applied for estimating the orientation of wave fronts in the ionosphere
(or atmosphere) and may be used for determination of the important
parameters of internal waves including the internal frequency, direction and
magnitude of kinetic momentum. A new method of estimating the electron
density distribution in plasma layers (described in Sect. 3.2) should be a
subject of future comparison with the ionosondes and tomographic data.
Mass-scale measurements of coherent and incoherent component of the RO
signal (Sect. 3.3) and introduced (Sect. 4) combined phase-intensity
ionospheric index are important for investigation of the temporal, seasonal
and regional evolution of the layered and turbulent structures at different
altitudes in the ionosphere and atmosphere with a global coverage and can be
provided in near future with usage of extended volume of the RO data
obtained during 20 years (1995–2015) of experimental researches.