Coefficients in equation (1)

## 1. Introduction

The aim of this book chapter is to reconsider the fundamental principle of radio-occultation (RO) remote sensing and to find new applications of RO method.

The RO remote sensing can be performed with any two cooperating satellites located on opposite sides with respect to the Earth’s limb and moving to radio shadow. Several RO missions are working now aboard the Low Earth Orbit satellites. These missions provide global monitoring of the atmosphere and ionosphere of the Earth at different altitudes with high spatial resolution and accuracy. Their data are very important for meteorology, weather prediction. The RO data can be used to detect the climate changes, connections between the ionospheric, atmospheric processes, and solar activity, and to estimate conditions for radio navigation and radio location.

Up to now the RO inverse problem solution was based on the assumptions that the atmosphere and ionosphere are spherically symmetric and that the influence of turbulent and irregular structures on the retrieved vertical profiles of refractive index is insignificant [1,2]. The vertical profiles of refractive index are usually determined by measurement of the Doppler shift of radio wave frequency [1–3]. Information contained in the amplitude part of the radio-holograms was almost not addressed earlier, and this fact impeded separation of the contributions from layers and turbulent (small-scale) structures.

A new important relationship between the second-order time derivative (acceleration) of the phase path (eikonal), Doppler frequency, and intensity variations of the radio occultation (RO) signal was revealed by theoretical considerations and experimental analysis of the radio-holograms recorded onboard of the CHAMP and FORMOSAT-3 satellites [4–6]. Using the detected relationship, a possibility of determining the altitude, position, and inclination of plasma layers in the ionosphere from the RO data has been revealed [4–8]. The proposed calculation technique is simpler than the phase-screen [9] and back-propagation methods [10, 11]. The detected relationship is also important for estimation of the total attenuation of radio waves in a satellite communication link by combining the analysis of information contained in the amplitude and phase channels of the radio-holograms. The mentioned relationship makes it possible to convert the eikonal acceleration (or the time derivative of the Doppler frequency shift) into the refractive attenuation.

The total absorption of radio waves in the decimeter wavelength range at a frequency of 930 MHz was earlier determined experimentally [12, 13] in the “MIR” orbital station–geostationary satellites communication link. In those papers, attenuation was removed from the amplitude data with the use of the time dependence of the derivatives of the phase and Doppler frequency shifts. Measurements of the total absorption for determining the water content in the stratosphere and troposphere will be performed in the future radio-occultation missions [14] at three frequencies near the water-vapor absorption line at the wavelength of 1.35 cm. For analysis and processing of these measurement data, a technique [15, 16] using the integral Fourier operators (Canonical Transform (CT) and Full Spectrum Inversion Fourier analysis (FSI)) is proposed. In [16, 17] a radio-holographic technique of the total absorption measurements has been previously proposed. Following this technique, the refractive attenuation effect on the amplitude of the field transformed by an integral Fourier operator is ruled out by using the relationship between the refractive attenuation and the second-order time derivative of the phase difference of the recorded and reference signals.

Unlike the methods used in [15–17], the eikonal acceleration/intensity technique does not use any integral transform and can be directly employed for determining the total absorption of radio waves in the case of significant refractive attenuation under the condition of single path propagation. Moreover, the combined analysis of the eikonal acceleration and radio wave intensity makes radio vision of the atmospheric and ionospheric layers possible, i.e., allows the layers to be detected by observing correlated variations in the eikonal acceleration and intensity against the background of an uncorrelated contribution of turbulent inhomogeneities and small-scale structures and permits one to measure the layer parameters.

The book chapter is organized as follows. In section 2 the basic rules are given for describing radio waves propagation in a spherically symmetric medium including a new relationship for the refractive attenuation. In section 3 an advanced eikonal /intensity technique is introduced and applied to find the total absorption from analysis of the RO data. In section 4 a locality principle and its applications to determining the location, slope, and height of plasma layers in the ionosphere are described. Comparison with the back-propagation radio-holographic method is carried out. In section 5 the seasonal changes of the bending angle during four years of observations in Moscow and Kamchatka areas are described. Conclusions and references are given in section 6 and section 7, respectively.

## 2. Basic rules for radio waves propagation in a spherically symmetric medium

To obtain basic relationships describing the radio wave propagation in a spherically symmetric medium it is necessary to use a formula [18] for electromagnetic field

where

where

where

where

Two terms in the curly brackets of equation (5) formally differ by a factor – the imaginary unit

Under the geometric optics assumptions [18,19]:

the next equations are valid [18,19]:

where * ψ*and gradient of amplitude

It follows from the relationships (9) that the ray equation has a form [18,19]:

where

where

One can obtain a relationship for the refractive attenuation of radio wave by multiplying equation (10) by

It follows from (13)

According to the Gauss theorem the next relationship is valid along a ray tube:

where

The relationships (11), (12), and (15) present basic rules describing the ray direction and power conservation laws in the spherically symmetric medium. From (15) one can obtain important formula for the refractive attenuation when the transmitter and receiver are located in a medium with arbitrary values of the refraction index.

In the case of spherical symmetric medium one can consider according to [18,20] a ray tube having at point

where

where

The refractive attenuation

where

where

The next relationships connect central angle

Formula (23) is valid when the tangent point on the ray trajectory, where the ray is perpendicular to the gradient of refractivity, is absent [20]. Equations (22)-(24) allow transforming the formula (21):

where

Constant

Thus the refractive attenuation can be evaluated from (25) and (26) as:

The refractive attenuation

Previously refractive attenuation

where

Equations (28) for the refractive attenuation generalize the relationship (29) for the case when the transmitter and receiver are located in a spherically symmetric inhomogeneous medium. This relationship can be appropriate for RO data analysis during experiments provided in the planetary and Earth’s atmospheres and ionospheres.

## 3. Total absorption

A new important relationship between the second-order time derivative (acceleration) of the phase path (eikonal), Doppler frequency, and intensity variations of the radio occultation (RO) signal has been established by theoretical considerations and experimental analysis of the radio-holograms recorded onboard of the CHAMP and FORMOSAT-3 satellites [4, 6-8]. The detected relationship makes it possible to convert the eikonal acceleration (or the time derivative of the Doppler frequency shift) measured using the RO phase data into the refractive attenuation and then exclude it from the RO amplitude data to obtain the total absorption. The method of measuring of the total absorption from joint analysis of the RO amplitude and phase variations is described below.

Layout of a RO experiment in the transionospheric link using the high-stability, synchronized by atomic-clock, radio signals of GPS navigation system is shown in Figure 2. Point

and the eikonal increments

where

Equations (30) relate the refractive attenuation

The attenuation of intensity of radio waves

The experimental quantity * O*(Figure 2) in the

*plane. The quantity*GOL

Eqs. (30) - (32) permit one to estimate the total absorption of radio waves

The results of estimation of the total absorption * 1*and

*, respectively). Smooth curves*2

*on the left in Figure 3 show the approximation obtained by a least-squares method. Slow trends in the refractive attenuations*3

*7% in the experiment, and such a value corresponds to moderately disturbed conditions in the ionosphere. Relationships between the refractive attenuations retrieved from the amplitude and phase variations are important for estimation of the altitude dependence of the total absorption in the atmosphere. This dependence is shown by curve*.

*in Figure 3 (right panels). Smooth curve*1

*corresponds to the total absorption*2

Obtaining more exact information on total absorption requires averaging of significant variations in the experimental values of * 1*–

*correspond to the resulting vertical profiles*5

*–*6

*, Figure 4, middle panel, are related to the dependences*10

*–*1

*,*4

*were displaced for comparison along the vertical axis. All sessions (No. 122, 02:27 LT, 77.6 N 141.0 W; No. 173, 17:35 LT, 80.9 N 337.1 W; No. 0030, 20:59 LT, 77.9 N 83.5 W; No. 0159, 21:40 LT, 83.0 N 258.6 W; and No. 0203, 16:56 LT, 76.3 N 37.9 W) correspond to the north polar regions. At altitudes between 12 and 30 km, the profiles*6, 7, 9, 10

*and*a

*, Figure 4, left panel), which begins at different heights, is observed. This effect is notable for all curves*p

*–*6

*in Figure 4 (middle panel). The differences at the initial height of splitting can be related to slow variations in the radio signal amplitude. The existence of splitting is probably a signature of the presence of small, but tangible integral atmospheric absorption, whose magnitude is, on the average, close to the values mentioned in [21] and to the magnitude 0*10

*0096*.

*0024 dB/km of absorption per unit length measured in [12, 13, 22] (curves*.

*–*6

*). The total absorption is varied within the limits 0.034 – 0.081 in the altitude range 12–5 km (Figure 4, middle panel, curves*10

*–*6

*). The total absorption is near zero at the altitudes greater 12-15 km. The estimated value of the absolute statistical and systematic errors in the total absorption*10

In Figure 4 (right panel) the averaged values of the total absorption

The introduced method is a perspective tool for investigation of seasonal and annual variations and geographical distributions of the total absorption from RO data. Also this method is, possibly, may be applied to study the influence of the tropical hurricanes and typhoons on the altitude profiles of water vapor in the stratosphere and tropopause.

## 4. Locality principle and RO remote sensing

A possibility to find the total absorption from joint analysis the RO amplitude and phase data described in section 3 is an important consequence of general locality principle valid in the of RO remote sensing of spherical symmetric atmospheres and ionospheres of the Earth and planets. Up to now this principle is implicit unformulated property of the RO method.

Below the fundamental principle of local interaction of radio waves with a spherically symmetric medium is formulated and introduced in the RO method of remote sensing of the atmosphere and ionosphere of the Earth and planets.

In accordance with this principle, the main contribution to variations of the amplitude and phase of radio waves propagating through a medium makes a neighborhood of a tangential point where gradient of the refractive index is perpendicular to the radio ray.

A necessary and sufficient condition (a criterion) is established to detect from analysis of RO data the displacement of the tangential point from the radio ray perigee.

This criterion is applied to the identification and location of layers in the atmosphere and ionosphere by use of GPS RO data. RO data from the CHAllenge Minisatellite Payload (CHAMP) are used to validate the criterion introduced when significant variations of the amplitude and phase of the RO signals are observed at RO ray perigee altitudes below 80 km.

The detected criterion opens a new avenue in terms of measuring the altitude and slope of the atmospheric and ionospheric layers. This is very important for the location determination of the wind shear and the direction of internal wave propagation in the lower ionosphere, and possibly in the atmosphere.

The new criterion provides an improved estimation of the altitude and location of the ionospheric plasma layers compared with the back-propagation radio-holographic method previously used.

### 4.1. Application of GPS RO method to study the atmosphere and ionosphere

The radio occultation (RO) method employs the highly-stable radio waves transmitted at two GPS frequencies

Therefore the spatial distributions of sporadic E layers are important for investigating the connections of natural processes in the neutral and ionized components of the ionosphere. The location and intensity of sporadic E-layers plays a critical role for the quality of radio communications in the HF frequency band. The RO measurements in the atmosphere can be affected significantly by ionospheric contributions since the RO signals propagate through two different parts of the ionosphere.

Usually the ionospheric influence in the RO measurements may be described through a relatively slow change in the excess phase without noticeable variations in the amplitude of RO signals. This effect can be effectively reduced by a number of different methods of ionospheric correction [10,52,53].

However disturbed ionosphere may significantly change not only the phase but also the amplitude of the RO signals. Strong amplitude and phase frequency dependent variations in the RO signals are often surprisingly observed within the altitudes of the RO ray perigee

The altitudes of sporadic E-layers have been evaluated as the height of the RO radio ray perigee in recent times [28,39-41]. A relationship between the eikonal (phase path) and amplitude variations in the GPS/MET RO data has been analyzed in [53] and conclusions have been made that (i) the amplitude variations in distinction to the phase of RO signal have a strong dependency on the distance from observation point to the location of an ionospheric irregularity and (ii) the location of the irregularities in the low ionosphere may be determined by measuring the distance between the observation point up to a phase screen which should be located perpendicularly to the RO ray trajectory at its perigee.

A radio-holographic back-propagation method has been suggested and applied for location of the irregularities in E- and F-layers of the ionosphere [10,11]. A relationship between the derivatives of the phase, eikonal, Doppler frequency on time and intensity of radio waves propagating through the near Earth’s space has been detected from both theoretical considerations and experimental analysis of the RO radio-holograms [4,5,31,36-38,47]. The introduced eikonal acceleration technique can de used for locating layers in the ionosphere and atmosphere.

The aim of this section is to demonstrate the possibility of identifying the contributions and measuring parameters of the inclined plasma layers by means of an analytical criterion. A test of a suggested method is provided by use of CHAMP RO data.

### 4.2. Criterion for layer locating

The scheme of RO experiments is shown in Figure 2. A navigational satellite

The global spherical symmetry of the ionosphere and atmosphere with a common centre of symmetry is the cornerstone assumption of the RO method. Under this assumption a small area centered at tangent point

The quiet ionosphere introduces regular trends in the excess phases at two GPS frequencies which can be removed by the ionospheric correction procedure [25,53]. The contributions in the phase and amplitude variations of RO signals of the intensive sporadic E-layers at the altitude interval 90-120 km is significantly greater than the impact of the F-layer turbulent structures [25]. Impact of a regular layer on the RO signal depends on position relative to the RO ray perigee. The length,

The ratio

Under spherical symmetry condition

If the vertical width

The next connection between the excess phase path (eikonal)

where _{s}* /dt*may be evaluated from the orbital data. The first formula (36) has been derived under condition [37]:

where

where

Eqn. (36) and (40) are the basis of the proposed method for determining the total absorption by measuring the time dependence of the intensity and eikonal of the RO signal at one frequency [31]. This method is much simpler than the previously used method based on estimation of the refractive attenuation on the first derivative of the bending angle on the impact parameter. When the total absorption is absent, it follows from (36) and (38), if the center of symmetry is located at point

Relationship (41) establishes equivalence of the values

However the locality principle has more general meaning. Therefore it is necessary to extend the theory of the RO method to develop an appropriate technique to find the locations of the tangent points on the RO ray. This is an aim of the last part of this section.

In some cases the centers of spherical symmetry in the two parts of the ionosphere located on the path

where

where

This allows one to formulate the locality principle for remote sensing of layered spherically symmetric medium in the absence of absorption. A certain point of the radio ray is tangential if and only if the refractive attenuations found from the second derivative of the eikonal on time and intensity variations of the radio waves passed through the medium are equal. In this case both the intensity and the second derivative of the eikonal variations are mainly influenced by a small neighbourhood of the tangential point.

The principle of locality allows one to determine the location of a tangential point and to find the altitude, slope and displacement of a layer from the radio ray perigee. According to Eqn. (36), (43) it follows:

where the refractive attenuation

If the displacement of the center of spherical symmetry satisfies the following conditions:

then one can find from (45):

where

Let us consider the refractive attenuation variations as the analytical signals in the form:

where

After substitution (48) in (44) one can obtain:

The ratio

Equation (50) establishes a rule: location of a tangent point on the ray trajectory can be fulfilled using the analytical amplitudes of the refractive attenuation variations

Note, that equation (50) is valid when the distance of one of the satellites from the ray perigee T is many times greater than the corresponding value for the second one. This condition is fulfilled for the planetary RO experiments provided by use of the communication radio link spacecraft-Earth and GPS occultations [31].

Correction to the layer height * Δh*and its inclination

*with respect to the local horizontal direction can be obtained from the displacement*δ

where

Condition of the spherical symmetry with new center

where

where

Factor

Note, that equation (52) provides the Abel’s transform in the time domain

where

### 4.3. Analysis of CHAMP experimental data

To consider a possibility to locate the plasma layers we will use a CHAMP RO event 005 (November 19, 2003, 0 h 50 m UT, 17.3 S, 197.3 W) with strong quasi-regular amplitude and phase variations. The refractive attenuations of the CHAMP RO signals

orbital data. The refractive attenuation * b*and

*denote the corresponding values*c

The vertical gradient

The second example of the identification and location of sporadic plasma layer in the lower ionosphere is shown in Figure 7 for CHAMP RO event 211 (July 04 2003, 10 h 54 m LT, 2.1 N, 145.6 W) with intensive sporadic e layers. The refractive attenuations

at the left vertical axis), the layer slope

The introduced method appears to have a considerable potential to resolve the uncertainty between the part

### 4.4. Comparison of the eikonal acceleration/intensity technique with back-propagation radio-holographic methods

The analytic technique can be compared with the radio-holographic approach for locating plasma structure in the ionosphere introduced previously [10,11]. In general the radio-holographic back-propagation may be carried out using a Green function

where * ,*connecting the observation point

*and normal*,

The Green function

where,

To obtain the optimal values of the vertical resolution and accuracy in measuring physical parameters in the atmosphere and ionosphere, usually the Green function

The Green function

where,

The complex wave field

where,

where,

The central angle

The back-propagated field

The form of the eikonal

where, the refraction index

The stationary phase method can be applied to evaluate the back-propagating field. For the stationary point, the following equation holds:

After substitution (65)-(68) into (69) one obtains:

From equation (70) the impact parameters

The derivatives

From (64)-(75) the following formula for

Under condition:

the next relationship follows from eqns (72), (76), (77):

where,

The simplest form of the Green function (58) has been used [10,11] to locate plasma layers in the E- and F- regions in the ionosphere. In this case

From condition (80) the curve

#### 4.5. Locality principle and its importance for RO remote sensing

Locality principle allowed designing new analytic technique for locating the inclined layered structures (including sporadic E_{s} layers) in the ionosphere. The location of the ionospheric layers including their altitude, displacement from the RO ray perigee and slope relative to the horizontal direction can be determined using the introduced criterion that compares the refractive attenuations found from the RO amplitude and phase data. Depending on the sign of the refractive attenuations the displacement of a plasma layer from the RO ray perigee should be positive (in the direction to a GPS satellite and vise versa). The magnitude of the displacement can be found from a ratio of the refractive attenuation’s difference to the magnitude of the refractive attenuation from the RO phase data. The altitude and slope of a plasma layer can be found from the known value of its displacement.

Therefore the standard estimation of a layer’s altitude as a height of RO ray perigee should be revised due to underestimation of the altitude of inclined plasma structures in the lower ionosphere.

The current radio-holographic back-propagation method implicitly uses the relationship between the eikonal acceleration and intensity variations of RO signals to locate irregularities in the ionosphere. The accuracy of this method depends on the form of the Green function used for the back-propagation. If the Green function corresponding to the propagation in the free space is used, then the inaccuracy of back-propagation method is proportional to the bending angle. The analytic technique is simpler and more precise than the previously published back-propagation method.

By use of the introduced criterion the RO method is capable to locate and determine the direction and magnitude of the gradient of electron density in the lower ionosphere. The gradient of the electron content indicates the direction of the different kinds of wave fronts in the ionosphere. In the particular case of the internal gravity waves (GW) the inclination of the wave vector to the vertical direction can be used to find the angular frequency and the parameters of GW.

The introduced criterion and technique extended the applicable domain of RO method to remote sense the waves in the lower ionosphere. This conclusion has a general importance for the planetary and terrestrial radio occultation experiments in a broad range of frequencies.

## 5. Bending angle: Seasonal changes

The RO method has important radio meteorological application. Previously the radio meteorological parameters (refractive angle, refractive attenuation, phase path excess, total absorption, and other) have been recalculated from the temperature, humidity and pressure delivered from the current meteorological observations. Nowadays the RO method directly measured the bending angle, refractive attenuation, phase path excess, total absorption, etc.) from the amplitude and phase delay of RO signal. Thus the RO radio meteorological observation are very important for estimation of condition for radio wave propagation, radio navigation, and radio climate in the near Earth space.

In this section the seasonal change of the bending angles as an important radio meteorological parameter will be considered.

Atmospheric refraction caused by gradients of the refractive index of air leads to a deviation of the direction of radio wave propagation from straight line connecting transmitter and receiver. Practical problems require to study variations of the bending angle, refractive attenuation and other radio parameters as functions of the coordinates of transmitter and receiver. When the altitude of a radio link is low, changes in the vertical profiles of temperature, pressure and humidity introduce main contribution in the refraction effects. Meteorological parameters depend on the climate and weather in different geographical positions, which was the cause of origin of radio meteorology – a branch of radio science which used the weather information for analysis of the electromagnetic waves propagation conditions in radio communication and radar applications [38,57,58]. The vertical and horizontal distributions of the pressure, temperature, and humidity found from meteorological measurements are approximated by use of different models to find the altitude and spatial dependences of the refractive index, bending angle, refractive attenuation and absorption of radio waves. However the meteorological measurements are local, and relevant parameters are variable, which inevitably leads to discrepancy between the measured and calculated values of the bending angles.

The innovative RO method is a new important tool for direct measurements of the radio meteorological parameters and for investigation of radio climate of the Earth at different altitudes in the atmosphere with a global coverage. In contrast to previously used goniometric methods with a narrow antenna pattern or interferometers for measuring refraction effects and their variations in radio links, the RO method directly determines with high accuracy the bending angle from measurements of the Doppler frequency of radio wave. The measured bending angle does not depend on the wavelength, orbits of satellites, and characteristics of the transmitting and receiving devices. The measured bending angle is delivered with high accuracy without any assumptions concerning the structure of the atmosphere, and can be regarded as an independent quantitative radio meteorological parameter in different regions of the Earth. It is essential that the spatial and temporal distributions of refractive properties can be obtained over a long period of time, which will contain daily, seasonal, and long-term radio climatic changes in the atmosphere. This information can be applied for detailed analysis of radio wave propagation conditions along the Earth’s surface.

The aim of this section is to establish the applicability of the bending angle as an indicator of the global state of the atmosphere. The annual and seasonal variations of the refractive parameters above Russia and some territories are analyzed and discussed.

### 5.1. Method of measurement

In determining the angle of refraction by the radio occultation method, the measured parameters of coherent radio waves with the frequencies

In analyzing the space–time variations of the bending angle, the results of 4252 occultation atmospheric soundings performed from June 2006 to July 2010 in the region of European Russia with coordinates of 50°N to 60°N and 30°E to 40°E were used. The extent of this region is 1100 km along the meridian and ~600 km along the latitude circle. In this region, three to seven measurement sessions were conducted every day.

The method for determining the bending angle

The sources of systematic error are related to the effect of the ionosphere and multipath propagation. The influence of the ionosphere is eliminated using two frequency ionospheric correction [53]. However, the ionospheric correction cannot completely remove the bending angle fluctuations caused by small scale electron concentration irregularities. In sensing the upper stratosphere, the bending angle errors caused by this factor can be as high as

The refractive angle is determined from the measurements of the signal frequency, a quantity that can be measured with a maximum accuracy. The results of the analysis made in [61] show that, in the middle latitude atmosphere, at heights of 5 to 30 km, the discrepancy between the measured and calculated (with the use of various models of the atmosphere) bending angles does not exceed ±1%. The height profile of the bending angle may contain inaccuracies related to the errors in height measurements. At the initial stage of data processing, the dependence of the refractive angle

### 5.2. Mean bending angle vertical profile

In analyzing space–time refraction variations, one should eliminate the influence of the regular component. To this end, a model of the bending angle vertical profile

If the bending angle is expressed in milliradians and the height is expressed in kilometers, the coefficients in the exponent (81) have the values given in Table 1.

^{-1} | ^{-2} | ^{-3} | ||

3.226 3.611 | –0.154 –0.166 | 3.765 × 10^{–3}4.128 × 10 ^{–4} | –1.487 × 10^{–3}–6.374 × 10 ^{–6} |

The height profiles of the refractive angles measured in the two regions and calculated with model (1) are compared in Table 2. Root mean square values of

0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 25 30 | 23.96 23.57 23.03 22.50 21.95 19.94 16.16 13.99 12.26 10.88 9.76 8.82 7.99 7.17 5.36 3.83 2.81 2.07 1.51 0.69 0.31 | 2.19 2.25 2.23 2.19 2.15 1.91 1.43 0.98 0.64 0.45 0.38 0.35 0.32 0.34 0.42 0.21 0.15 0.11 0.08 0.04 0.02 | 24.41 23.68 22.98 22.30 21.65 18.75 16.32 14.28 12.25 11.06 9.76 8.63 7.64 6.75 5.25 3.87 2.82 2.06 1.50 0.68 0.31 | –0.45 -0.11 0.06 0.20 0.29 0.19 -0.16 -0.29 -0.29 -0.18 -0.01 0.18 0.35 0.41 0.11 -0.04 -0.01 0.01 0.01 0.002 -0.003 |

Of interest is the distribution of these variations. The refractive angle disturbances at the corresponding heights are distributed, with respect to the mean value of

### 5.3. Seasonal and diurnal bending angle variations

The bending angle rapidly decreases from ~24 mrad to ~0.3 mrad as the ray perigee height

As an illustration of slow disturbances in the height profile of the bending angle, let us consider the vertical profiles of the deviations of the refractive angles, observed in individual measurement sessions, from the values calculated with model (81), i.e.,

Shown in Figure 8 are four profiles

Curves in Figure 4 | Date January 2007 | UTC, h, min | N, deg | E, degr. | Curves in Figure 4 | Date January 2007 | UTC, h, min | N, deg | E, deg |

1 2 3 4 | 01 01 06 10 | 04:34 04:44 05:10 04:05 | 57.34 53.99 54.03 58.03 | 39.9 39.31 34.66 33.55 | 5 6 7 8 | 10 20 19 20 | 04:17 16:19 06:26 06:41 | 58.29 56.51 53.41 57.54 | 41.71 42.56 35.62 41.76 |

In addition to slow (seasonal and diurnal) variations in the refractive angle, significant rapid fluctuations caused by atmospheric irregularities are observed. Using the results of 200 measurement sessions, the fluctuations of the bending angle observed in winter and in summer without separating them in time of day were analyzed. It is necessary to eliminate the regular and slow variations in order to estimate the rapid fluctuations. To this end, a filtering procedure was used that involves subtraction of function

Changes in the conditions of refraction in regions with different climates and on different time scales will have both individual and general laws. To detect these patterns let us look at the changes of refraction in a homogeneous area of the climatic conditions at different time intervals and analyze the seasonal and annual changes. To reduce the influence of spatial factors let us to limit the area in which the seasonal variations in the angle of refraction are investigated, and to select the cell size of approximately 400x400 km^{2}, extending in latitude from 54.0 N to 58.0 N and longitude from 35.0 E to 41.0 E. The center of this area is located in the vicinity of Moscow. In the period from January 1, 2007 and November 30, 2009 in this area was carried out 1232 RO soundings of the atmosphere. In each month of the year from 25 to 29 soundings at different times of day were held. Let us consider the seasonal changes of the bending angle at two altitudes: in the middle stratosphere at 15 km, and in the upper troposphere at 9 km (Figure 8). Note that the most accurate radio occultation measurements in the stratosphere and upper troposphere have been provided at these altitudes. At the altitude 17 km in the middle stratosphere, there is a positive trend, i.e. strengthening of refraction with time. This increase is nearly equal to 0.07. mrad during four years. In contrast to the stratospheric region in the upper troposphere at an altitude of 9 km in the period under review there was a negative trend in refraction, whose value is amounted to 0.11 mrad. When reducing the height, this trend is weakening and at the altitude 4 km the long-term trend of refraction is practically not observed.

The future task is to investigate the trends as functions of time and geographical position in different climatic zones for longer periods. However it is clear is that the angle of refraction is a sensitive indicator of the state of the troposphere – stratosphere system. Seasonal changes in the refractive properties observed in the stratosphere and the troposphere are evident, but they manifest themselves in different ways. In the stratosphere, there are quasi-harmonic changes with a period of 12 ± 0.5 months and the amplitude of about 0.12 mrad relative to the average trend. Maximum values of the bending angle occur in late July - early August, and the minimal during February - March. Seasonal changes in the upper troposphere also contain a component with a period of 12 ± 0.5 months, but they are opposite to the phase variations in the stratosphere. Their amplitude is in average 0.23 mrad. The maximum refraction occurs in March near the vernal equinox, and the minimum - in August. The influence of a weak quasi-monochromatic component is seen in the middle troposphere at an altitude of 4 km in the bending angle variations. Maximum values of the bending angle is ~ 15 mrad are observed, usually in late summer - early autumn, and in the rest of the year they are 3-5 mrad. This behavior corresponds to refraction in the middle and lower troposphere due to weather changes, which essentially smoothes the effect of changing seasons of the year.

## 6. Conclusions

The fundamental principle of local interaction of radio waves with a spherically symmetric medium is formulated and introduced in the RO method of remote sensing of the atmosphere and ionosphere of the Earth and planets.

In accordance with this principle, the main contribution to variations of the amplitude and phase of radio waves propagating through a medium makes a neighborhood of a tangential point where gradient of the refractive index is perpendicular to the radio ray.

A necessary and sufficient condition (a criterion) is established to detect from analysis of RO data the displacement of the tangential point from the radio ray perigee.

This criterion is applied to the identification and location of layers in the atmosphere and ionosphere by use of GPS RO data. RO data from the CHAllenge Minisatellite Payload (CHAMP) are used to validate the criterion introduced when significant variations of the amplitude and phase of the RO signals are observed at RO ray perigee altitudes below 80 km.

The new criterion provides an improved estimation of the altitude and location of the ionospheric plasma layers compared with the back-propagation radio-holographic method previously used.

The detected criterion opens a new avenue in terms of measuring the altitude and slope of the atmospheric and ionospheric layers. This is important for the location determination of the wind shear and the direction of internal wave propagation in the lower ionosphere, and possibly in the atmosphere.

The locality principle makes it possible to convert the eikonal acceleration (or the time derivative of the Doppler shift) into refractive attenuation. This is important for estimation of the total absorption of radio waves on the satellite-to-satellite transionospheric communication paths. This dependence is also important for measuring the water vapor content and atmospheric gas minorities in the future radio-occultation missions in view of the possibility to remove the refractive attenuation effect from the amplitude data. The advantages of the proposed method were tested by analysis of the CHAMP satellite radio-occultation data.

The obtained results indicate that measurements of the total absorption on radio occultation paths can potentially be used for monitoring of the atmospheric-oxygen content provided that the transmitter and receiver gain calibration is substantially improved. It follows from the above analysis that the comparison of the refractive attenuations retrieved from the amplitude and phase variations of a radio-occultation signal is necessary for the detection of layered structures in the atmosphere.

The total absorption, refractive attenuation, bending angle, bending angle, and index of refraction are important radio meteorological parameters which can be measured directly with a high accuracy by the radio occultation method. The prolonged radio occultation data base is very important for determination of the radio climate changes at different altitudes in the atmosphere with a global coverage.

### Acknowledgement

This work was supported in part by Russian Fund of Basic Research (grant No. 10-02-01015-a), Program of Presidium RAS No. 22, Program PSD- IY.13 of the Branch of Physical Sciences of the Russian Academy of Sciences, by Grants of the National Science Council of Taiwan NSC 101-2111-M-008-018; 101-2221-E-008-0, by Australian Research Council Project ARC-LP0883288 of the Department of Industry, Innovation, Science, and Research of Australia International Science Linkage under Project DIISR/ISL-CG130127, and by the Australia Space Research Program Project endorsed to research consortiums led by the Royal Melbourne Institute of Technology University. The authors are grateful to the GeoForschungsZentrum (Potsdam, Germany) for the data obtained during the CHAMP satellite mission.