A new formulation of the previously introduced principle of locality is presented. The principle can be applied for modernization of the radio occultation (RO) remote sensing of the atmospheres and ionospheres of the Earth and other planets. The principle states that significant contributions to variations of the intensity and phase of the radio waves passing through a layered medium are connected with influence of the vicinities of tangential points where the refractivity gradient is perpendicular to the radio ray trajectory. The RO method assumes spherical symmetry of the investigated medium. In this case, if location of a tangent point relative to the spherical symmetry centre is known, the time derivatives of the RO signal phase and Doppler frequency variations can be recalculated into the refractive attenuation. Several important findings are consequences of the locality principle: (i) if position of the centre of symmetry is known, the total absorption along the ray path can be determined at a single frequency; (ii) in the case of low absorption the height, displacement from the radio ray perigee, and tilt of the inclined ionospheric (atmospheric) layers can be evaluated; (iii) the contributions of the layered and irregular structures in the RO signal can be separated and parameters of layers and turbulence can be measured at a single frequency using joint analysis of the intensity and phase variations. Specially for the Earth's troposphere, the altitude distributions of the weak total absorption (about of 1–4 db) of the radio waves at GPS frequencies corresponding to possible influence of the oxygen, water vapour, and hydrometeors can be measured with accuracy of about 0.1 db at a single frequency. In accordance with the locality principle, a new index of ionospheric activity is introduced. This index is measured from the phase variations of radio waves passing through the ionosphere. Its high correlation with the S4 scintillation index is established. This correlation indicates the significant influence of locally spherical symmetric ionospheric layers on variations of the phase and intensity of the RO signal passing through transionospheric communication links. Obtained results expand applicable domain of the RO method as a powerful remote sensing technique for geophysical and meteorological research.

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

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.

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

Scheme of radio occultation measurements.

Comparison of the refractive attenuations

When absorption is absent, the refractive attenuation

Left plot – Comparison of the refractive attenuations

Left plot – Comparison of the polynomial approximations of
refractive attenuations

Next important findings that follow the locality principle are addressed below.

If location of the symmetry centre is known (for example, when the point

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

Polynomial approximations of the refractive attenuations

If centre of symmetry does not coincide with the expected location – point

The spherical symmetry of a medium with new centre

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

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

Left, top – Comparison of the refractive attenuations

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

Parameters of coherent and incoherent components.

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)

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

According to the locality principle, a global correlation
between the phase and intensity variations of the RO signal can exist. Index S4 (I),
as measured from intensity variations, should be correlated with index S4
(F), defined by the second derivative of time of the eikonal of the RO
signal at GPS frequencies F1 and F2. According to the principle of locality
in the case of spherical symmetric medium, the following connections can
exist:

Figures 7 and 8 show the results of correlation of index S4(I), defined by
the variations of the intensity I at the frequency F1 with indices S4(F1),
S4(F2) measured from the second derivative of the phase paths excess at
frequencies F1, F2 during FORMOSAT-3 RO events held in January and February
2012. Circles in Figs. 7 and 8 correspond to the experimental values of
index S4 (I) (vertical axis) and S4 (F1), S4(F2) (horizontal axis), respectively. The
solid curves in Figs. 7 and 8 are regression lines and have been found by the
least squares method. The correlation coefficient of index S4(I) to S4(F1)
and S4(F2) varies in the intervals 0.69 to 0.78 and 0.70–0.75,
respectively. The correlation coefficient of index S4(I) with combined index
[S4(F1)

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

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.

We are grateful to Taiwan Centre for Space and Remote Sensing Research for access to the FORMOSAT-3 RO data. This work was supported in part by program no. 9 of the Presidium of the Russian Academy of Sciences (RAS), program IV.13 of the Physical Sciences Division RAS, and grant No. 13-02-00526-a from Russian Foundation of Basic Research. We are grateful to R. R. Salimzyanov for help in preparing the manuscript. We send special thanks to the referees for their fruitful remarks and suggestions which helped us to improve our paper. Edited by: J. Y. Liu