### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Phase Path, Refractive Angle, and Refractive Attenuation
- 3. Refractive Attenuation and Reflectivity Cross Section
- 4. Characteristics of the Direct and Reflected Signals
- 5. Eikonal Acceleration and Refractive Attenuation
- 6. Influence of the Boundary Layer on Parameters of the Reflected Signals in MIR/GEO RO Experiments
- 7. Conclusions
- Acknowledgments
- References

[1] The relationships between the Doppler frequencies, eikonal acceleration, and refractive attenuations of the direct and reflected signals are established for bistatic and radio occultation experiments. These connections allow recalculating the Doppler shifts and the phase delays to the refractive attenuation (reflectivity cross section) and open a new avenue for potentially measuring the total absorption in the atmosphere at low elevation angles. The fundamental characteristics of bistatic remote sensing of the atmosphere and Earth's surface such as the phase delay, reflection coefficient, reflectivity cross section, and Doppler shift of the reflected signals relative to the direct signals are obtained in analytical forms by taking into account the refraction and absorption effects in the atmosphere. Difference in the Doppler frequencies of the reflected and direct signals is proportional to the difference of the modified refractive index at the radio ray perigee and at the Earth's surface. The obtained analytical results are in good agreement with the measurements data obtained during the MIR/GEO (wavelengths 2 and 32 cm), and CHAMP (wavelengths 19 and 24 cm) radio occultation experiments. Detecting the reflected signals in radio occultation data has opened new perspectives for bistatic monitoring of the atmosphere and Earth's surface at low elevation angles. Experimental results of the propagation effects at low elevation angles are of great importance for fundamental theoretical investigation of radio waves propagation.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Phase Path, Refractive Angle, and Refractive Attenuation
- 3. Refractive Attenuation and Reflectivity Cross Section
- 4. Characteristics of the Direct and Reflected Signals
- 5. Eikonal Acceleration and Refractive Attenuation
- 6. Influence of the Boundary Layer on Parameters of the Reflected Signals in MIR/GEO RO Experiments
- 7. Conclusions
- Acknowledgments
- References

[2] Nowadays, the developments of radio means and its data processing facilities have advanced the exploration of the atmosphere and Earth's surface from space using the bistatic radio-holographic (or interferometric) scheme. In the bistatic scheme, a receiver is located at an extended distance from a transmitter of radio waves. Radio occultation (RO) and radio tomographic bistatic techniques using the high-stability and coherence properties of GPS signals have been recently successfully applied to the global monitoring of the atmosphere and ionosphere at different altitudes [*Ware et al.*, 1996; *Kursinski et al.*, 1997; *Hajj and Romans*, 1998; *Kunitsyn and Tereshchenko*, 2003; *Melbourne*, 2004; *Rocken et al.*, 1997; *Liou et al.*, 2010]. The radio-holographic method can improve the vertical resolution and alleviate potential multipath effects of the RO technology [*Pavelyev*, 1998; *Hocke et al.*, 1999; *Gorbunov*, 2002; *Jensen et al.*, 2003]. The eikonal acceleration technique is proposed [*Liou and Pavelyev*, 2006] and developed for identifying and locating the layered structures of the atmosphere and ionosphere [*Pavelyev et al.*, 2010]. Based on forty years of radio occultation (RO) experiments it is now recognized that the phase (eikonal) acceleration of radio waves is as important as the Doppler frequency, as derived from analysis of high-stability Global Positioning System (GPS) RO signals. Eikonal acceleration technique allows one to convert the phase and Doppler frequency changes into refractive attenuation variations. From such derived refractive attenuation and amplitude data one can estimate the integral absorption of radio waves [*Pavelyev et al.*, 2010]. In this paper we will show that application of the eikonal acceleration technique can be used for analyzing reflections from terrestrial surfaces.

[3] The bistatic spaceborne radar scheme has been proposed earlier for remotely sensing the Earth's surface from space [e.g., *Milekhin et al.*, 1986; *Martín-Neira*, 1993]. The key advantages of the bistatic radio-holographic scheme in comparison with the monostatic radar technique are: (1) the capability of radio imaging of the Earth surface at various aspect angles; (2) the capability of using the signals of communication and television satellites; (3) the high sensitivity to various types of anomalies of the radio wave scattering and propagation and, as a result, the possibility for observing particular phenomena both on the Earth's surface and in the atmosphere, especially at small grazing angles; and (4) the technical feasibility for a joint action of the receiver and the transmitter, that enlarges the width of the observed surface area and offer a new opportunity to use continuous signals when sounding [*Milekhin et al.*, 1986; *Pavelyev et al.*, 1996].

[4] Sounding the Earth's surface using the bistatic radar technique allows solving the remote sensing tasks, such as: (1) periodical measurements of the bistatic characteristics of the Earth's surface; estimation of their regional, seasonal and annual variations, determination of the characteristics having the highest sensitivity to the environmental characteristics, anthropogenic impacts; (2) continuous monitoring the parameters of the atmospheric structures (e.g., tropical and extratropical cyclones, dust storms, etc.) using radio wave propagation effects in the vicinity of radio shadow zone at low elevation angles; (3) obtaining the radio imagery of the areas selected to analyze the technical feasibility of the satellite system for all-weather ecology monitoring of the Earth. In this paper we are mainly concern with reflections at small elevation angles, however some of our results are relevant for broad range of the incidence angles.

[5] The first bistatic experiments at low elevation angles were conducted by using the radio communication link orbital station “MIR”–geostationary satellites at wavelengths of 2 and 32 cm, respectively [*Rubashkin et al.*, 1993; *Pavelyev et al.*, 1996, 1997]. Highly stable, synchronized by atomic clocks, signals of Global Positioning System (GPS) can be used for spaceborne bistatic remote sensing [*Martín-Neira*, 1993]. GPS bistatic surface reflections were observed from space during GPS/MET and CHAMP RO experiments [*Beyerle and Hocke*, 2001; *Igarashi et al.*, 2001; *Beyerle et al.*, 2002; *Pavelyev et al.*, 2002]. Bistatic incoherent, quasi-specular GPS reflections arriving from a relatively wide cone around a nominally specular direction were registered at high-elevation-angle geometry both from space [*Lowe et al.*, 2002] and from aerial platforms [*Komjathy et al.*, 2000]. The review of results relevant to this field can be found in the work of *Gleason et al.* [2009]. Therefore, bistatic scheme using the Global Navigational Satellite System (GNSS) signals is regarded as an innovative remote sensing technique for studying the Earth's environment and could complement investigations using radiometers and radars. This technique has a constant source of coherent high-stable radio waves transmitted from GNSS satellites (GPS, GALILEO, GLONASS) which illuminate the Earth. In contrast to the monostatic radar, the bistatic remote sensing experiments are currently fulfilled at elevation angles near the direction of the specular reflections [*Gleason et al.*, 2005; *Lowe et al.*, 2002]. This allows obtaining significant information about the bistatic scattering characteristics of terrestrial surfaces [*Milekhin et al.*, 1986; *Martín-Neira*, 1993; *Pavelyev et al.*, 1996, 1997, 2004; *Helm*, 2008] and for surface altimetry purposes [*Cardellach et al.*, 2004].

[6] The aim of this paper is (1) to generalize the eikonal acceleration technique for bistatic reflections, and (2) to obtain analytical equations, with accounting for the refraction and absorption effects in the atmosphere, the fundamental bistatic characteristics of the Earth's as a planet observed from space. In this study we derive the following fundamental characteristics: the phase delay, reflection coefficient, reflectivity cross section, and Doppler shift of reflected signals relative to the direct signal in analytical forms. The paper is organized as follows: in section 2, the relationships for the phase path, refractive angle, and the Doppler shift for the direct and reflected signals are presented. In section 3, the relationship between the refractive attenuation and reflectivity cross section are introduced. In section 4, the results of numerical evaluation of the characteristics of the direct and reflected signals: refractive angle, phase delay and Doppler shift are described. The relationships between the eikonal acceleration and refractive attenuation of the direct and reflected radio waves are analyzed and validated using experimental RO CHAMP data in section 5. Influence of the boundary layer on the parameters of reflected signals are provided in section 6. Quasi-specular character of reflections (small values of the random surface slopes) is the implicit assumption of an analytical model proposed in this paper. This means that the width of the Doppler frequency spectrum of reflections is small as compared with the difference of the Doppler shifts of direct and reflected signals. According to analysis of quasi-specular (noncoherent) reflections provided by *Milekhin et al.* [1986], this assumption does not significantly influence on the value of the reflection cross section and other refractive parameters considered in this paper. In this paper we consider only coherent part of the reflections for estimating the phase or eikonal delays. The noncoherent reflections may introduced dispersion in the estimation of the eikonals and delays calculated for the case of small Rayleigh parameter.

### 2. Phase Path, Refractive Angle, and Refractive Attenuation

- Top of page
- Abstract
- 1. Introduction
- 2. Phase Path, Refractive Angle, and Refractive Attenuation
- 3. Refractive Attenuation and Reflectivity Cross Section
- 4. Characteristics of the Direct and Reflected Signals
- 5. Eikonal Acceleration and Refractive Attenuation
- 6. Influence of the Boundary Layer on Parameters of the Reflected Signals in MIR/GEO RO Experiments
- 7. Conclusions
- Acknowledgments
- References

[7] The geometry of bistatic radiolocation experiments is shown in Figure 1. Radio waves are emitted by transmitter onboard of satellite located at point G and then they propagate along the two paths: 1 and 2 (Figure 1). The direct and reflected radio signals from a transmitter aboard a satellite G arrive after propagation along the paths 1 and 2 at a receiver onboard a low Earth orbital (LEO) satellite (point L, Figure 1).*R*_{L}, *R*_{G} are the distances from the center of spherical symmetry (point O) to the transmitter G and receiver L, respectively, *R*_{0} is the distance GL, *h* is the height of the ray perigee (Figure 1). Point D is the specular reflection point located on a smooth sphere with radius *a* corresponding to the average Earth's surface in the RO region (Figure 1). The angle between the directions OG and OL is equal to θ. The central angle θ is common for both trajectories GL (1) and GDL (2). In the case of spherical symmetry, θ is linked to the refractive angles *ξ*_{d} (*P*_{d}), *ξ*_{r} (*P*_{r}) and impact parameters *p*_{d}, *p*_{s} corresponding to the direct and reflected radio waves via equations [*Kalashnikov et al.*, 1986]

where *n* (*R*_{G}), *n* (*R*_{L}) and *n* (*a*) are the refractive indexes at points G, L, and D, respectively, *p*_{s} is the impact parameter corresponding to the line of sight GL, *p*_{s} is the distance between the center of spherical symmetry and the straight GL, *p*_{s}, *p*_{r} are the impact parameters corresponding to the direct GTL and reflected GDL rays, *p*_{d} = *const*, *p*_{r} = *const* along the rays GTL and GDL because of spherical symmetry of the atmosphere, *ψ* is the grazing angle between the horizontal direction and reflected ray with vertex at point D (Figure 1). The second equation (3) describes dependence of the impact parameter *p*_{r} on *ψ*. In addition to angle *ψ*, we introduce a similar parameter *α* which is linked to the impact parameter of the direct signal by following relationship:

Formula (4) can be obtained from equation (3) by means of analytic continuation of the grazing angle into the complex plane *ψ* = ±*iα*, and, vise versa, equation (3) may be obtained from (4) by substitution *α* = ±*iψ*. Therefore the variables *p*_{r}, *p*_{d} and the refractive angles *ξ*_{d} (*p*_{d}), *ξ*_{r} (*p*_{r}) corresponding to the direct and reflected signals are connected by means of analytic continuation in an abstract space of the impact parameters. Below the variables *p*_{r}, *p*_{d} and *ξ*_{d} (*p*_{d}), *ξ*_{r} (*p*_{r}) are considered under the same values of central angle θ, in this case the magnitudes of *ψ* and *α* are significantly different. The phase path excesses (eikonals) of the reflected and direct signals *S*_{d,r} may be described by equations

where κ_{r} (*p*_{r}), κ_{d} (*p*_{d}) are the main refractive part of the phase path excess connected with influence of the vertical gradients of refractivity in the atmosphere along the trajectories of the direct and reflected rays, *S*_{s} (*p*_{s}) is the phase path in the free space along the line of sight GL. The phase path excesses (eikonals) *S*_{d,r} are equivalent to the carrier phase delay relative to the propagation phase delay along the distance GL. Below we introduce the eikonal acceleration as the second derivative of the eikonals *S*_{d,r} with respect to time. The eikonal acceleration is proportional to the first derivative of the Doppler frequency with respect to time [*Pavelyev et al.*, 2010].

[8] The next equations are valid for the refractive angles *ξ*_{r} (*p*_{r}), *ξ*_{d} (*p*_{d}) and main refractive part of the phase path excess κ_{r} (*p*_{r}), κ_{d} (*p*_{d}) [*Kalashnikov et al.*, 1986]

where *r*_{0} is the minimal distance of the direct ray trajectory LG from the center of spherical symmetry (point O). The lower limits *r*_{0} and *a* in the integrals (8), (9) correspond to the direct and reflected radio waves, respectively. The next link between the refractive angle and the main refractive part of the phase path excess follows from equations (8) and (9)

Below parameters *n*(*R*_{G}), *n*(*R*_{L}) are supposed to be equal to unity. Equations (5), (6) describe the phase delays of the reflected and direct signals as functions of the impact parameters *p*_{d,r} and the distances *r*_{0}, and *a*. The refractive angles *ξ*_{d} (*p*_{d}), *ξ*_{r} (*p*_{r}), and the main refractive part of the phase path excess κ_{d} (*p*_{d}), κ_{r} (*p*_{r}) are dependent on the altitude profile of the refractive index *n* (*r*). One can find the difference of the Doppler shifts *F*_{d,r} corresponding to the reflected and direct signals from equations (5)–(7):

Equations (11) and (13) are valid for the general case of noncircular orbits of the GPS and LEO satellites. Equations (11) and (13) can be simplified under condition

which is valid if (*p*_{d,r} − *p*_{s}) ∣*p*_{s}∣. This inequality is valid for all practical RO situations. Under indicated approximation one can obtain from equations (11) and (13)

where *λ* is the wavelength relevant to the carrier frequency of the emitted signals. Under the assumptions that the atmosphere is spherically symmetric and a large-scale relief is absent, i.e., *n* (*a*), *a* = *const* in the region of the measurements, then expression (17) may be simplified as

According to (16), (18) the difference of the Doppler frequencies of the direct and reflected signals Δ*F* is equal to

Equations (5), (6) and (19) can be used for evaluating the difference of the phase path delays and Doppler frequency of the direct and reflected signals. In the case of a low grazing angle *ψ*, equation (19) may be transformed to the form

where *h* is the height of the ray perigee related to the direct signal, *M* (*r*) is the magnitude of the modified refractive index at the altitude *r*. Equation (20) is approximately fulfilled under condition *ψ*^{2} ≪ 1. The difference between the modified refractive index at the height *h* and at the Earth's surface is proportional to the difference of the Doppler frequencies corresponding to the reflected and direct signals.

[9] The simplified equation (19) describes proportional dependence of the difference of the Doppler frequencies of the direct and reflected signals and difference of their impact parameters. This dependence is valid in the broad range of the grazing angle *ψ* 0 ≤ *ψ* ≤ *π*/2 under assumption of quasi-specular reflections.

### 3. Refractive Attenuation and Reflectivity Cross Section

- Top of page
- Abstract
- 1. Introduction
- 2. Phase Path, Refractive Angle, and Refractive Attenuation
- 3. Refractive Attenuation and Reflectivity Cross Section
- 4. Characteristics of the Direct and Reflected Signals
- 5. Eikonal Acceleration and Refractive Attenuation
- 6. Influence of the Boundary Layer on Parameters of the Reflected Signals in MIR/GEO RO Experiments
- 7. Conclusions
- Acknowledgments
- References

[10] As previously derived by *Pavelyev and Kucherjavenkov* [1978] and *Pavelyev et al.* [1997], the refractive attenuations *X* (*p*_{d,r}) of the direct and reflected signals are described by the same analytical expressions:

where is the partial derivative of the central angle with respect to the impact parameters *p*_{d,r}. The magnitude of is evaluated under constancy of the distances *R*_{L}, *R*_{G}:

From (21) and (22), the refractive attenuation of direct signal *X*_{d} is equal to:

In the case *R*_{G} ≫ *R*_{L}, *R*_{0} ≈ *R*_{G}, *p*_{d} ≈ *p*_{s}equation (24) corresponds to the well known expression derived previously [*Fjeldbo et al.*, 1966; *Eshleman et al.*, 1980]. We define the reflection coefficient *η*^{2} from the Earth's surface as a ratio of the intensity *W*_{r} of the reflected radio waves to the intensity of radio waves in the free space *W*_{0} at point L. According to this definition the refractive attenuation *X* (*p*_{r}) of reflected signal may be described by the following equation:

where *χ*_{r} is the total absorption of the radio waves along path 2 (Figure 1), *V* (*ψ*) is the Fresnel's coefficient describing the reflection of radio waves from a plane surface. The expression for *X* (*p*_{r}) may be obtained after differentiation and substitution of θ from (2) to (21):

where variables *R*_{DL}, *R*_{DG} are linked to *ψ* and *p*_{r} by equations:

[11] In the case when refraction is absent:

Variables *R*_{DG}, *R*_{DL} from (28) are the distances from the specular point D to the transmitter and receiver, respectively, and expression (26) for the refractive attenuation *X* (*p*_{r}) is simplified:

According to the definition of the reflectivity cross section *σ* of the Earth's as a planet observed from space [e.g., *Milekhin et al.*, 1986], the magnitude of *σ* is connected to the reflection coefficient *η*^{2}:

where *R*_{GD}, *R*_{LD} are the distances along the straights lines GD and LD from the specular point D to the transmitter G and receiver L, respectively:

The distance *R*_{DG}, _{DL}(31) coincides with variables *R*_{DG}, *R*_{DL} in the case when refraction is absent.

[12] Comparing equation (25) with equation (30) gives the magnitude of *σ*:

In the case when the refraction and absorption in the atmosphere are absent, the magnitude of *σ* coincides with well known expression [e.g., *Yakovlev*, 2003]:

Therefore equations (26) and (32) are the geometrical optics presentation of the reflectivity cross section of the Earth's *σ* for a general case of strong refraction effect in the broad interval of the grazing angle 0 ≤ *ψ* ≤ *π*/2. It is important that the refractive attenuation *X* (*p*_{r}), reflection coefficient ∣*V*^{2} (*ψ*)∣, and the total absorption *χ*_{r} are included in the expression for the cross section (32) as independent multiplicative parameters. Therefore application of the eikonal acceleration technique allows excluding of the refractive attenuation from (32) and opens a possibility to evaluate the total absorption of radio waves at small elevation angles.

### 4. Characteristics of the Direct and Reflected Signals

- Top of page
- Abstract
- 1. Introduction
- 2. Phase Path, Refractive Angle, and Refractive Attenuation
- 3. Refractive Attenuation and Reflectivity Cross Section
- 4. Characteristics of the Direct and Reflected Signals
- 5. Eikonal Acceleration and Refractive Attenuation
- 6. Influence of the Boundary Layer on Parameters of the Reflected Signals in MIR/GEO RO Experiments
- 7. Conclusions
- Acknowledgments
- References

[13] Analytical model which describes the phase delay, reflection coefficient, reflectivity cross section, and Doppler shift of reflected and direct signals is used together with the electrodynamic model of the refractive index in the atmosphere introduced previously [*Pavelyev et al.*, 1996, 1997; *Pavelyev*, 1998]. The vertical profiles of the refractivity *N* (*h*) and its vertical gradient in the boundary layer of the atmosphere calculated from an exponential model introduced by *Pavelyev et al.* [1996] are shown in Figure 2. Computation of curves 1–7 was performed using the values of the near-surface refractivity *N*_{0} and its vertical gradient indicated in Figure 2 (left). For curves 1–7 the magnitude of *N*_{0} was equal to 340 N units (Figure 2, left and right). The data in Figure 2 correspond to a very broad interval of the atmospheric conditions in the boundary layer of the atmosphere under an assumption of monotonic dependence of the modified refractivity index *M* (*r*) on height.

[14] Dependence of the refractive angles *ξ*_{r} (*p*_{r}), *ξ*_{d} (*p*_{d}) on the grazing angle *ψ* related to the reflected (curves 1–7) and direct (curves 8–14) signals corresponding to the common values of central angle θ is shown in Figure 3 (left). Curves 1–7 and 8–14 in Figure 3 (left) correspond to the same values and *N*_{0} which are shown in Figure 2. The refractive angles *ξ*_{r} (*p*_{r}), *ξ*_{d} (*p*_{d}) have the same values at the boundary of radio shadow *ψ* = 0 where the impact parameters *p*_{r}, *p*_{d} are equal to *n* (*a*)*a*. Significant dependence on the vertical gradient of refractivity prevents application of the Laplace theorem for evaluation of the bending angle *ξ*_{r} (*p*_{r}). The changes of the refractive angle *ξ*_{d} (*p*_{d}) with the grazing angle *ψ* are faster than variations of *ξ*_{r} (*p*_{r}). This behavior is a result of strong dependence of variable *α* on the grazing angle *ψ* (curves 1–8, Figure 3, right). Curve 8 in Figure 3 (right) describes dependence *α* (*ψ*) when the atmosphere is absent. Curves 1–7 in Figure 3 (right) indicate a significant influence of the refraction effects on the function *α* (*ψ*).

[15] For estimating the possibility to detect reflections in the GPS RO data, the difference of the phase delays (equivalent to the code phase delay in the case of nondispersive medium) of the reflected and direct signals *S*_{r} − *S*_{d} as a function of the altitude of the ray perigee *h* in path 1 (Figure 1) is shown in Figures 4 (left) and 4 (right) for magnitudes of *N*_{0} equal to 340 N units (left) and 480 N units (right), respectively. Curves 1–7 in Figure 4 correspond to the same values shown in Figure 2 above. Curve 8 corresponds to the case of free space, when the atmospheric influence is absent. The difference *S*_{r} − *S*_{d} changes in the range of 0 … 250 m at the altitude intervals of 0 < *h* < 20 km (Figure 4, left and right). This difference is well below the chip length of the continuous C/A code of GPS signals that equals to 300 m. Therefore in the indicated altitude interval *h* the GPS RO receiver's output contains both the reflected and direct signals. Results of application of the analytical model for calculation of the Doppler shifts of the reflected and direct signals as a function of the grazing angle and height are shown in Figure 5 (right). Curves 1–7 in Figures 5 (left) and 5 (right) and curves 9–15 in Figure 5 (right) correspond to the same values *s*_{r} − *s*_{d} and *N*_{0} which are used in Figure 2. Curves 8 correspond to the case of free space (Figure 5). Curves 1–8 correspond to the case when the GPS and LEO satellites are moving in the same direction, the dotted curves 9–15 describe the opposite case (Figure 5, right). Curves 1–8 in Figure 5 (left) represent the connection between the magnitudes of grazing angle *ψ* and height *h*. Numerical analysis indicates that there is a near-linear relationship between the difference of the Doppler shifts *F*_{r} − *F*_{d} and the height *h* of the ray perigee. The height interval 0 < *h* < 15 km where the difference *F*_{r} − *F*_{d} is well below 50 Hz indicates that the reflected GPS signals can be registered with sampling frequency of 50 Hz by the currently used GPS RO satellites receivers.

[16] Below we will derive an important relationship that connects the derivative of the Doppler frequencies with respect to time and the refractive attenuation of the direct and reflected radio waves that was introduced earlier only for direct signals by *Liou and Pavelyev* [2006]. If the atmosphere is spherically symmetric, and a large-scale surface undulation is absent, under the condition

one can obtain according to equations (16) and (18) the relationships:

where *m* is a slowly changing function of time that can be found from the satellites trajectory data. By using equation − ≈ [*X*_{d,r} (*t*) − 1] [*Liou et al.*, 2007], which is valid for the direct and reflected signals, one can obtain from (35)–(37)

It follows from equation (38) that the derivative of the Doppler frequencies *F*_{d,r} with respect to time is proportional to the difference of the refractive attenuation of the direct and reflected signals, respectively. Usually the value *X*_{d} is significantly greater than *X*_{r}. So in the free space when *X*_{d} = 1 the derivative of difference Δ*F* with respect to time is nearly equal to a constant because *m* is a slowly changing parameter

Equation (38) allows computing the refractive attenuations *X*_{d,r} directly from the phase and/or frequency data. This can used to exclude the refractive attenuation from the amplitude data and estimating the total absorption of radio waves at paths 1 and 2 (Figure 1). The value *X* (*p*_{r}) accounts for the generalized spherical divergence factor and the refractive attenuation of the reflected waves along path 2. This opens the way to measure the reflection coefficient of the Earth's surface which includes the total absorption of radio waves along the raypath 2 (Figure 1).

[17] The accuracy of equations (38), (39) depends on the width of the Doppler spectrum of the direct and reflected signals. The width of the Doppler spectrum of reflected signals diminishes at small elevation angles. The multipath propagation in the lower troposphere diminish applicable domain of equation (38). However, there exists an interval of elevation angles were both equations (38) and (39) are valid. In the bistatic experiments the ratio *η*_{e}^{2} (*ψ*) of the intensities of the reflected and direct signals is measured. This ratio is related to parameter *X* (*p*_{r}) as follows

where *χ*_{r}, *χ*_{d} are the total absorption coefficients of radio waves on paths 1 and 2 (Figure 1); Φ_{r} (*ψ*) is the factor that accounts for the influence of the large-scale terrain features and shadowing; and *V*^{2} (*ψ*) is the reflection coefficient of a plane wave incident at a grazing angle *ψ* on a plane atmosphere-ground interface. The coefficient *V*^{2} (*ψ*) depends on the permittivity ɛ of the Earth's ground and polarization of the transmitting antenna. The coefficient *V*^{2} (*ψ*) is related to the grazing angle *ψ* for the case of two orthogonal circular polarizations and homogeneous ground with permittivity ɛ as follows

where ∣*V*_{r,l}^{2} (*ψ*)∣ is the intensity reflection coefficient from a plane surface corresponding to the right hand (RHCP) and left hand (LHCP) circular polarizations. Minus sign equation (42) corresponds to the LHCP case. Coefficient *η*_{e}^{2} (*ψ*) is more convenient for describing the experimental data than *η*^{2} (*ψ*) because it allows cancellation of slow changes in the receiver gain and antenna parameters. The dependences of the reflection coefficient *η*_{e}^{2} (*ψ*) on the grazing angle *ψ* for the cases of the RHCP and LHCP circular polarizations of a GPS RO receiving antenna are shown in Figures 6 (left) and 6 (right), respectively, for the case of LEO satellite deployed at a circular orbit with the height 800 km. It is assumed that a transmitter emits radio waves with RHCP polarization. Curve 1 corresponds to ice (ɛ = 3.0), curves 2 and 3 are related to dry and wet clay (ɛ = 4.0+j0.4, ɛ = 15.0+j5.4) with 3% and 15% moisture content, respectively, and curve 4 represents the sea surface (ɛ = 75+j52). The magnitudes of the permittivity were chosen according to *Klein and Swift* [1977], *Milekhin et al.* [1986], and *Lang et al.* [2008]. Sea surface and ice reflections correspond to most and least intensive reflections at LHCP polarization in the 40° … 90° interval and vice versa at RHCP polarization in the 2° … 30° range of the grazing angles. The contrast between the sea and ice is very high at both RHCP and LHCP polarization almost in the whole range of 0°…90° of the grazing angles. The contrast between the wet and dry clay (curves 2 and 3) is also high.

### 5. Eikonal Acceleration and Refractive Attenuation

- Top of page
- Abstract
- 1. Introduction
- 2. Phase Path, Refractive Angle, and Refractive Attenuation
- 3. Refractive Attenuation and Reflectivity Cross Section
- 4. Characteristics of the Direct and Reflected Signals
- 5. Eikonal Acceleration and Refractive Attenuation
- 6. Influence of the Boundary Layer on Parameters of the Reflected Signals in MIR/GEO RO Experiments
- 7. Conclusions
- Acknowledgments
- References

[18] Analysis of the CHAMP RO data demonstrates validity of the relationship (38) for direct signal. A typical example is shown in Figure 7 (RO session 0187, 20 November 2003). The vertical profiles of the phase path excesses (eikonals) measured at frequencies L1 and L2 are shown on a logarithmic scale by curves 1 and 2 (Figure 7, left). For convenience an artificial bias of 1 m was introduced into the eikonals. The eikonals L1 and L2 change from 2 to 4 m at 110 km height to 1000 m at 1 km height. Curve 3 describes the altitude dependence of the atmospheric contribution, isolated from the ionospheric impact by use of linear combination of the eikonals L1 and L2 (Figure 7, left). The atmospheric part of eikonal changes over a broad dynamic range from 0.8 mm at the 75 km altitude to 1 km near the Earth's surface (Figure 7, left). In contrast, the attenuation *X*_{a} changes in a steep dynamic range from 0.05 to 1.1 over the altitude interval from 2 km to 110 km (Figure 7, right, curve 1). Despite this difference, the relationship (38) indicates an intrinsic connection between the eikonal and attenuation of RO signals. This can be seen by comparing the attenuation *X*_{a} and refractive attenuation *X*_{p} computed from the amplitude and eikonal data (Figure 7, right, curves 1 and 2). In order to convert the phase data, the phase acceleration *a* has been estimated numerically according to equation (38) as the second derivative of the phase path excess with respect to time *t* by using a fixed time interval value of Δ*t* = 0.42 s for differentiation. The relationship (38) widens the applicable domain of RO method. In particular the RO method can be applied for estimating the total absorption of radio waves in the atmosphere [*Pavelyev et al.*, 2010]. The attenuation *X*_{a} (*t*) and refractive attenuation *X*_{p} (*t*) recalculated from the amplitude and phase data using equations (10) and (12) are shown by curves 1 and 2 in Figure 8 (right) (CHAMP RO session 0028, 20 November 2003). The smooth curves 3 in Figure 8 (right) indicate the altitude dependence of the refractive attenuation corresponding to the exponential altitude profile of the refractivity in the atmosphere with accounting for the total path absorption effects calculated by using the theoretical and experimental results reported by *Kislyakov and Stankevich* [1967] for decimeter-range radio waves. Excellent correspondence is observed between the functions *X*_{a} (*t*) and *X*_{p} (*t*) changing from 0 db at 40 km to −10 db and −15 db at 5 km (curves 1 and 2 in Figure 8, right). There is also a good correlation between the high-frequency part of variations in *X*_{a} (*t*) and *X*_{p} (*t*). The total absorption Y(h) can be found from a ratio Y(h) = X_{a}(h)/X_{p}(h). To obtain dependence Y(h) the vertical profiles of X_{a}(h) and X_{p}(h) (curves 1 and 2 in Figure 8, right) have been approximated by polynomials using a least squares method. In Figure 8 (left), the vertical profiles of the approximated and modeled refractive attenuations X_{p}(h), X_{a} (h) X_{m} (h), and the total absorption Y(h) found from experimental data and models are shown by curves 1–3 and 4, 5, respectively. At the altitudes between 14 and 30 km the vertical profiles X_{p}(h) and X_{a} (h)(curves 1 and 2) are nearly coinciding and have good correspondence with the standard profile X_{m} (h) (curve 3). Below 14 km altitude the vertical profiles X_{p}(h) and X_{a} (h) begin split (curves 1 and 2). The splitting obviously indicates an influence of the atmospheric total absorption, which in average is near to the values described by *Kislyakov and Stankevich* [1967], *Matyugov et al.* [1994], and *Pavelyev et al.* [1997]. The total atmospheric absorption changes between 0–30% in the altitude interval 14–5 km (Figure 8, left, curves 4, 5). The described results indicate a possibility of the RO total absorption measurements at GPS frequencies.

### 6. Influence of the Boundary Layer on Parameters of the Reflected Signals in MIR/GEO RO Experiments

- Top of page
- Abstract
- 1. Introduction
- 2. Phase Path, Refractive Angle, and Refractive Attenuation
- 3. Refractive Attenuation and Reflectivity Cross Section
- 4. Characteristics of the Direct and Reflected Signals
- 5. Eikonal Acceleration and Refractive Attenuation
- 6. Influence of the Boundary Layer on Parameters of the Reflected Signals in MIR/GEO RO Experiments
- 7. Conclusions
- Acknowledgments
- References

[19] Scheme of the bistatic MIR/GEO RO experiment [*Pavelyev et al.*, 1996] is shown in Figure 1. MIR/GEO team used a transmitter installed on the orbital station “MIR” (point L) and receivers installed on GEO satellites (point G) at *λ* = 32 and 2 cm (monochromatic signals at both frequencies). RO signals have been transmitted from GEO satellites to an Earth-based station for analysis. Open-loop phase tracking method was used for removing the Doppler effect due to orbital motion of the satellite and expected atmospheric influence. Then a spectral method was applied for detecting reflected signals in the RO data. The Doppler spectra of RO signal reflected from the surface of the Indian Ocean are shown in Figure 9 (left) (wavelength *λ* = 32 cm). The Doppler frequency (expressed in Hz) is shown at horizontal axis, the spectral intensity of the reflected signals (normalized to the spectral intensity of the direct signal) is given on the vertical axis. The data are shown in Figure 9 (left) in succession with the time interval between neighboring spectra equal to 1/3 s. From Figure 9 (left), the spectral power density distribution of the reflected signals depends on time and can be connected to the features in long-scale slope variations of the sea surface and physical properties of boundary layer of the atmosphere.

[20] Reflected signal has been observed also at wavelength 2 cm during MIR/GEO experiment (RO event 28 May 1998). Time-Doppler frequency story of signal reflected from the Aral Sea (MIR/GEO data, wavelength 2 cm) is shown in Figure 9 (right). A weak reflected signal (sharp strip above, near frequency 20 KHz) is observed together with the more intense direct signal (broad bright strip below near frequency 25 KHz). Difference of Doppler frequencies of the reflected and direct signals gradually diminishes as the time increases. The reflected signals disappear after passing of a specular point from Aral Sea because of the roughness of the land surface. Results of calculations of difference of the Doppler shifts of reflected and direct signals are given in Figure 10 (left). Curves 1–7 correspond to the refractivity *N*_{0} = 340 N units and vertical gradient of the refractivity index −80.2; −70.6; −59.7; −50.8; −43.5; −37.5; and −32.5 N units/km, respectively. Curve 8 corresponds to the case of absence of the atmosphere. The results of modeling are in good correspondence with the RO experimental data (Figure 10, left and right). The RO region located above the Indian Ocean at 31.2°S latitude and 68.2°E longitude. In Figure 10 (right), the dependence of the Doppler frequencies difference Δ*F* (*t*) (Hz) of the direct and reflected signals is shown as a function of time *t* for RO event 25 December 1990. Curve 1 describes the experimental data, curves 2, 3 correspond to two models of Δ*F* (*t*). The first model describes the function Δ*F* (*t*) for the case of free space propagation (curve 2). The second model shows the influence of real atmosphere with the magnitudes of parameters of boundary layer equal to *N*_{0} = 350 N units, = −38 (N units/km) (curve 3). As follows from Figure 10, the Doppler difference Δ*F* (*t*) depends mainly on the vertical gradient of the refractivity index near the radio shadow. Increasing in the vertical gradient causes slower diminishing of Δ*F* (*t*) near radio shadow. This effect also can be seen in Figure 10 (left) for the difference between the Doppler frequencies of reflected and direct signal Δ*F* (*t*) which has different slope depending on the vertical gradient of refractivity in the boundary layer. It follows from analysis of data shown in Figure 10 that temporal dependence of the Doppler difference Δ*F* (*t*) may be informative for evaluation of vertical gradient of the refractivity index in the boundary layer.

[21] From *Pavelyev et al.* [1996] the reflection coefficient depends on the vertical gradient of refractivity in the boundary layer. Theoretical dependence of the power of the reflected signals on the vertical gradient of the refractivity is given in Figure 11 (left). The central angle θ (expressed in degrees) is shown at the horizontal axis. Power attenuation of the reflected signal normalized relative to the free space level is shown on the vertical axis in Figure 11 (left). Curve 1 corresponds to the case when the atmosphere is absent. Curves 2–5 are related to the vertical gradients that are equal to = −35.4 N units/km; = −43.0 N units/km; = −50.0 N units/km; = −57.0 N units/km. The angular position of radio shadow is different for the indicated values of vertical gradient and changes, respectively, from 102.3° (absence of the atmosphere, = 0) to 103.5°; 103.75°; 103.9°; 104.1°. From Figure 11 (left) the power of the reflected signals near the boundary of the radio shadow is a function of the central angle and vertical gradient . Thus the amplitude of the reflected signal near the radio shadow is an indicator of the vertical gradient of the refractivity in the boundary layer of the atmosphere. In the decimeter range of radio waves, the atmospheric absorption along the path 1 is small and the total absorption coefficient *χ*_{d} is near to unity. However, the radio waves propagate along the path 2 through all altitudes in the atmosphere at near grazing angles and are more sensitive to the absorption effect. The reflection coefficient *η*^{2} (*t*) was computed from measured powers of the reflected and direct signals, *W*_{r} (*t*) and *W*_{d} (*t*). The measured reflection coefficients *η*^{2} (*t*) are compared with their theoretical (calculated) equivalents in Figure 11 (right). The values of the experimental reflection coefficients are shown on the vertical axis in Figure 11 (right). The elapsed time from the start of the session in seconds is indicated on the horizontal axis. Curves 1–3 represent the numerically calculated reflection coefficient in the presence of the atmosphere for *N*_{0} = 320 N units and = −42.0 N Units/km. These magnitudes of *N*_{0} and agree with the standard refractive index profile in the atmosphere [*Bean and Dutton*, 1966]. Curve 4 describes *η*^{2} (*t*) in the absence of the atmosphere. Curves 1, 2, 3, and 4 were calculated for a seawater conductivity of *σ*_{W} = 2.7, 4.0; and 7.0 mhos/m, respectively. In accordance with *Akindinov et al.* [1976], the real part of the permittivity was taken to be equal to 79. Curve 5 was constructed for *σ*_{W} = 4 mhos/m, with accounting for the total absorption in atmospheric oxygen. Curve 6 is an experimental one. The time instant when the MIR orbital space station entered the geometrical radio shadow relative to the geostationary satellite corresponds to an intersection of curve 4 with the horizontal axis (Figure 10, right). The experimental reflection coefficient reached a maximum in the radio shadow zone and is subjected to significant variations. Outside the radio shadow zone, curve 5 satisfactorily fits in average the experimentally measured reflection coefficient *η*^{2} (*t*). Comparison of curve 5 with curves 1–3 shows that the total absorption by the atmosphere over the measurement site was Γ = 4.8 ± 1.2 dB. This corresponds to an average absorption coefficient of 0.0096 ± 0.0024 dB/km for the path length of radio waves in the atmosphere equal to 500 km. This value closely agrees with the absorption in atmospheric oxygen as measured by the radio occultation technique in the work of *Kislyakov and Stankevich* [1967] and also with the theoretical and experimental values reported by *Matyugov et al.* [1994].