Search

The method of reconstructing the global distribution of statistical parameters of internal gravity waves (IGWs) in the atmosphere has been tested using the data of amplitude fluctuation measurements of a radio signal in satellite radio occultation (RO) observations. In our previous work, for the chosen model of the spatial IGW spectrum, its relationship with the spectra of amplitude fluctuations was obtained, an algorithm for reconstructing the model parameters was developed, and the reconstruction errors were estimated. The reconstructed parameters of the vertical IGW spectrum are the external scale, which separates the unsaturated large-scale waves from saturated small-scale waves, and the structural characteristic, which determines the spectral amplitude of saturated waves. These parameters are used to calculate the variance of temperature fluctuations and the potential energy of the waves. This article presents the altitude–latitude distribution of IGW parameters in the stratosphere according to the measurements of the COSMIC experiment in 2011. The characteristic features of these distributions are noted, and the results are compared with the data of other measurements.

In this paper, we discuss a method for reconstructing the parameters of internal gravitational waves (IGWs) in the stratosphere from amplitude fluctuations in satellite radio-occultation observations. In the formation of radio-signal fluctuations, IGWs play the main role in these observations, while the contribution of isotropic turbulence is negligible. We consider methodological issues: (1) the choice of a model for the spatial spectrum of internal waves, (2) the derivation of relations connecting the statistical parameters of the IGW spectra and amplitude fluctuations in the approximation of a phase screen and weak fluctuations, (3) the development of a reconstruction algorithm, and (4) the estimation of possible errors. The reconstructed parameters are the outer or dominant scale and the structural characteristic of the vertical IGW spectrum, which determines the spectral amplitude in the saturation mode. The operating range covers altitudes from 28 km to the upper boundary of the tropopause. The error estimates for the reconstruction algorithm are 10–20% for the outer scale and 20–40% for the structural characteristic.

The phase-screen (split-step) method is widely used for modeling wave propagation in inhomogeneous media. The method of plane phase screens is best known. However, for modeling a 2D problem of radio occultation sounding of the Earth’s atmosphere, the method of cylindrical phase screen was developed many years ago. In this paper, we propose a further generalization of this method for the 3D problem on the basis of spherical phase screens. In the paraxial approximation, we derive the formula for the vacuum screen-to-screen propagator. We also infer the expression for the phase thickness of a thin layer of aisotropic random media. We describe a numerical implementation of this method and give numerical examples of its application for modeling a diverging laser beam propagating on a 25-km-long atmospheric path.

We develop a method for the reconstruction of the statistical parameters of internal gravity wave (IGW) in the atmosphere from the amplitude fluctuations of signals acquired in radio occultation observations. We discuss the choice of the model of the IGW spatial spectrum, derive the relationships between the IGW spectra and the amplitude fluctuation spectra, develop the reconstruction algorithm and estimate its uncertainties. The IGW spectrum parameters to be retrieved are the outer (dominant) scale and the structure characteristic, the latter determining the power of saturated IGW. The method was tested on the COSMIC (Constellation Observing System for Meteorology, Ionosphere and Climate) observations acquired in the winter and summer seasons 2011. The IGW parameters were reconstructed in the height range from the upper boundary of the tropopause to 28 km. We show the altitude-latitudinal distributions of the IGW parameters and two integral characteristics: the variances of the temperature fluctuations and the IGW potential energy. We perform a detailed comparison of our results with those obtained from radiosonde and radio occultation observations. The developed method can be applied for the global monitoring of IGW parameters and activity in the middle atmosphere.

On September 23, 1968, the Cosmos-243 satellite was launched into orbit with four radio telescopes directed to the nadir on board. They were designed to measure the microwave radiation of the Earth’s surface and its atmosphere at wavelengths of 0.8, 1.35, 3.4, and 8.5 cm. The onboard infrared radiometer measured radiation in the band of 10–11 µm in the same solid angle as the radio telescopes. This experiment, which was initiated by scientists from the Institute of Radioengineering and Electronics (IRE) and Institute of Atmospheric Physics (IAP) and, in particular, academicians V.A. Kotel’nikov and A.M. Obukhov, broke new ground in the remote sensing of the Earth from space, which is being actively developed.

A brief description of the MTVZA-GYa microwave radiometer installed on the Meteor-M No. 2 Russian polar orbiting meteorological satellite is presented. The procedures of bias correction and quality control indispensable in satellite data assimilation are considered. The operational system of global data assimilation by the Hydrometcenter of Russia is briefly described. Results of numerical experiments on the assimilation of MTVZA-GYa observations in six temperature-sensitive channels are given. It is demonstrated that the assimilation of these data significantly improves the accuracy of short-range weather forecasts in the Southern Hemisphere. The effect of the use of MTVZA-GY data in the Northern Hemisphere is neutral.

A new reference occultation processing system (rOPS) will include a Global Navigation Satellite System (GNSS) radio occultation (RO) retrieval chain with integrated uncertainty propagation. In this paper, we focus on wave-optics bending angle (BA) retrieval in the lower troposphere and introduce (1) an empirically estimated boundary layer bias (BLB) model then employed to reduce the systematic uncertainty of excess phases and bending angles in about the lowest 2 km of the troposphere and (2) the estimation of (residual) systematic uncertainties and their propagation together with random uncertainties from excess phase to bending angle profiles. Our BLB model describes the estimated bias of the excess phase transferred from the estimated bias of the bending angle, for which the model is built, informed by analyzing refractivity fluctuation statistics shown to induce such biases. The model is derived from regression analysis using a large ensemble of Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) RO observations and concurrent European Centre for Medium-Range Weather Forecasts (ECMWF) analysis fields. It is formulated in terms of predictors and adaptive functions (powers and cross products of predictors), where we use six main predictors derived from observations: impact altitude, latitude, bending angle and its standard deviation, canonical transform (CT) amplitude, and its fluctuation index. Based on an ensemble of test days, independent of the days of data used for the regression analysis to establish the BLB model, we find the model very effective for bias reduction and capable of reducing bending angle and corresponding refractivity biases by about a factor of 5. The estimated residual systematic uncertainty, after the BLB profile subtraction, is lower bounded by the uncertainty from the (indirect) use of ECMWF analysis fields but is significantly lower than the systematic uncertainty without BLB correction. The systematic and random uncertainties are propagated from excess phase to bending angle profiles, using a perturbation approach and the wave-optical method recently introduced by Gorbunov and Kirchengast (2015), starting with estimated excess phase uncertainties. The results are encouraging and this uncertainty propagation approach combined with BLB correction enables a robust reduction and quantification of the uncertainties of excess phases and bending angles in the lower troposphere.

The retrieval of profiles of meteorological variables from radio occultation observations requires knowledge of bending angle profiles up to heights of no less than 60–70 km. Because of the residual error of the ionospheric correction, retrieved profiles become too noisy by a height of about 40 km. In order to invert the bending angle profiles, the statistical optimization is used. This makes it possible to construct an optimal linear combination of the a priori estimate of the average bending angle profile and a posteriori noisy estimate based on observations. The estimate of the average bending angle profile for the given coordinates and the time of year is usually based on the climatological atmospheric model. MSIS and CIRA models that have been used for this purpose are now obsolete and do not describe the global changes in the atmospheric state. The model of average bending angles BA–IAP (Bending Angle–Institute of Atmospheric Physics) is built based on the processing of the array of COSMIC radio occultation observations during 2006–2013. The proposed model is statistically validated based on the COSMIC database. It is shown that our model describes the average bending angle profiles more accurately than the MSIS model.

Based on more than 4500 sessions of radio transillumination of Earth’s atmosphere along the satellite–atmosphere–satellite path obtained in the COSMIC experiment, the distribution along latitude and over local time of the spatial spectra of variations in the ionospheric phase delay and signal amplitude has been analyzed. The spatial spectra have been calculated for two height ranges, i.e., 60–80 and 80–100 km. In the phase signal spectrum within the height range 80–100 km, the second maximum in the vicinity of a frequency of 7–8 rad/km is clearly seen. Its diurnal and latitudinal behavior and its decrease towards high latitudes in both hemispheres can also be seen. In the height range of 60–80 km, this maximum is hardly observed. Although solar flares can lead to substantial local changes in the electron concentration, no substantial difference in the behavior of the spectral densities of the amplitude and phase delay at long limb paths was observed within these two height ranges on days of active and quiet sun. The latter fact makes it possible to develop a united algorithm of optimal ionospheric correction of the radio occultation data independent of solar activity.

Radio occultation sounding method were invented as a way of researching of Solar system planetary atmospheres. After it was applied to analysis of the Earth's atmosphere and it is still effective research instrument.

We present the results of processing and analysis of more than 4500 events of radio occultation sounding of the Earth’s atmosphere observed in the course of the COSMIC experiment on the limb path ‘satellite-ionosphere-satellite’. Events observed in December 2011 (when a number of solar flares occurred) and in January 2012 (when a strong solar proton event took place) were analyzed. It is shown that small-scale variations of electron density increase in polar latitudes, equatorial region, and midlatitudes of the southern hemisphere in January 2012. In the same period, an increase of large-scale variations of electron density is observed during daylight hours in the equatorial region and in the southern hemisphere. No noticeable distinctions in comparison with days of quiet Sun were observed in December 2011.

Asymptotic methods of calculating the propagation of centimeter radio waves in a neutral atmosphere in space-space paths are considered. The methods are based on the technique of Fourier integral operators. The approximations that allow the representation of the corresponding operators as compositions of nonlinear coordinate changes, multiplications by reference signals, and Fourier transformations are constructed. The approximations are based on a technical procedure using a linearized canonical transform. This approach makes it possible to devise fast numerical algorithms. Numerical simulations are conducted with the use of realistic global gridded fields of meteorological parameters including the turbulence. The numerical simulations show high accuracy and efficiency of the proposed methods.

The differences between the refraction angles measured and calculated for the reanalyses of the European Centre for Medium-Range Weather Forecasts were statistically analyzed on the basis of 64 radio occultation events recorded by the Microlab-1 satellite. It is shown that, for minimum ray heights below 20 km, the main contribution to the differences is made by spatial refractive-index fluctuations neglected by the model. The power spectral density of these fluctuations is mainly concentrated within the vertical wave-number range 0.5–10 rad/km. For heights above 30 km, the deviations are mainly determined by ionospheric disturbances and may vary several times during changes of the site and time of observations. This suggests that the results of satellite radio-occultation sounding of the neutral atmosphere can be used as an indirect quantitative estimate of local discrepancies between the actual field of the refractive index and its values calculated on the basis of a hydrodynamic atmospheric general circulation model.

We investigate the capabilities of the back-propagation and radio-holographic methods for the interpretation of data of radio occultation sounding of ionospheric perturbations in regions of sporadic E-layers. Several occultations of Microlab-1 satellite are analysed, where multipath propagation from E-layers is detected. It is shown that the back-propagation technique can be used for processing radio occultation data in multipath zone in order to derive vertical profiles of the refraction angle. The holographic technique allows for the detection of complicated horizontal structures of ionospheric inhomogeneities, where the standard Abel inversion technique cannot be used for the derivation of vertical profiles of electron concentration.

This paper describes the results of the statistical comparison of the inversions of GPS/MET radio occultation data with analysis data of the European Centre for Medium-Range Weather Forecast (ECMWF). The Prime Time 4 period (February 2-16, 1997) GPS/MET data were analyzed. The data analysis algorithms include the derivation of refraction angles from the measured phase excess, the back-propagation method for handling data in multipath regions, ionospheric correction and noise reduction, and the Abel inversion of the refraction angle profiles and the derivation of dry temperature. We use a forward model of radio occultation experiments in order to produce artificial occultation data for the ECMWF global fields of atmospheric variables. The forward model exists in two variants, which are geometric optics propagator and wave optics propagator. The wave optics propagator allows for the simulation of diffraction effects and multipath propagation, which made this operator the choice for the validation. The artificial occultation data are processed by exactly the same inversion algorithm as the GPS/MET data. This allows for the adequate comparison of the retrieved temperatures. For regions with dense observational networks (Europe, China, and United States), the comparison between the satellite data inversions and the model analysis fields is characterized by biases inside 0.5 K and mean square deviations of 1.5-2 K. In the Southern Hemisphere, where observational data are sparse, the bias can reach 2 K, and the mean square deviation is 2-3 K. Because the quality of GPS/MET data must not be different in different hemispheres, the difference in the bias is assumed to be based on the used ECMWF analysis data.

Two methods for the determination of refraction angle profiles from radio occultation measurements in multipath areas are analyzed and compared: (1) the radio-optic method based on the analysis of the local spatial spectra of the measured wave field and (2) the back propagation of the received wave field to a single-ray region. The basic limitations of the radio-optic method are (1) the restriction of the resolution of refraction angle profiles due to the uncertainty relation of refraction angle and impact parameter and (2) diffractive effects in subcaustic areas, where the spatial spectra cannot be interpreted in terms of geometric optical rays. The basic limitation of the back propagation method is related to ray and caustic structures, which may not contain single-ray areas. It is shown that strong refraction reduces the uncertainties of refraction angle and impact parameter. On the other hand, strong refraction or superrefraction is responsible for complicated caustic structures that cannot be resolved by the back propagation technique. The two methods are thus complementary to each other and can be combined for processing lower tropospheric occultation data. This analysis is corroborated by numerical simulations based on global fields of atmospheric variables from analyses of the European Centre for Medium Range Weather Forecast.

The first results of the GPS/MET programme of sounding the Earth's atmosphere by means of a Low Earth Orbit satellite implemented with a Global Positioning System (GPS) receiver indicated that in the lower troposphere, the phase and amplitude of the signal received undergo strong oscillations. This effect is a consequence of multipath propagation resulting from the complicated structure of the atmospheric refractivity in the tropical regions. In this paper, we propose a method of processing of the phase measurements based on the theory of diffraction. The method was tested in processing GPS/MET radio soundings. Comparison of the reconstructed profiles of the temperature and humidity with those derived from radio soundings and extracted from the NMC Global Atmospheric Circulation Model indicates validity of the method proposed.

Refractometric sounding of the Earth's atmosphere using a multiorbit system of Global Positioning System (GPS) and low-Earth orbit (LEO) satellites is modeled on the basis of data from the model of global atmospheric circulation (ECHAM3). The errors of the tomographic reconstruction of meteorological fields are investigated because of insufficient measured refractometric data. For a system of 100 LEO and 18 GPS satellites the estimates of errors of geopotential and temperature is obtained, which are 3—5 m and 0.3 K, respectively, in the pressure range 10—200 mbar.

Perturbation theories for the rays and for the eikonal are discussed. It is shown that these theories give completely equivalent results in the calculation of any arbitrarily chosen value. The problem of refraction in an arbitrary inhomogeneous medium is considered as an example.

[1] Microwave occultation using centimeter- and millimeter-wave signals between low Earth orbit (LEO) satellites (LEO microwave occultation, LMO) is an advancement of GPS radio occultation (GRO) exploiting in addition to refraction also absorption of signals. Beyond the successful GRO refractivity profiling capability, which leaves a temperature-humidity ambiguity in the troposphere where moisture cannot be neglected, LMO enables joint retrieval of pressure, temperature, and humidity profiles without auxiliary background information. Here we focus on the upper troposphere/lower stratosphere and advance the LMO method in two ways: (1) we introduce a new retrieval algorithm for processing LMO excess phase and amplitude data from multiple frequencies, complementing existing GRO retrieval algorithms, and (2) we employ the algorithm in an ensemble-based end-to-end performance analysis and assess the accuracy of pressure, temperature, and humidity profiles retrieved from the LMO data. The end-to-end simulations were carried out under quasi-realistic conditions for a day of LEO-LEO occultation events, based on a high-resolution atmospheric analysis of the European Centre for Medium-Range Weather Forecasts (ECMWF) and accounting for scintillation noise from turbulence and instrumental errors. The new algorithm was found robust, fast, and versatile to adequately process LMO data under all conditions from dry and clear to moist and cloudy air as contained in the ECMWF analysis. The retrieved pressure, temperature, and humidity profiles were generally found unbiased and within target accuracy requirements, set by scientific objectives of atmosphere and climate research going to be supported by the data, of

The paper contains a description of the numerical simulation of remote sensing of the Earth's atmosphere using a system of synchronously orbiting satellites based upon methods of refractional tomography. Materials of the Global Atmospheric Research Program were used for the simulations. It vaas shown that a system of 8 or 12 satellites linked to each other with communication lines passing through the atmosphere potentially allows reconstruction of the pressure field with sufficient precision for meteorology

[1] We investigated amplitude fluctuations and differential transmission errors for radio occultation (RO) measurements performed in X/K band for Low-Earth Orbiter – Low-Earth Orbiter (LEO–LEO) cross-links. We performed a series of high-resolution numerical simulations using a quasi-realistic anisotropic turbulence model with European Centre for Medium-Range Weather Forecast (ECMWF) ERA40 reanalyses as the background fields. Three test cases were simulated: high, middle, and low latitudes, with turbulence intensity profile being estimated based on high-resolution radio sonde data. The numerical end-to-end simulations were based on the multiple phase screens technique (forward modeling) and the canonical transform amplitude ratioing method (differential transmission retrieval). The anisotropy coefficient was varied from 3 to 50. We found the dependences of the scintillation index on anisotropy in the area of weak fluctuations to be consistent with previous theoretical studies: scintillation index is approximately proportional to the square root of the anisotropy coefficient. For strong fluctuations this dependence becomes weaker. The type of dependence of differential transmission errors from anisotropy is more sensitive to the fluctuation strength. Generally, it increases with anisotropy, but the dependence saturates for stronger anisotropy and becomes flat or even slightly reverse. At all fluctuation levels, including the strongest ones, the differential transmission error is found smaller than 5% (smaller than 1–2.5% for all weak and moderately strong fluctuations) for ∼1 km height resolution, reinforcing the results of the previous studies that X/K band RO phase delay and transmission are a promising future source for accurate temperature and humidity profiling.

[1] The multiple phase screens technique is often used for modeling wave propagation and radio occultation sounding of the atmosphere. The last step of this procedure is the propagation from the last phase screen to the observation orbit of the spaceborne receiver. This step was formerly performed by the computation of multiple diffractive integrals, which impairs the numerical efficiency of the algorithm. We introduce an asymptotic method of wave propagation in vacuum from the last phase screen to a generic observation path. The exact solution is written in the form of the Zverev Transform, which belongs to the class of Fourier Integral Operators (FIO). The phase function of such an operator can be derived from the geometric optical equations. We construct an approximation for the Zverev Transform, utilizing the linearization of the equation of the geometric optical ray propagation in the vicinity of a smooth model of the ray structure. This permits the design of a fast numerical algorithm based on an FFT. Numerical simulations are performed for microwave propagation through a model atmosphere based on realistic gridded global fields of meteorological parameters. The new method based on the Linearized Zverev Transform is compared with the standard combination of multiple phase screens and diffractive integrals and with the asymptotic forward modeling for the propagation of centimeter waves in the atmosphere on limb paths. This method is especially useful for modeling LEO–LEO microwave occultations at frequencies of 9–30 GHz. The simulations demonstrate a high accuracy and numerical efficiency of the proposed method.

We investigate algorithms for filtering noisy radio occultation data with atmospheric multipath behavior in order to improve the accuracy of the inversion based on canonical transform/full spectrum methods. The noise filtering procedure uses the compression of the signal spectrum by multiplying it with a reference signal, Fourier filtering of the narrow-banded signal, and decompression of its spectrum.We study filtering in the time domain and in the impact parameter domain. Our results show that the inversion methods are able to handle additive white noise and high resolution bending angle profiles are obtained. For comparison we also present results where the inversion of multipath behavior is based on using phase data only.

Canonical Transform (CT) and Full-Spectrum Inversion (FSI) methods can be used in processing radio occultation (RO) data to retrieve transmission profiles, which are necessary for the retrieval of atmospheric humidity. LEO-LEO occultations with X/K band frequencies can provide transmission data along the wing of the 22 GHz water vapor absorption line. The retrieval of transmission requires spherical symmetry of the atmospheric refractivity. This condition is broken in presence of turbulence, which can result in significant errors in transmission. We suggest the computation of the differential transmission from the differential CT/FSI-amplitude to correct for the effect of turbulence and horizontal gradients. The efficiency of the method is tested by realistic numerical end-to-end simulations. We modeled turbulence based on power form of the spectrum. The simulations demonstrate that the error in the retrieved transmission can be significant in a single frequency channel, while the differential transmission can be retrieved with high accuracy.

We discuss the recent achievements in the application of asymptotic methods based on Fourier Integral Operators (FIOs) for inversion and direct modeling of radio occultations. We show that FIOs can be derived from the first principles: stationary phase principle and energy conservation. We discuss accurate and approximate solutions for the kernel of the FIOs. The approximations can be used for designing very efficient numerical algorithms, where the FIOs are reduced to a composition of multiplying with reference signals and Fourier transforms. Another application is inversion algorithms using an FIO that retrieves the geometric optical ray structure of wave fields. Asymptotic methods of forward modeling are based on inverse FIOs that map the geometric optical ray structure to wave fields. Such algorithms are very fast and significantly reduce numerical inaccuracies, which arise in computation of multiple diffractive integrals.

[1] Processing of radio occultation data requires filtering and quality control for the noise reduction and sorting out corrupted data samples. We introduce a radio holographic filtering algorithm based on the synthesis of canonical transform (CT2) and radio holographic focused synthesized aperture (RHFSA) methods. The field in the CT2-transformed space is divided by a reference signal to subtract the regular phase variation and to compress the spectrum. Next, it is convolved with a Gaussian filter window and multiplied by the reference signal to restore the phase variation. This algorithm is simple to implement, and it is numerically efficient. Numerical simulation of processing radio occultations with a realistic receiver noise indicates a good performance of the method. We introduce a new technique of the error estimation of retrieved bending angle profiles based on the width of the running spectra of the transformed wavefield multiplied with the reference signal. We describe a quality control method for the discrimination of corrupted samples in the L2 channel, which is most susceptible to signal tracking errors. We apply the quality control and error estimation techniques for the processing of data acquired by Challenging Minisatellite Payload (CHAMP) and perform a statistical comparison of CHAMP data with the analyses of the German Weather Service (DWD). The statistical analysis shows a good agreement between the CHAMP and DWD error estimates and the observed CHAMP–DWD differences. This corroborates the efficiency of the proposed quality control and error estimation techniques.

[1] Canonical transform (CT) and full-spectrum inversion (FSI) methods can be used in processing radio occultation data to retrieve transmission profiles, which are necessary in addition to bending angle profiles for the retrieval of atmospheric humidity independent of temperature. Low Earth orbiter–low Earth orbiter cross-link occultations with X/K band frequencies can provide transmission data along the wing of the 22 GHz water vapor absorption line. The computation of transmission assumes spherical symmetry of the atmospheric real and imaginary refractivity. This condition is broken in the presence of turbulence, which can result in significant errors in transmission. We suggest the computation of the differential transmission from the differential CT/FSI amplitude to correct for the effect of turbulence. The efficiency of the method is tested by quasi-realistic numerical end-to-end simulations. We explicitly modeled turbulence as random refractivity fluctuation field based on power form of the fluctuation spectrum. The simulations demonstrate that the error in the retrieved transmission can be significant in a single frequency channel, while the differential transmission can be retrieved with high accuracy. The new method does not impose any significant restriction on the frequency difference between the channels and provides very good differential transmission sensitivity if spacings of a few gigahertz are chosen (e.g., 10 and 17 GHz or 17 and 20 GHz).

[1] We analyze and compare two approaches to processing radio occultation data: (1) canonical transform method and (2) full spectrum inversion method. We show that these methods are closely related and can be explained from two view points: (1) both methods apply a Fourier transform like operator to the entire radio occultation signal, and the derivative of the phase of the transformed signal is used for the computation of bending angles, and (2) they can be explained from a signal processing view point as the location of multiple tones constituting the complete signal. The full spectrum inversion method is a composition of phase correction and Fourier transform, which makes the numerical algorithm computationally more efficient as compared to the canonical transform method. We investigate the relative performance of the two methods in simulations using a wave optics propagator. We use simple analytical models of the atmospheric refractivity as well as radiosonde data in order to reproduce complex multipath situations. The numerical simulations as well as the analytical estimations indicate that a resolution of 60 m (or even higher) can be achieved.

[1] Fourier integral operators (FIOs) are used for constructing asymptotic solutions of wave problems and for the generalization of the geometrical optics. Geometric optical rays are described by the canonical Hamilton system, which can be written in different canonical coordinates in the phase space. The theory of FIOs generalizes the formalism of canonical transforms for solving wave problems. The FIO associated with a canonical transform maps the wave field to a different representation. Mapping to the representation of ray impact parameter was used in the formulation of the canonical transform (CT) method for processing radio occultation data. The full-spectrum inversion (FSI) method can also be looked at as an FIO associated with a canonical transform of a different type. We discuss the general principles of the theory of FIOs and formulate a generalization of the CT and FSI techniques. We derive the FIO that maps radio occultation data measured along the low Earth orbiter orbit without first applying back propagation. This operator is used for the retrieval of refraction angles and atmospheric absorption. We give a closed derivation of the exact phase function of the FIO obtained in the “phase matching” approach by Jensen et al. [2004] We derive a novel FIO algorithm denoted CT2, which is a modification and improvement of FSI. We discuss the use of FIOs for asymptotic direct modeling of radio occultation data. This direct model is numerically much faster then the multiple phase screen technique. This is especially useful for simulating LEO-LEO occultations at frequencies of 10–30 GHz.

In the present work, we study a canonical transform that directly maps the measured field to the impact parameter representation without first carrying out a back-propagation. This canonical transform is determined to first order in a small parameter that measures the deviation of the satellite orbit from a circle. When the parameter is equal to zero, i.e., for circular orbits, our canonical transform reduces to a Fourier transform. In the general case, the form of the generating function is such that it does not directly allow an implementation as an FFT-like algorithm. However, using approximations the direct canonical transform mapping yields fast, efficient numerical implementations.

Wave optics algorithms are applied for processing radio occultation data, especially in the lower troposphere in order to reconstruct the geometric optical ray structure of the wave field. Wave field diffracted by strong inhomogeneities in the lower troposphere and by long propagation distance from the planet limb to the space-borne receiver has a complicated structure. Its amplitude and phase are determined by multiple interfering rays and diffraction effects on caustics. There are two groups of the methods of the interpretation of radio occultation data. The first group (radio holographic technique) uses short term Fourier analysis of the wave field multiplied by the reference signal, in small sliding apertures. The maxima of local spectra localize rays. The second group uses a global Fourier Integral Operator, which transform the wave field to a single-ray representation. These method include back propagation and canonical transform methods. Refraction angles are then computed from the phase of the transformed wave field. The both groups of the methods are complementary to each other. Short term Fourier analysis is convenient for data visualization and identification of different kinds of problems in the data. The analysis based on Fourier Integral Operators is facilitated for operational data processing. We show examples of processing simulated radio occultation data using the radio holographic and canonical transform techniques.

[1] We performed statistical comparisons of data from Challenging Minisatellite Payload (CHAMP) with European Centre for Medium-Range Weather Forecasting (ECMWF) analyses. Our data processing technique includes: (1) radio holographic (radio optics) and canonical transform analysis of wave fields, (2) visualization of the local spatial spectra of wave fields for interactive data quality control, (3) ionospheric correction and noise reduction based on statistical optimization, (4) Abel inversion, and (5) comparison of CHAMP and ECMWF data based on forward modeling and inversion of artificial radio occultation data. In the tropics and midlatitudes at heights below 10 km, the CHAMP-ECMWF differences are significantly greater than the GPS Meteorology (GPS/MET)-ECMWF differences. Because big systematic negative differences between observed and simulated refraction angles are also observed in this region, we can explain that by poor quality of the CHAMP data in multipath regions due to signal tracking errors. Elsewhere, systematic CHAMP-ECMWF differences do not exceed 1 K in the tropics and polar latitudes, and 0.5 K in midlatitudes, and deviations of CHAMP data from ECMWF analyses are smaller than those of GPS/MET data. This can be explained by enhancement in the latest ECMWF analyses.

[1] The atmospheric propagation of GPS signals under multipath conditions and their detection are simulated. Using the multiple phase screen method, C/A-code modulated L1 signals are propagated through a spherically symmetric refractivity field derived from a high-resolution radio sonde observation. The propagated signals are tracked by a GPS receiver implemented in software and converted to refractivity profiles by the canonical transform technique and the Abel inversion. Ignoring noise and assuming an ideal receiver tracking behavior, the true refractivity profile is reproduced to better than 0.1% at altitudes above 2 km. The nonideal case is simulated by adding between 14 and 24 dB of Gaussian white noise to the signal and tracking the signal with a receiver operating at 50 and 200 Hz sampling frequency using two different carrier phase detectors. In the upper troposphere and stratosphere the fractional refractivity retrieval error is below 0.3% for 50 Hz sampling and below 0.15% for 200 Hz sampling. In the midtroposphere down to altitudes of about 2 km, phase-locked loop tracking induces negative fractional refractivity biases on the order of −1 to −2% at 50 Hz sampling frequency. Modifications to the receiver tracking algorithm significantly improve the retrieval results. In particular, replacing the carrier loop's two-quadrant phase extractor with a four-quadrant discriminator reduces the refractivity biases by a factor of 5; increasing the sampling frequency from 50 to 200 Hz gains another factor of 2.

Sounding Earth’s atmosphere by GPS radio occultation is a key objective of the Challenging Minisatellite Payload (CHAMP) satellite launched in July 2000. First results from CHAMP indicate that temperature profiles with good accuracy (about 1.5 K) and high vertical resolution can be derived from the mid-troposphere to the upper stratosphere. In the lower troposphere, however, the inversion process is frequently complicated by the occurrence of multi-path propagation rendering solutions using retrieval techniques based on single-ray propagation worthless. Recently, the canonical transform (CT) method has been introduced by one of us (M.E.G.) to solve the problem of calculating bending angle profiles within sub-caustic zones. The CT method uses the connection between geometrical and wave optics. The geometric optical rays are described by the Hamilton system in the corresponding phase space. In geometrical optics disentangling individual rays in multi-path zones can be performed by means of a canonical transform such that the projection of the ray manifold to the new geometric coordinate is unique. The corresponding asymptotic transform of the wave field to the single ray representation is then given by the Fourier integral operator associated with the canonical transform. Multiple phase screen simulation studies using refractivity profiles derived from radiosonde measurements demonstrate that the CT method accurately reproduces the true refraction angle profile. In contrast, the standard single-ray retrieval yields dry temperatures deviating by more than 10 K from the observed values. About 4000 CHAMP occultation profiles observed between 14 May and 10 June 2001 have been processed with the CT method. We present first results and compare these with corresponding profiles obtained by the standard retrieval.

[1] One of the most significant error sources in radio occultation soundings are residual errors of the ionospheric correction originating from small scale ionospheric turbulence and receiver noise. These errors degrade the quality of retrievals above 35–40 km. We describe a combined algorithm of the ionospheric correction and noise reduction including the following important details: (1) fitting of climatological model, (2) dynamical estimation of signal and noise covariances using radio occultation signals above 50 km, and (3) statistically optimized retrieval of neutral atmospheric and ionospheric refraction angles from L1 and L2 measurements using background refraction angles and signal and noise covariances (optimal linear combination). This method allows for the estimation of the error covariances of the retrieved neutral atmospheric and ionospheric refraction angles. We show some examples of GPS/MET data processing and perform a statistical comparison with the retrievals obtained by UCAR and DMI/IGAM algorithms. Our algorithm was previously used in the statistical validation of GPS/MET data on the basis of ECMWF analyses.

[1] The methods of processing radio occultation data in multipath zones which were used up to now have very strong restrictions of the applicability. In this paper, we introduce a new approach to the problem of deciphering the ray structure of wave fields in multipath zones using the short-wave asymptotic solution of the wave problem. In geometric optics a canonical transform resolves multipath by introducing new coordinate and momentum in such a way that different rays are distinguished by their coordinates. The wave field is processed by a Fourier integral operator associated with the canonical transform. The transformed wave function can then be written in the single-ray approximation, which allows for the determination of refraction angles from the derivative of the eikonal. The new method retains all the advantages of the back propagation such as the removal of effects of diffraction in free space and the enhancement of the vertical resolution in retrieved profiles, but it has much wider applicability limits. The method is convenient for operational applications. We discuss a fast numerical implementation of the method and present the results of numerical simulations confirming the applicability of the method.

[1] Numerical simulations of radio occultation experiments indicate that the effect of multipath propagation and diffraction must play a very significant role in the lower troposphere. Two methods of processing radio occultation data in the multipath zone were previously discussed: the radioholographic (radio-optics) method and the back propagation method. Both methods, however, have their restriction. The radio-optics method cannot accurately take into account the effects of diffraction in subcaustic zones. The back propagation method does not work for complicated caustic structures. In this paper we describe a combined method. The radio-optics analysis is applied both for the radio occultation data measured along the low Earth orbiter orbit and for the back propagated data. The computed spatial spectra are then averaged. This allows one to get rid of diffraction effects in subcaustic zones to a significant extent. The method is validated in numerical simulations on the material of global fields of atmospheric variables from analyses of the European Centre for Medium-Range Weather Forecasts. An example of processing a real GPS/MET occultation is given.