A new method is introduced to locate the layered structures in the ionosphere based on simultaneous observations of radio wave temporal intensity and phase variations in trans-ionospheric satellite-to-satellite links. The method determines location of the tangent point on the trans-ionospheric ray trajectory where gradient of refractivity is perpendicular to the ray trajectory and the influence of a layered structure on radio wave parameters is maximal. This new technique was applied to the measurements provided during CHAMP radio occultation (RO) mission. For the considered RO events, the locations of the inclined plasma layers in the lower ionosphere are found and the electron density distributions are retrieved. The method is checked by measuring the location of the tangent point on the ray trajectory in the neutral gas in the atmosphere. The results showed a fairly good agreement.
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 RO experiments carried out by use of the coherent signals emitted by the Global Positioning navigational System (GPS) provide many possibilities to deduce the physical properties of the atmosphere and ionosphere [Kursinski et al., 1997]. Of particular importance are new ways to investigate the location of the layered plasma structures in the ionosphere. This paper describes for the first time those specific characteristics of the phase and intensity variations, which provide an independent technique for estimating the location of the layered structures in the ionosphere. The cornerstone of the RO inversion technique is based on an implicit assumption of global spherical symmetry of the atmosphere and ionosphere with the centre, which nearly coincides with the centre of the Earth. The location of the layered structure is assumed at a position of a ray perigee on the ray trajectory [Hajj et al., 2002]. The horizontal gradients of the refractivity produced by the inclined plasma layers in the ionosphere can change the location of the centre of spherical symmetry [Wickert et al., 2004] and one can observe the significant variations of the amplitude and phase of the RO signals in the 40–80 km height interval of the RO ray perigee, in which the expected contributions from the neutral gas or the electron density in the RO signal changes are negligible. Previously Gorbunov et al.  and Sokolovskiy et al.  proposed a radio-holographic back-propagation method to locate plasma irregularities in the ionosphere. In this paper, we introduce a simpler approach which can be used to find the location and estimate the parameters of the inclined plasma layers in the ionosphere.
2. Main Relationships
 The scheme of the GPS radio occultation (RO) experiment is shown in Figure 1. Point O is the centre of the spherical symmetry of the Earth's neutral atmosphere. Radio waves emitted by a GPS satellite (point G) arrive at a receiver onboard the LEO satellite (point L) along the ray GTL where T is the tangent point in the atmosphere. At the point T, the gradient of refractivity N(h) is perpendicular to the RO ray trajectory GTL (Figure 1). A cornerstone of the RO method is the assumption that the point T coincides with the RO ray perigee [e.g., Hajj et al., 2002], and, therefore, at point T the ray's distance from the Earth's surface h is minimal. The projection of the RO ray perigee on the Earth's surface determines the geographical co-ordinates of the RO region.
 Records of the RO signal along the LEO trajectory at two GPS frequencies f1 = 1575.42 MHz and f2 = 1227.6 MHz contain the amplitudes A1(t) and A2(t), respectively, along the phase path excesses Φ1(t) and Φ2(t) of the radio field as functions of time t. The vertical velocity of the occultation beam path v⊥ is about of 2 km/s. This value of v⊥ is many times greater than those corresponding to the motion of the layers in the ionosphere and atmosphere. Therefore, the RO signal contains quasi-instantaneous radio image of the Earth environment in the RO region.
 In the case of spherical symmetry of the ionosphere and atmosphere, there are simple relations between the phase path excess Φ(p) and the refraction attenuation of radio waves X(p) [Pavelyev et al., 2004; Liou et al., 2005]
where κ(p) is the main refractivity part of the phase path excess, ξ(p) = −dκ(p)/dp is the refraction angle, θ(p) is the central angle, p, ps are the impact parameters of the ray trajectory GTL, and line of site GDL, respectively, R0, R1, R2 are the distances GDL, OG, and OL, respectively, and L(p) is the distance GABL which consists of the distances d1 (GA), d2 (BL), and arc AB (Figure 1). The distance d2 is nearly equal to the distance TL because the smallness of refraction effect. Under condition ∣p − ps∣ ≪ ps, the time derivative dΦ(p)/dt has a form:
where d1s and d2s are the distances GD and DL, respectively (Figure 1). Under condition:
the second time derivative of F(p) can be obtained from (4):
Condition (6) and equation (7) are valid because ps(t) and dps/dt are slowly changing in the RO measurements. By use of equation dp/dt − dps/dt ≈ [X(t) − 1]dps/dt [Pavelyev et al., 2004], one can obtain from (7):
Equation (8) establishes an equivalence between the variations of the phase acceleration a = d2Φ(t)/dt2 and the refraction attenuation X(t). Parameters m and dps/dt may be evaluated from the orbital data if the locations of the spherical symmetry centre O and its projection D on the line of sight GDL are known (Figure 1):
where v and w are the velocity components of the GPS and LEO satellites, respectively, which are perpendicular to the straight line GL in the plane GOL. Equations (8)–(11) can be used to find the distance LD d2s from simultaneous observation of the phase and intensity variations of the radio wave:
The horizontal gradients in the ionosphere can displace the centre of the spherical symmetry from its standard position - point O to point O′ (Figure 1). Therefore, the tangent point T will be also displaced from the RO ray perigee along the ray trajectory to point T′. As a consequence, parameter m will also change its magnitude. The equations (8)–(12) may be used in the case of local spherical symmetry with new centre O′ under the assumptions: (1) absence of multi-path propagation and (2) validness of inequality (6). In the 3-D case, the new centre O′ may not belong to the plane GOL. In this case, one must know for estimation of parameter m the dihedral angle between the planes GLO and GLO′.
 Therefore, if the magnitude of the parameter m will be estimated from the experimental data, then it is possible to find the new value of distance T′L d′2, and thus determine the location of the new tangent point T′ relative to the RO ray perigee. We assume that m is a slowly changing function of time. If the noise is very small the parameter m can be determined directly from equation (8) as a ratio:
In the presence of noise, the value m(tk) corresponding to some instant of time tk can be determined from the RO data as a ratio of average of the squared refraction attenuation variation and phase acceleration variations:
where 2M is a number of samples for averaging, and X(ti), a(ti) are the current values of the refraction attenuation and phase acceleration variations at the time instant ti. Equation (14) is valid, if there is a full correlation between the refraction attenuation and the phase acceleration according to equation (8). There are different sources of the amplitude and phase variations of the RO signal (e.g., turbulence, multipath propagation, etc.) which do not obey equation (8). However, the amplitude and phase variations corresponding to the inclined layered structures in the atmosphere and ionosphere must obey relationship (8). Therefore, the parameter m can be determined from the correlation relationship:
Equations (14) and (15) give the upper and lower boundaries of the parameter m, respectively. Then, after use of equations (8)–(12), we estimate the upper and lower boundaries of the distance d = d′2 − d2 = d′2 − (R22 − p2)1/2.
3. Analysis of CHAMP RO Data
 A description of the CHAllenging Mini Payload (CHAMP) GPS RO mission is given by Wickert et al. . The data archives used in this work may be accessed at: http://isdc.gfz-potsdam.de/champ. To validate equation (8), we consider the CHAMP RO event 0069, September 21, 2003, 05 h 33 m of local time with geographical co-ordinates 40.4°N 19.6°W (Figure 2). Curves 1 and 2 in Figure 2 indicate the refraction attenuations Xp and Xa retrieved from the phase acceleration a and the amplitude variations, respectively. Smooth curves 3 indicates the result of modeling of the refraction attenuation by use of the exponential model of refractivity in the neutral atmosphere [Pavelyev et al., 1996]. Curves 4 and 5 in Figure 2 describe the displacement of point T d from the RO ray perigee calculated from the relationships (9), (10) by use of equation (14) (curve 4) and (15) (curves 5). One can observe a good correspondence between the refraction attenuations Xa and Xp retrieved from the phase acceleration a and the amplitude variations, respectively, in the 10–30 km height interval, where curves 4 and 5 are coinciding, and the location of point T corresponds to position of the RO ray perigee (Figure 2). The displacement d of point T is below 100–200 km in this altitude interval. This coincidence validates the suggested method for the location of point T. In the 35–80 km and 88–96 km altitude intervals, the curves 4 and 5 do not coincide. This indicates the different origin of the phase acceleration and refraction attenuation variations (Figure 2). The correlation between a and X − 1 does exist in the 80–86 km and 98–106 km height intervals (Figure 2) where the RO signal variations are owing to two sporadic E-layers in the ionosphere (curves 1 and 2 in Figure 2).
 The sporadic E-layers contributions to variations of the RO signal are considered in detail in Figure 3 for two CHAMP RO events 0069, September 21, 2003, 40.4°N 19.6°W (top), and 0169, July 05, 2003, 29.4°S 232.9°W (bottom). Two sporadic E-layers are seen in the event 0069 (Figure 3, top) with centers at the 82.5 and 103.5 km heights of the RO ray perigee. The replies from sporadic E-layers are often observed in the RO experiments at the 80–90 km altitudes of the RO ray perigee [e.g., Wu et al., 2005]. However, this altitude interval is unusual for sporadic E-layers [e.g., Kelley, 1989]. Thus, the first sporadic E-layer with the RO ray perigee height equal to 82.5 km is displaced from the ray perigee (Figure 2, top). The displacement can be estimated by using curves 3 and 4 (Figure 3, top). Estimation of d by use of the curve 4 found from the correlation between a and X − 1 (equation (15)) gives the value d about −530 km. For comparison with this result, we evaluate in addition the displacement d by the back-propagation method, which is described in detail by Gorbunov et al.  and Sokolovskiy et al. . Curves 5 and the error bars in Figure 3 correspond to the estimation of the displacement d by the back-propagation method. As follows from Figure 3, the considered methods give the results coinciding in their intervals of accuracy. Therefore, the first sporadic E-layer is located on the RO ray trajectory between the point D and L (Figure 1) at a distance about 530 km from the RO ray perigee.
 It is possible to find the height h′(T′) and the inclination δ of a plasma layer in the ionosphere from results of determination of its distance d from the RO ray perigee by use of the relationship [Wickert et al., 2004]
where r0 is the Earth radius, and h is the height of the point T. The inclination of the first layer to the local horizon is about 8.9° and the corrected height above the Earth surface is in the 96–98 km interval (Figure 3, top). The second sporadic E- layer is disposed near the RO ray perigee because both curves 3 and 4 give the same values d, which are nearly equal to 0 (Figure 3, top). For the RO event 0169, the centre of the sporadic E-layer is located at 119.5 km altitude and practically coincide with the RO ray perigee since the distance d (curve 3 and 4 in Figure 3, bottom) is equal to 0. The accuracy in the determination of value d, which may be estimated as a difference between curves 3 and 4 in Figure 3, is about 30–100 km. Therefore, the suggested method has a promise to be an effective tool for the localization of the sporadic E-layers and inclined plasma layers in the ionosphere.
4. Estimating the Plasma Parameters
 For the estimation of the plasma density in the inclined ionospheric layers, one may apply the method of the solution for the inverse problem described by Liou et al. . According to their procedure, the vertical gradient of the refractivity and electron density altitude distributions have been retrieved from the amplitude variations of the RO signal. Then, after integrations, the vertical distribution of the electron density is determined.
 Results of restoration of the electron density variations ΔNe(h) and its gradient dNe(h)/dh from the amplitude variations of the RO signal are given for the considered RO events 0069 and 0169 in Figure 4 (top and bottom, respectively). The refraction attenuations Xa and Xp retrieved from the phase acceleration a and the amplitude variations, respectively, are shown by curves 1 and 2 in Figure 4. Curves 3 and 4 in Figure 4 demonstrate the retrieved variations ΔNe(h) and dNe(h)/dh as functions of the corrected (by use of equation (14)) height h. Curves 1 and 2 agree very well, especially for the event 0169, when both curves 1 and 2 are nearly coinciding (Figure 4). This is a new validation of the equation (8) for the case of sporadic E-layers in the ionosphere. The electron density perturbations ΔNe(h) changes mainly in the ±5 · 109–±25 · 109 [el/m3] interval, and its vertical gradient dNe(h)/dh in the ±20 · 109 [el/m3km]–±50 · 109 [el/m3km] interval. The estimated heights and plasma parameters are usual values for the sporadic E-layer in the ionosphere [e.g., Kelley, 1989].
 As demonstrated above, the new phase acceleration/refraction attenuation ratio technique appears to provide a new method to locate the layered structures in the ionosphere. This method is validated by locating the tangent point in the atmosphere and ionosphere by means of analysis of the CHAMP RO data. Recent GPS/MET, CHAMP, SAC-C, GRACE, and future FORMOSAT-3 RO missions provide a growing data-base for determining the location and estimation of the electron density distribution in the layered plasma structures in the ionosphere. Comparison with ionosondes data is desirable for further derivation and revealing the boundary of application of the suggested method. The application of this and other new techniques will generate a more extensive body of information on plasma structures and natural processes in the ionosphere and their connection with processes in magnetosphere and in interplanetary space.
 We are grateful to the Center of Geophysical Studies, Potsdam, for the presented CHAMP experimental data on radio occultation. This work was supported by the National Science Council of Taiwan, grant NSC 94-2811-M-008-055, the Russian Foundation for Basic Research, project 06-02-17071, and Russian Academy of Sciences, programs OFN-16 and OFN-17.