By now, a series of advanced wave optical approaches to the processing of radio occultation (RO) observations are widely used. In particular, the canonical transform (CT) method and its further developments need to be mentioned. The latter include the full spectrum inversion (FSI) method, the geometric optical phase matching (PM) method, and the general approach based on the Fourier integral operators (FIOs), also referred to as the CT type 2 (CT2) method. The general idea of these methods is the application of a canonical transform that changes the coordinates in the phase space from time and Doppler frequency to impact parameter and bending angle. For the spherically symmetric atmosphere, the impact parameter, being invariant for each ray, is a unique coordinate of the ray manifold. Therefore, the derivative of the phase of the wave field in the transformed space is directly linked to the bending angle as a single-valued function of the impact parameter. However, in the presence of horizontal gradients, this approach may not work. Here we introduce a further generalization of the CT methods in order to reduce the errors due to horizontal gradients. We describe, in particular, the modified CT2 method, denoted CT2A, which complements the former with one more affine transform: a new coordinate that is a linear combination of the impact parameter and bending angle. The linear combination coefficient is a tunable parameter. We derive the explicit formulas for the CT2A and develop the updated numerical algorithm. For testing the method, we performed statistical analyses based on RO retrievals from data acquired by the Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) and collocated analysis profiles of the European Centre for Medium-Range Weather Forecasts (ECMWF). We demonstrate that it is possible to find a reasonably optimal value of the new tunable CT2A parameter that minimizes the root mean square difference between the RO retrieved and the ECMWF refractivity in the lower troposphere and allows the practical realization of the improved capability to cope with horizontal gradients and serve as the basis of a new quality control procedure.

The first step in the development of the wave optical (WO) approach to the processing of radio occultation (RO) observations was made by

Later works

The further development of the WO approach based on the representation view relied upon the concept of the canonical transform (CT) originating from classical mechanics

In both cases, there is a strict mathematical representation (quantum mechanics or wave optics) and its asymptotic solution (classical mechanics or geometrical optics). While the evolution of de Broglie waves of probability or electromagnetic waves is described by the Hamilton operator, the evolution of rays or classical trajectories of particles is described by the Hamilton system. The Hamilton operator is obtained from the Hamilton function by the substitution of the momentum operator instead of classical momentum. Accordingly, for the classical problem, the phase space is introduced, the dimension of which equals double the geometric dimension because to each geometrical coordinate, we can conjugate the corresponding momentum. For the wave problems, momentum is understood as the ray direction vector.

The canonical transforms arise when we consider the class of the transforms of the phase space that conserve the canonical form of the Hamilton dynamical system. It was first demonstrated by

The application of the CT approach for the RO observation processing was pioneered by

The idea of the CT approach is as follows. Given the observations or RO complex signal

The multipath propagation problem consists in the decomposition of the signal equal to the sum or different sub-signals to retrieve the ray structure of the observed field. The solution of this problem discussed in the aforementioned papers consisted in the transform of the observed wave field

The idea explored in the present manuscript consists in the further development of the CT approach by using a generalized transform with the coordinate

The paper is organized as follows. In Sect. 2, we discuss the canonical transform in wave optics and quantum mechanics in general terms, including a brief review of FIOs. Based on this context, we discuss in Sect. 3 the application of the CT method for RO and introduce the particular phase space and the specific choice of coordinates, as well as the new generalization, adding an affine transform with a tunable parameter for improving the coping capability with horizontal gradients. In Sect. 4, we discuss the practical modifications needed to readily advance existing numerical implementations of the CT algorithm and present results of our performance evaluation from processing actual observed Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) RO data, including how to find an optimal value of the tunable parameter to minimize the retrieval errors in the lower troposphere. Section 5 finally provides the summary and main conclusions of the paper.

The canonical transforms (CTs) in classical mechanics are a class of transforms of the coordinates and momenta conserving the Hamiltonian form of the dynamical equation

We assume that the wave field can be represented in the standard form:

The amplitude

When discussing the CTs, it is necessary to bear in mind that most of the relations have an asymptotic nature, in which

Therefore, for a specific class of signals, including quasi-monochromatic ones and their superpositions, it is possible to introduce a phase space

Consider RO observations. The original signal corresponds to a range of rays starting at the transmitter and the phase space

The outstanding and still simple idea of

Considering now

In terms of FIO,

Radio occultation observation geometry with relevant geometrical variables indicated (for description see Sect. 3.1)

Here we discuss the application of the CT technique for the analysis of RO observations (Fig.

The RO observation geometry is schematically represented in Fig.

The first approach of processing RO data, belonging to the class of CT, was back propagation (BP)

A much better coordinate for the new representation should be the impact parameter

The complicated nature of the BP+CT algorithm stimulated further studies

In order to arrive at the phase function of the FIO of the second type, consider the expression for the derivative of the phase of the observed wave field:

To find an approximate solution that significantly reduces the computational costs at the expense of an insignificant reduction of accuracy, the representation of the approximate impact parameter was introduced. The impact parameter

All the modifications of the CT approach discussed above relied upon impact parameter

Impact parameter multipath, old coordinate (impact parameter) lines, and modified coordinate lines.

The variations in the ray impact parameter, which is no longer an invariant coordinate in the ray space, seem to undermine the elegant idea of the CT approach. Still, the CT method can be applied using the same formulas, but the coordinate

More importantly, horizontal gradients may result in multivalued ray manifold projections when using the effective impact parameter

A typical multivalued bending angle profile

The modified canonical transform Eq. (

We denote the generalized FIO

It is possible to arrive at a quantitative estimate of

Modification of existing numerical algorithms may not be so straightforward as it follows from the above mathematical considerations. In order to avoid this, it is possible to complement an existing implementation of any WO-based numerical algorithm by an additional affine transform.

We will now derive the transform between

The generating function of transform (

The above derivation allows for one more generalization. We can consider

Statistics for latitude band

Statistics for latitude band

Statistics for latitude band

Our implementation of the CT2A algorithm was based on the existing program code with the addition of the parameter

In our numerical validation, we retrieved COSMIC refractivity profiles

Statistics for latitude band

Statistics for latitude band

Statistics for latitude band

Figures

These results indicate that for the latitudes of

Statistics for latitude band

Statistics for latitude band

Statistics for latitude band

The statistics for different

Statistics for latitude band

Statistics for latitude band

This indicates that CT2A acts as a QC procedure not involving any external data and only based on the internal properties of observed signals. On average, CT2A provides a higher cut-off height which is estimated from the CT amplitude by correlating it with the

In this study, we discussed the general idea of the canonical transform (CT) method and provided a new generalization adding more flexibility for application in RO processing. CTs in classical mechanics (geometrical optics) are implemented in quantum mechanics (wave optics) by linear operators with oscillating kernels. Such operators are referred to as Fourier integral operators (FIOs). During the past century, this approach acquired a solid theoretical basis. In numerous mathematical monographs, one finds the advanced theory of FIOs. The central role in this theory is played by the concept of the ray manifold and its projections.

In quantum mechanics and wave optics, FIOs were employed for the quantization procedure, i.e., the construction of the asymptotic quantum (quasi- or semiclassical) solutions on the basis of the classical (geometric optical) ones. The idea of the CT method for processing RO observations is the reverse: the reconstruction of the geometric optical solution from the wave optical one, which can be referred to as the dequantization.

Although there have been many modifications, like original CT combined with back propagation (BP), full spectrum inversion (FSI), phase matching (PM), and CT of type 2 (CT2), there is no essential difference between these FIO-based methods. The difference consists of the approximation of the phase function of the FIO leading to the corresponding approximate representation of the impact parameter and bending angle and in the specific implementation (such as cut-off, filtering, and quality control procedures). All these methods map the wave field into the representation of the impact parameter

The implementation of this idea in the real, non-spherically symmetric atmosphere encounters some difficulties. First, in the strict sense, there is no such quantity as the impact parameter as a unique variable any more, but it is still possible to operate with the effective impact parameter derived from Doppler frequency shift using the same relations as for a spherically symmetric medium. This quantity can be implemented in the observation operator for the variational assimilation of RO observations, canceling errors due to horizontal gradients. However, the above property of the impact parameter, which is supposed to be a unique coordinate of the ray manifold, does not always hold for the effective value. In some cases, the situation referred to as the impact parameter multipath may occur, resulting in retrieval errors in atmospheric profiles derived from RO data.

In order to mitigate this fundamental shortcoming, we introduced a generalization of the CT approach. We used a generalized definition of the coordinate in phase space, defined as a linear combination of the impact parameter and bending angle. Because this can be understood as an affine transform of the phase space, we coined the abbreviation CT2A for the new method. This transform has a parameter

We implemented the CT2A algorithm by modifying our existing program code for the CT2 method. In order to evaluate its statistical performance under real RO observation conditions, including challenging horizontal gradients in the lower troposphere, we processed a large ensemble of COSMIC RO data for the year 2008 on the 1st and 15th day of every month, adding up to a total of about 60 000 RO events. We used the total relative difference in COSMIC from collocated ECMWF analysis profiles over the lower troposphere as the metric for this evaluation and the tuning parameter estimation.

For the latitudes of

Overall, these results suggest that the CT2A method is not only theoretically an innovative generalization of the CT or FIO class of methods but also practically a valuable advancement for RO processing in that it can improve the capability to cope with challenging horizontal gradient conditions in the lower troposphere and serve as the basis of a new QC procedure.

The COSMIC data used in this study are freely available at CDAAC website.

KBL formulated the problem, introduced the initial idea of the study, participated in theoretical discussions, and contributed to finalizing the paper. MG performed the theoretical derivations, implemented the algorithm numerically, performed the statistical study, and wrote the initial draft of the paper. GK participated in theoretical discussions and contributed to finalizing the paper.

The authors declare that they have no conflict of interest.

Michael Gorbunov is grateful for the Russian Foundation for Basic Research (grant no. 20-05-00189 A) for the financial support. Gottfried Kirchengast acknowledges support, including partial co-funding of the work of Michael Gorbunov, from the Aeronautics and Space Agency of the Austrian Research Promotion Agency (FFG-ALR) under the Austrian Space Applications Programme (ASAP) project ATROMSAF1 (proj. no. 859771) funded by the Ministry for Transport, Innovation, and Technology (BMVIT). Kent B. Lauritsen has been supported by the Radio Occultation Meteorology Satellite Application Facility (ROM SAF), which is a decentralized operational RO processing center under EUMETSAT. The authors acknowledge Taiwan’s National Space Organization (NSPO) and the University Corporation for Atmospheric Research (UCAR) for providing the COSMIC data.

This research has been supported by the Russian Foundation for Basic Research (grant no. 20-05-00189) and the Aeronautics and Space Agency of the Austrian Research Promotion Agency (grant no. 859771).

This paper was edited by Peter Alexander and reviewed by two anonymous referees.