We discuss the relationships that link the observed fluctuation spectra of the amplitude and phase of signals used for the radio occultation sounding of the Earth's atmosphere, with the spectra of atmospheric inhomogeneities. Our analysis employs the approximation of the phase screen and of weak fluctuations. We make our estimates for the following characteristic inhomogeneity types: (1) the isotropic Kolmogorov turbulence and (2) the anisotropic saturated internal gravity waves. We obtain the expressions for the variances of the amplitude and phase fluctuations of radio occultation signals as well as their estimates for the typical parameters of inhomogeneity models. From the GPS/MET observations, we evaluate the spectra of the amplitude and phase fluctuations in the altitude interval from 4 to 25 km in the middle and polar latitudes. As indicated by theoretical and experimental estimates, the main contribution into the radio signal fluctuations comes from the internal gravity waves. The influence of the Kolmogorov turbulence is negligible. We derive simple relationships that link the parameters of internal gravity waves and the statistical characteristics of the radio signal fluctuations. These results may serve as the basis for the global monitoring of the wave activity in the stratosphere and upper troposphere.

For the first time the regular radio occultation (RO) monitoring of the Earth's atmosphere was
implemented with the aid of the low Earth orbiter (LEO)
Microlab-1, which was equipped with a receiver of highly stable GPS signals at
wavelengths of

The stability of GPS signals, complemented with its global coverage and high
vertical resolution, draws the attention of researchers to the study of
inhomogeneities in atmospheric refractivity in addition to the retrieval of
mean profiles

In the radio band, the amplitude fluctuations are much smaller than in the
visible band; therefore, the weak fluctuation theory may be applicable down
to altitudes of several kilometers. The main limitation is due to the
humidity fluctuations, whose role becomes significant in the troposphere. The
upper boundary of the measurable fluctuation of RO signals is about 30–35

The aims of this paper are to clarify the role of the two inhomogeneity types
and to evaluate their actual contributions in the amplitude and phase of RO
signals. Our analysis is based on the phase screen approximation and the weak
fluctuation theory. In the framework of these approximations, we obtain
simple analytical relationships for the variance of fluctuations of radio
signals for anisotropic and isotropic inhomogeneities. At this stage of our
study, we confined the analysis of experimental data to height range from 25
down to 4

For RO signal analysis, we employ the following approximations:

a two-component model of the 3-D spectrum of the atmospheric refractivity fluctuations;

the approximation of the equivalent phase screen;

the first-order approximation of the weak fluctuation (the Rytov approximation).

For the description of the wave propagation, we define the characteristics of
the random media by its 3-D spectrum of the relative fluctuations of
refractivity

Stellar occultations indicated that the atmosphere is characterized by two
types of density fluctuations: (1) large-scale anisotropic ones and (2) isotropic ones

Both components of the spectrum have a power-law interval with the power of

For

More discussion on the model

To obtain the value of the structure characteristic

For

Due to the exponential decay of air density with the altitude, a ray
propagating in the atmosphere is mainly affected by the vicinity of the ray
perigee, with the effective size along the ray of about several hundred
kilometers. The distance from the perigee to the LEO is much greater, about
3000

The amplitude fluctuations are considered weak if their variance is less than
unity

Because the velocity of the ray immersion in satellite observations is large compared to the atmospheric motions associated with the refractivity inhomogeneities, it is possible to apply the hypothesis of “frozen” inhomogeneities for mapping measured temporal spectra of signal fluctuations into spatial spectra.

The approximations of phase screen and weak fluctuations allow deriving simple expressions for the statistical moments of RO signal fluctuations. In this section, we will discuss the relationships that link the fluctuations of RO signals with those of atmospheric refractivity for IGW and turbulence models as well as the mean profiles of variances of RO signal fluctuations.

For a satellite-to-satellite path, using the approximations of phase screen
and weak fluctuation, it is possible to derive the following 2-D correlation
functions in the observation plane (

Taking the Fourier transform, we arrive at the following expressions for the
2-D fluctuation spectra of the received signal:

In the general case, the relationship between the 2-D spectrum of the eikonal
fluctuations in the phase screen

Formula (

Moderate anisotropy

Strongly anisotropic inhomogeneities

The variance of the logarithmic amplitude fluctuation is determined by the
scales of the order of the Fresnel zone

In this case, the vertical fluctuation spectrum of

In the observations, we obtain a 1-D realization of the signal along the
receiver trajectory. During a RO event, the changes of the satellite
positions are small with respect to their distance from the phase screen.
Moreover, the fluctuation correlation scale along the ray significantly
exceeds the correlation scale in the transverse direction

For isotropic inhomogeneities, the characteristic frequencies are determined
by the corresponding scales and oblique velocity

For the fluctuations of logarithmic amplitude in both models of the 3-D
spectrum of inhomogeneities, the principle scale is the Fresnel scale

In the case of strongly anisotropic inhomogeneities,

For a moderate anisotropy,

In order to analyze the influence of anisotropy upon amplitude fluctuations,
consider the ratio of Eqs. (

The main contribution into phase fluctuations comes from inhomogeneities with
vertical scales close to the outer scale. Therefore, it is possible to use
the geometric optical approximation for Eqs. (

For the variance of phase fluctuations for strong anisotropy

For a moderate anisotropy

For a strong anisotropy,

For the case

For the case of a strong anisotropy

For isotropic inhomogeneities,

The profiles were evaluated for a GPS–LEO system with orbit altitudes of
20 000 and 800

The structure characteristic of the relative fluctuations of refractive index
was specified for the model of saturated IGWs in a dry atmosphere, according
to Eq. (

RMS fluctuations of logarithmic amplitude, eikonal, incident angle, and single-point correlation of logarithmic amplitude and phase for the model of saturated IGWs (blue lines) and for the model of turbulence (red lines).

For the isotropic turbulence at heights of 4–15

Figure

For the turbulence model, we assumed the outer scale to be equal to 1

It is known that local profiles of atmospheric inhomogeneities exhibit large
natural variability. Furthermore, even their average profiles significantly
vary depending on latitude, season, orography, and region. The turbulence
structure characteristic for different observations, even in a free
atmosphere, may vary by up to 2 orders of magnitude

The most important difference between turbulence and saturated IGWs is the
anisotropy of the latter. The variances of RO signal parameter fluctuations,
being single-point characteristics, do not contain an immediate information
on the anisotropy of the 2-D field of RO signal fluctuation in the observation
field. This information can be extracted from an ensemble of 1-D spectra of RO
signal fluctuations measured at different obliquity angles, when categorized
according to frequency or to vertical wave number. For turbulence, due to its
isotropy, the fluctuation frequencies

Amplitude fluctuation spectra for the lower stratosphere:

Amplitude fluctuation spectra for the upper troposphere:

Figures

The amplitude fluctuation spectra are represented as the product of
wave number and spectral density, normalized to the variance. Such a product
will be hereinafter referred to as the spectrum, as distinct from the
spectral density. The spectra indicate a maximum corresponding to the Fresnel
scale. The theoretical spectra for both inhomogeneity types have asymptotics
with a slope of

Figures

The measured RMS values of the relative fluctuations of the amplitude in the
stratosphere are 0.08–0.20, which is in a fair agreement with the IGW model
(Fig.

The normalized eikonal fluctuation spectra for the lower
stratosphere:

The normalized eikonal fluctuation spectra for the upper
troposphere.

The phase in RO observations is presented as the excess phase, which equals
the difference between the full eikonal and the straight-line
satellite-to-satellite distance. We will refer to the excess phase as the
eikonal. Double-frequency observations allow for the exclusion of the
ionospheric component of the eikonal under the assumption that the
trajectories of the two rays coincide. The ionospheric corrected eikonal
consists of two components: (1) the neutral atmospheric eikonal evaluated as
the integral along a straight ray, which in Sect. 3 was denoted as

Figures

For the stratosphere, the measured RMS values of the eikonal fluctuations are
3–10

The atmospheric inhomogeneity models have not only different anisotropy but
also a different slope,

Nevertheless, Figs. 2–5 indicate that the diffractive decays of the experimental spectra are in a better agreement with the IGW model, as compared to the turbulence model.

In this study, we discussed the 3-D spectra of atmospheric inhomogeneities of two types: (1) isotropic Kolmogorov turbulence and (2) anisotropic saturated IGWs. For RO observations, in the approximations of the phase screen and weak fluctuations, we derived the relationships that link the observed 1-D fluctuation spectra of the amplitude and phase with empirical 3-D inhomogeneity spectra. This allowed us to obtain the analytical expressions for the variances of the amplitude, phase, and ray incident angle fluctuations as well as the single-point amplitude-phase correlation for both inhomogeneity types. The theoretical estimates of the variances of RO amplitude and phase fluctuations for different values of the parameters of atmospheric inhomogeneity model, including the structure characteristics and vertical scales, for middle latitudes in the stratosphere and upper troposphere, indicate that the major contribution into RO signal fluctuations comes from saturated IGWs. The contribution of the Kolmogorov turbulence, under these conditions, is small. Even taking into account a significant spread of possible values of the structure characteristics and typical scales of inhomogeneities, it is hard to expect that this can compensate the difference between the IGWs and turbulence in this altitude range. Moreover, the averaging of RO signal fluctuations along the whole ray inside the atmosphere damps the influence of intermittence, which is typical for turbulence under stable stratification conditions.

For anisotropic inhomogeneities we employ an empirical model of saturated
IGWs (Eq.

The use of the simple model (

Following the ideas of

Joint observations of the amplitude and phase of RO signals open new pathways
in the development and application of radio holographic methods. These
methods allow enhancing the retrieval accuracy and resolution

From GPS/MET data acquired on 15 February 1997, we evaluated the variances
and spectra of the relative fluctuations of amplitude and the fluctuations of
phase for the lower stratosphere, comprising the altitudes from 25

In comparison with the visible band, the radio band is characterized by a
much greater Fresnel scale

Satellite observations of stellar occultations indicate that in the visible
band, at the perigee height about 30

The statistical analysis of eikonal fluctuations is aggravated by the fact
they are nonstationary, and one of the main problems is the determination of
the mean profile. We evaluated the eikonal spectrum using two different mean
profiles: (1) the model profile and (2) the profile obtained by the sliding
averaging of the eikonal profile over an altitude windows with a half-width
of

For strongly anisotropic inhomogeneities, RO signal fluctuations are
determined primarily by the vertical structure of inhomogeneities and,
accordingly, by the vertical velocity of the ray immersion for different
obliquity angles

The main result of this study consists in the statement that at altitudes
above 4–5

In contrast,

Along with a wide spectrum of saturated IGWs, separate quasi-monochromatic
perturbations are detected from spikes in stellar scintillation spectra

In this study, we presented simple relationships and theoretical estimates of
the amplitude and phase variances of RO signal for typical parameters of 3-D
spectra based on two models: (1) the Kolmogorov turbulence and (2) saturated
IGWs. For GPS/MET observations in the
altitude range of 4–25

The code used in this study does not belong to the public domain and cannot be distributed.

GPS/MET radio occultation data are freely available. To get
access to them, it is necessary to sign up at the website of the CDAAC:

The authors declare that they have no conflicts of interest.

The work of Valery Kan and Michael E. Gorbunov was supported by the Russian Foundation for Basic Research, grant 16-05-00358. The work of Viktoria F. Sofieva was supported by the EU project GAIA-CLIM. We are very grateful to the two reviewers for their appropriate and constructive suggestions. Edited by: William Ward Reviewed by: two anonymous referees