[3] 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.

[4] Records of the RO signal along the LEO trajectory at two GPS frequencies *f*_{1} = 1575.42 MHz and *f*_{2} = 1227.6 MHz contain the amplitudes *A*_{1}(*t*) and *A*_{2}(*t*), respectively, along the phase path excesses Φ_{1}(*t*) and Φ_{2}(*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.

[5] 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*) = −*d*κ(*p*)/*dp* is the refraction angle, θ(*p*) is the central angle, *p*, *p*_{s} are the impact parameters of the ray trajectory GTL, and line of site GDL, respectively, *R*_{0}, *R*_{1}, *R*_{2} are the distances GDL, OG, and OL, respectively, and *L*(*p*) is the distance GABL which consists of the distances *d*_{1} (GA), *d*_{2} (BL), and arc AB (Figure 1). The distance *d*_{2} is nearly equal to the distance TL because the smallness of refraction effect. Under condition ∣*p* − *p*_{s}∣ ≪ *p*_{s}, the time derivative *d*Φ(*p*)/*dt* has a form:

where *d*_{1s} and *d*_{2s} 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 p_{s}(t) and dp_{s}/dt are slowly changing in the RO measurements. By use of equation *dp*/*dt* − *dp*_{s}/*dt* ≈ [*X*(*t*) − 1]*dp*_{s}/*dt* [*Pavelyev et al.*, 2004], one can obtain from (7):

Equation (8) establishes an equivalence between the variations of the phase acceleration *a* = *d*^{2}Φ(*t*)/*dt*^{2} and the refraction attenuation *X*(*t*). Parameters *m* and *dp*_{s}/*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 *d*_{2s} 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′.

[6] 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*(*t*_{k}) corresponding to some instant of time *t*_{k} can be determined from the RO data as a ratio of average of the squared refraction attenuation variation and phase acceleration variations:

where 2*M* is a number of samples for averaging, and *X*(*t*_{i}), *a*(*t*_{i}) are the current values of the refraction attenuation and phase acceleration variations at the time instant *t*_{i}. 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} − *d*_{2} = *d*′_{2} − (*R*_{2}^{2} − *p*^{2})^{1/2}.