IMPLEMENTATION OF FULL SPECTRUM INVERSION METHOD …831390/FULLTEXT01.pdf · IMPLEMENTATION OF FULL SPECTRUM INVERSION METHOD TO RETRIEVE BENDING ANGLE BASED ON SIMULATED DATA Arkadiusz

IMPLEMENTATION OF FULL SPECTRUM INVERSION METHOD TO RETRIEVE BENDING ANGLE

BASED ON SIMULATED DATA

Arkadiusz Lebkowski Krzysztof Nadolny

This thesis is presented as part of Degree of

Master of Science in Electrical Engineering

Blekinge Institute of Technology

May 2013

Blekinge Institute of Technology School of Engineering Department of Electrical Engineering Supervisor: Prof. Mats Pettersson Examiner: Dr. Benny Lövström

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Abstract

This master thesis was done in cooperation with RUAG Space AB. Currently RUAG is working on designing the next generation instrument using Radio Occultation (RO) for climate monitoring and atmospheric sounding. RO is a method used to measure the temperature, pressure and humidity of the Earth's atmosphere. It is based on the fact that radio signal is refracted when it passes through the atmosphere. The level of refraction depends on the properties of the atmosphere. To derive refraction profile, the bending angle needs to be calculated by using the Doppler shift of the signal and actual positions of the emitter GPS (Global Positioning System) satellite and receiver LEO (Low Earth Orbit) satellite. Several methods of the bending angle retrieval have been developed, but these algorithms have different performance on multipath, partial loss of the input data or calculations based on the noisy data. In this project Full Spectrum Inversion (FSI) method is implemented. It is shown in this work that FSI can accurately reproduce the bending angle profile on simulated occultation data. In this report different filters are also tested and the best filters reported. FSI is able to disentangle multipath area, shortens the calculation time and provides accurate bending angle profile. Key words: atmosphere, bending angle, Full Spectrum Inversion, GPS, LEO, radio occultation, satellites, weather forecasting.

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Acknowledgement

We are very grateful for all help of several people without which this master thesis would not have been possible. We would like to use this section to thanks them.

At first we wish to thank our supervisor prof. Mats Pettersson for all the support he gave us to make this thesis possible. We are thankful for the meetings on which we could better understand our problems and for the possibility to visit the RUAG company and meet people working there.

We also would like to thank the RUAG company team, especially Anders Carlström and Magnus Bonnedal for guidance, support and encouragement from the initial to the final level of the thesis. We are very grateful for all tips and solutions to our problems we were faced with. Arkadiusz Lebkowski Krzysztof Nadolny

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Table of Content

Abstract ................................................................................................................................................................. 3

Acknowledgement ................................................................................................................................................. 5

Table of Content ................................................................................................................................................... 7

List of figures ........................................................................................................................................................ 9

List of tables ........................................................................................................................................................ 11

List of abbreviations ........................................................................................................................................... 13

1. Introduction ............................................................................................................................................... 15

1.1. RUAG Space AB ............................................................................................................................................... 16

1.2. Problem Description .......................................................................................................................................... 16

1.3. Scope of Thesis Work ........................................................................................................................................ 17

1.4. Division of Thesis .............................................................................................................................................. 17

1.5. Outline of the Thesis .......................................................................................................................................... 17

2. Background and Related Work ................................................................................................................ 18

2.1. Related work ...................................................................................................................................................... 18

2.2. Radio occultation and the basics of the subject .................................................................................................. 18

3. Implementation of FSI method based on simulated data (Case 9) ........................................................ 22

3.1. Description of input data .................................................................................................................................... 22

3.2. Implementation and filtering .............................................................................................................................. 25

3.2.1. Implementation .......................................................................................................................................... 25

3.2.2. Filtering ...................................................................................................................................................... 33

3.3. Data analysis and methodology ......................................................................................................................... 37

3.4. Test results and conclusion ................................................................................................................................ 38

4. Implementation of FSI method for more demanding cases of input data (Case 42) ............................ 41

4.1. Presentation of input data ................................................................................................................................... 41

4.1.1. Similarities and differences between the two input data sets ..................................................................... 41

4.1.2. Case 42 ....................................................................................................................................................... 41

4.2. Implementation of FSI method on Case 42 ........................................................................................................ 46

4.2.1. Fixing the discontinuity of φNCO ................................................................................................................. 47

4.2.2. Radio holographic filtering (RHF) ............................................................................................................. 47

4.2.3. Corrections of satellite's orbits (CSO) due to Earth's oblateness ................................................................ 50

4.2.4. Horizontal gradients correction (HGC) ...................................................................................................... 51

4.3. Results ............................................................................................................................................................... 56

5. Summary and conclusions ......................................................................................................................... 63

6. Future work ................................................................................................................................................ 65

Appendix A ......................................................................................................................................................... 66

Reference list ....................................................................................................................................................... 72

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List of figures Figure 1.1. Block diagram of radio occultation technique, where L1 and L5 are the normalized GPS carrier frequencies ................................................................................................................................ 15 Figure 2.1. Variables used in the calculations ...................................................................................... 20 Figure 2.2. Two-dimensional model of oblate Earth and spherical Earth with the most important parameters given ................................................................................................................................... 21 Figure 3.1. Block diagram of GPS signal processing in OL mode ....................................................... 23 Figure 3.2. Absolute value of Anav(t) with and without noise (CNR equals 65 dBHz) ......................... 24 Figure 3.3. Absolute value of Anav(t) with and without noise (CNR equals 55 dBHz) ......................... 24 Figure 3.4. Absolute value of Anav(t) with and without noise (CNR equals 45 dBHz) ......................... 25 Figure 3.5. Amplitude U(ω) of Fourier transform in whole range of angular frequency ..................... 27 Figure 3.6. Phase of Fourier transform u(ω) in whole range of angular frequency illustrating a problem with phase unwrapping at an angular frequency of 1.275 rad/s. ............................................ 27 Figure 3.7. Phase of the Fourier transform u(ω) in whole range of angular frequency ........................ 28 Figure 3.8. Amplitude of the Fourier transform U(ω) in useful range ................................................. 28 Figure 3.9. Phase of the Fourier transform u(ω) in useful range .......................................................... 28 Figure 3.10. Reception time t(ω) determined from the Fourier transform phase ................................. 29 Figure 3.11. Value of GPS and LEO satellite separation angle Θ(t) in time t ...................................... 30 Figure 3.12. Θ(t) in discrete time domain and its approximated continuous function ......................... 30 Figure 3.13. Θ(t) in discrete time domain and its approximated continuous function, over a short time period .................................................................................................................................................... 31

Figure 3.14. Calculated ( ) in frequency domain ........................................................................... 31 Figure 3.15. Flow chart of all variables and equations used in FSI method implementation ............... 33 Figure 3.16. Amplitude of the Doppler signal in frequency domain .................................................... 35 Figure 3.17. Bending angle at different heights for CNR-65 dBHz ..................................................... 38 Figure 3.18. Bending angle at different heights for CNR-55 dBHz ..................................................... 38 Figure 3.19. Bending angle at different heights for CNR-45 dBHz ..................................................... 39 Figure 3.20 Difference between calculated and reference bending angle profiles in function of height for different CNR ratios ........................................................................................................................ 39 Figure 4.1. Positions of the satellites when for rising LEO satellite. (Blue circle indicates GPS satellite, brown circle LEO satellite) .................................................................................................... 42 Figure 4.2. Positions of the satellites for setting LEO satellite. (Blue circle indicates GPS satellite, brown circle LEO satellite) .................................................................................................................. 42 Figure 4.3. Magnitude of the Anav in time domain for rising LEO satellite .......................................... 43 Figure 4.4. Angle of the Anav in time domain for rising LEO satellite .................................................. 44 Figure 4.5. Magnitude of the Anav in time domain for setting LEO satellite......................................... 44 Figure 4.6. Angle of the Anav in time domain for setting LEO satellite ................................................ 44 Figure 4.7. Phase φNCO of occultation signal for rising LEO satellite .................................................. 45 Figure 4.8. Phase φNCO of occultation signal for setting LEO satellite ................................................. 45 Figure 4.9. Processing steps in the program ......................................................................................... 46 Figure 4.10. Phase φNCO of occultation signal after filtering for rising LEO satellite ......................... 47 Figure 4.11. Scheme of RHF ................................................................................................................ 48 Figure 4.12. Filtered and original value of the Doppler signal phase for rising LEO satellite ............. 49 Figure 4.13. Filtered and original value of the Doppler signal phase for rising LEO satellite over a short time interval ................................................................................................................................. 49 Figure 4.14. Filtered and original magnitude of the Doppler signal for rising LEO satellite ............... 50 Figure 4.15. Horizontal gradients ......................................................................................................... 51 Figure 4.16. Impact parameter a(ω) affected by horizontal gradients. At 3.9 105 the effect of horizontal gradient can be seen ............................................................................................................ 52

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Figure 4.17. Values of the signal components frequencies determined as Fourier Transform domain and value of the signal component phase derivative ............................................................................ 53 Figure 4.18. Values of the signal components frequencies determined as Fourier Transform domain and value of the signal component phase derivative in short time period ............................................ 54 Figure 4.19. Function which shows whether all conditions are fulfilled “0”, or not “1” ..................... 55 Figure 4.20. Impact parameter with substituted parts .......................................................................... 55 Figure 4.21. Flow chart of all variables and equations in FSI method implementation for Case 42 .... 56 Figure 4.22. Bending angle vs. height based on data from rising LEO satellite (Case 42). This case is without RHF and CSO .......................................................................................................................... 57 Figure 4.23. Bending angle vs. height based on data from setting LEO satellite (Case 42). This case is without RHF and CSO .......................................................................................................................... 57 Figure 4.24. Bending angle vs. height based on data from rising LEO satellite (Case 42). This case is with RHF and without CSO ................................................................................................................. 58 Figure 4.25. Bending angle vs. height based on data from setting LEO satellite (Case 42). This case is with RHF and without CSO ................................................................................................................. 58 Figure 4.26. Bending angle vs. height based on data from rising LEO satellite (Case 42). This case is with RHF and CSO ............................................................................................................................... 59 Figure 4.27. Bending angle vs. height based on data from setting LEO satellite (Case 42). This case is with RHF and CSO ............................................................................................................................... 59 Figure 4.28. Bending angle vs. height based on data from rising LEO satellite (Case 42). This case is with RHF, CSO and HGC .................................................................................................................... 60 Figure 4.29. Bending angle vs. height based on data from setting LEO satellite (Case 42). This case is with RHF, CSO and HGC .................................................................................................................... 60 Figure 4.30. Bending angle vs. height based on simulated input data (Case 9) ................................... 61 Figure 4.31. Bending angle vs. height based on another set of more demanding case of input data (Case 5) ................................................................................................................................................. 62 Figure A.1. Occultation geometry ........................................................................................................ 66 Figure A.2. Position of the occultation point and perigee point for the lowest part of occultation ...... 67 Figure A.3. Perigee position in geodetic coordinate system ................................................................ 70

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List of tables Table 3.1. Values for antenna gain ......................................................................................... 23 Table 3.2. RMS bending angle errors for Butterworth filter. Each value*10E-3 [rad] ......... 34 Table 3.3. Filter settings used in further calculations ............................................................. 34 Table 3.4. RMS bending angle errors [rad] for different noise levels applied to Anav(t) ........ 35 Table 3.5. Filter settings used in further calculations ............................................................. 36 Table 3.6. RMS bending angle errors [rad] for filtering phase of the FFT ............................. 36 Table 3.7. RMS bending angle errors [rad] for filtering the phase φNCO ................................ 37 Table 3.8. RMS bending angle errors [rad] for filtering bending angle ................................. 37 Table 3.9. RMS bending angle errors for different level of noises after filter application ..... 40 Table 4.1. RMS error results based on the following solutions .............................................. 61

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List of abbreviations C/A Coarse/Acquisition CNR Carrier to Noise Ratio CSO Corrections of Satellite’s Orbits CT Canonical Transform CT2 Canonical Transform 2 DMI Danish Metrological Institute ECMWF European Centre for Medium-Range Weather Forecasts FFT Fast Fourier Transform FIO Fourier Integral Operators FSI Full Spectrum Inversion GNSS Global Navigation Satellite System GO Geometrical Optics GPS Global Positioning System HGC Horizontal Gradients Correction LEO Low Earth Orbit MEO Medium Earth Orbit NCO Numerically Controlled Oscillator PM Phase Matching PO Physical Optics RHF Radio Holographic Filtering RMS Root Mean Square RO Radio Occultation WOP Wave Optics Propagator

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1. Introduction

Climate on Earth is changing. The changes are more visible in present times, which may be caused by the global warming effect. The climate change is one of the biggest challenge that humanity is facing in nature and the atmosphere is crucial to monitor for this effect. The modern human society therefore needs to monitor these changes globally. An evolving method to do accurate measurements covering whole planet is a technique called radio occultation (RO). This approach uses LEO (Low Earth Orbit) satellite to track signals from GPS (Global Positioning System) satellite. The signal is refracted by the atmosphere and bending angle can be calculated. By monitoring changes in the signal we can derive atmospheric profiles of humidity, temperature and pressure. These data can be used for many applications, such as weather forecasting and climate change monitoring.

To retrieve bending angle, which is our desirable parameter, two fundamental methods can be used [4], Geometrical Optics (GO) and Physical Optics (PO). GO methods are based on the geometrical principles such as refraction phenomena or reflection angle, which are considered in terms of a ray. Such a concept exclude phenomena like interference and diffraction, thus it is not able to predict or resolve all aspects during bending angle retrieval.

Most PO methods are based on the frequency analysis of electromagnetic waves and can be successfully used in the multipath area, where the signal is refracted into multiple signal rays. PO can be based on one of the following techniques [4]:

- Full Spectrum Inversion (FSI) [5] - Phase Matching (PM) [19] - Canonical Transform 2 (CT2) [20] In this master thesis only FSI method was implemented, as it was chosen by RUAG in

the project. The method has high accuracy and is relatively straightforward which makes it easy to implement and to use.

Figure 1.1. Block diagram of radio occultation technique, where L1 and L5 are the normalized GPS carrier frequencies

high altitude climatology and Abel inversion

auxiliary meteorological data

navigation bit stream, I and Q correlator values, and Doppler phase shift

for L1 and L5 frequencies

iono-free bending angle

refractivity

temperature, water vapor, pressure

satellites positions

radio holographic methods, multipath

ionospheric corrections

corrections due to Earthoblateness

associated bending angles L1 and L5 bending angle

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In Figure 1.1 there is a graph showing steps required to obtain atmospheric profiles [1]. In the grey rectangle there are steps from the processing chain to which this master thesis applies.

1.1. RUAG Space AB

RUAG is an international group of companies associated with aerospace and defense technology. The activities are oriented into two market segments: civil and military. After acquisition of Saab Space and Austrian Aerospace in 2008 and purchase of Oerlikon Space in 2009, RUAG Space became the largest and independent deliverer of space components and subsystems in Europe. RUAG Space operates in three countries: Switzerland, Sweden and Austria and has a total of 1100 employees.

This master thesis was proposed by RUAG Space AB. The company is mostly specialized in on-board satellite equipment with high reliability, like i.e. computer systems, antennas and separation systems for space launchers. The company has two departments that are located in Göteborg and in Linköping. RUAG Space AB in Sweden is employing a total of 378 people [2].

1.2. Problem Description

The main problem discussed in this report applies to the implementation of the FSI

method for radio occultation data. In the first part of the thesis, most problems concern physical realization of the FSI method in the Matlab program, such as data conversions, incorrect data length, etc. To remove noise from the signal, appropriate filter types has to be chosen to ensure the best accuracy in relation to reference data. In the second part, FSI is implemented on more demanding cases of radio occultation data. Noise contained in the signal is causing serious distortions. Also in the more demanding cases of input data, influence of the Earth's oblateness and horizontal gradients has to be considered and solutions proposed. All these problems must be solved in order to retrieve the proper bending angle profile. In this master thesis we focus on the following research questions: RQ1: What is the impact of noise at different levels in the radio occultation signal on the final bending angle result in FSI method?

- Radio occultation signals are affected by different kinds of noise, i.e. ionosphere noise, amplifier noise, horizontal gradients etc. Their impact on the final bending angle results may vary.

RQ2: What is the impact of the spherical Earth assumption in the FSI method, since the implementation needs to process the oblate Earth profile to accurate bending angle results?

- The Earth’s equatorial radius is equal to 6378 km, while the polar radius is equal to 6357 km [15]. When the oblate Earth is considered, the crucial factor are the positions of the satellites. Depending on their location, the effective Earth curvature will be possible to predict, accounting for almost 21 km of difference between the lengths of the equatorial radius and polar radius.

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1.3. Scope of Thesis Work

In the first part of this report an implementation of FSI method and accomplishing accurate solution for filtering noisy data was completed. The purpose was to provide as precise results as possible in comparison to reference model. No other ways to provide faster and more accurate results with other methods were investigated, only a simple comparison of 5 types of filters used on input data and fft phase.

In the second part FSI was used in more complex input data. It was found that more advanced filters had to be used in order to obtain the information from the noisy input signal. In addition, there were data gaps in the input data, so the other purpose was to restore the information and to provide results that are similar to results obtained in part one of this work. The oblateness of the Earth’s was also taken into account, as well as satellites corrections and finding the local centre of Earth’s curvature. To improve the results, radio holographic filtering and solution for horizontal gradients were applied.

1.4. Division of Thesis

A group of two members was necessary because this subject requires a lot of effort to understand and resolve appearing problems. There was a need to read many papers to understand the general idea and many hours of studying the program code provided by RUAG. The member Arkadiusz Lebkowski was responsible more for a theoretical part while Krzysztof Nadolny was responsible more for practical part – program code.

The thesis can be divided into two parts – implementation and verification of FSI method to a simple simulated input data Case 9, and retrieving bending angle from more demanding cases of simulated input data Case 42 and Case 5.

1.5. Outline of the Thesis

This master thesis begins with a short presentation of the RUAG Space company,

description of the problem and division of the thesis. Chapter 2 presents the general idea of the subject with background and related work. Chapter 3 presents the algorithm of the FSI method, proposed solutions of implementation of complex equations for numerical calculations, and approach to the data filtering. In this section only Case 9 with artificially added white noise was considered. The chapter closes with the analysis of results and conclusions. Chapter 4 begins with the presentation of more demanding RO data, shows the differences between them and data presented in previous chapter. Next, implementation of the FSI method to the more complex data is explained, where necessary corrections, like radio holographic filtering, horizontal gradients problems, satellites corrections and more are introduced to the program and results of their improvements to the calculations are presented. Thesis ends with a summary in chapter 5 and in next chapter a brief description of future work.

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2. Background and Related Work 2.1. Related work

Radio occultation is relatively new method of weather forecasting. There are a several

methods of calculation the refractivity profile, e.g. Canonical Transform (CT) [21], FSI [5], PM [19], a method using Neural Network [22], or their combinations [24]. They are all developed with the aim to handle the problems related to multipath in the lower troposphere. The most efficient methods able to disentangle multipath are methods based on the Fast Fourier Transform. i.e.: CT and FSI. The CT is very accurate and allows to achieve very high resolution (30 – 60 m), but is complex and time consuming because of the back-propagation algorithm implemented in it. While the FSI method is easy to implement due to computational simplicity and under assumption of circular occultation geometry satisfactorily results with high spatial resolution can be found. The difference between CT and FSI in details has been investigated by Gorbunov in [25]. He also explored deeply the implementation of FFT in RO in [20]. Sokolovsky, in his paper from 2010 [26] studied the nature of RO in respect of dependence of the refractivity inversion bias on the cutoff height and noise.

Full Spectrum Inversion method implemented in this master thesis project is based on Jensen’s achievements. Algorithm and all equations included in the program was taken from his paper [5]. The part of the program responsible for identification of the noise impact, analysis and filtration was developed on the basis of solutions proposed in the RUAG Space design document [8] and [9] written by M. E. Gorbunov and K. B. Lauritsen. Introduced corrections of the satellites and local centre of curvature positions, was obtained from Danish Meteorological Institute (DMI) scientific paper [10].

2.2. Radio occultation and the basics of the subject

This master thesis subject is related to space application where satellites are used to

measure atmospheric conditions, like humidity, atmospheric pressure and temperature. Calculations of these parameters are based on occurrence of radio occultation between these two satellites – LEO and GPS, which are located on the opposite sides of the Earth.

First in the thesis we start with the description how the GPS system operates. The GPS system finds the position of the GPS receiver by calculating distances to at least three points at known coordinates – trilateration. In terms of GPS the known positions are the GPS satellites. Each satellite transmits navigation message which contains orbit parameters, so that satellites’ position can be calculated. To measure the distance between satellites and the unknown position, the term of “pseudorange” was introduced. When signal from GPS is transmitted, the transmission time is encoded in the signal using Coarse/Acquisition (C/A) code. C/A code is a pseudo random sequence of digits -1,1. Each satellite is associated with unique sequence of C/A code, so that it can be easily identified. The time is obtained from atomic clock onboard the GPS satellite. The reception time between two signals is calculated based on the difference of the transmission time encoded in the signals received by the receiver. Because atomic clocks are very expensive, it is economically unviable to install it in the receiver. The receiver contains a much cheaper quartz clock which is stable on short

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time-scales but drifts over long time-scales. Then the receiver synchronizes its internal clock to the ones in the GPS satellites and obtains very high time precision. The time difference between transmission and reception is a measure of the distance, namely “pseudorange” [3]. That is how the arbitrary position can be found, but how to use it for weather monitoring?

Electromagnetic wave passes through a medium from a distant emitter and is received by a receiver. The receiving instrument measures the phase and amplitude of the signal wave during an occultation episode. The measurement sequences can be utilized to gather physical properties of the intervening medium. The occultation in the atmosphere can be measured by two satellites, one being an emitter and one a receiver. The role of the receiver can also be carried out by an airplane. The changes in the phase and amplitude of the electromagnetic signal during the occultation episode are stored in the receiver and a profile of these variations is created. The data contained in these profiles provide information about the refractive properties of the intervening medium [12]. An analogous example to radio occultation can be measurements in seismology. An array of seismometers is located over a geographical radius to study a various types of seismic waves. When the earthquake occurs, the seismic waves arrive to each seismometer. Because of the use of the array, differential arrival times can be measured at different seismic station in the array. Spectral properties of the waves can also be handled. From these observations a different paths of these waves and some physical properties of the medium can be obtained [12].

In the example considered in this report the whole system is as follows - RO is a technique which requires 2 satellites – GPS (or other GNSS – Global Navigation Satellite System) transmitter on MEO (Medium Earth Orbit) at about 20 000 km and GPS receiver on LEO orbit to maximum distance of around 2 000 km. These two satellites are located on opposite sides of the planetary limb at the known positions. When the GPS satellite transmits the signal in direction of the LEO satellite, rays penetrate the Earth atmosphere. Like the light is refracted by various transparent materials (water, glass, diamond), the same happens to the electromagnetic wave in the atmosphere. A slight change in the density, humidity, temperature modifies the refractive index of the atmosphere. This causes the signal to bend and changes its frequency due to Doppler effect. The Doppler frequency of received GPS signal is the main measurement of the LEO satellite. It relies on the onboard frequency reference oscillator. Frequency bias on the reference oscillator on the receiver and on the transmitter biases the Doppler measurement. This biases are solved for by tracking all transmitted signals through a ground based network of receivers. Then, in the ground processing the geometric Doppler caused by satellites movements is separated from the atmospheric Doppler [4]. When the atmospheric Doppler is determined it is possible using one of the several methods to calculate the bending angle α of the rays as a function of height h. When the bending angle is known the refractivity profile is computed from which the temperature, pressure and humidity can be determined. The accuracy of calculation increases when more than one of GPS signals are used. GPS satellites transmit L1, L2 and L5 signals. Each of them has different carrier frequency:

- L1 – 1575,42 MHz - L2 – 1227,60 MHz - L5 – 1176,45 MHz

Our task was to implement FSI method in Matlab and to test the implementation to RO data that was provided by the company. The outcome from the Matlab program is an estimate of the bending angle which is dependable of atmospheric compositions and conditions. The

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simple scheme with the most important variables used in the program is presented in figure 2.1.

Figure 2.1. Variables used in the calculations

α – [rad] bending angle re – [m] the Earth radius rGPS – [m] distance between GPS satellite and the center of the Earth rLEO – [m] distance between LEO satellite and the center of the Earth a – [m] impact parameter - the distance between the local center of the Earth and the

closest point to the Earth surface located on the tangent line of the GPS signal ray Θ – [rad] is the separation angle between the LEO and the GPS as measured from the

centre of the Earth local centre of curvature h – [m] height defined as: a - re

Satellites’ positions are described in a coordinate system which has its origin in the centre

of the mass of the Earth. All calculations are based on the assumption that the rays are bent with respect to this centre of mass. Unfortunately it is not always the case when the oblate Earth is considered. The problem lies in the asymmetry of the Earth and the associated asymmetry of the atmosphere. For the spherical model the surface is always symmetrical, and centre of curvature indeed coincide with the center of the sphere. So that satellites positions and GPS rays are bent with respect to the same coordinate system. On the other hand for oblate model the principle of symmetry is fulfilled only for points placed on the equator and poles. For all other points symmetry is not preserved and the local centre of curvature does not coincide with the center of the mass of the oblate Earth. This problem is depicted on the Figure 2.2. When GPS signal ray traverses the atmosphere it is bent with respect to the local centre of Earth curvature, which changes its position depending on the course of the ray path. Since the ray path are bent with respect to the local centre of curvature (which may do not coincide with the centre of Earth mass) the satellites positions have to be moved to the same coordinate system (with origin in the local centre of curvature) in which the rays are bent.

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Figure 2.2. Two-dimensional model of oblate Earth and spherical Earth with the most important

parameters given

m – [m] major axis of the Earth cross-section, equal: 6378137 m n – [m] minor axis of the Earth cross-section, equal: 6356752 m z – z–axis of the coordinate system Rc – [m] local Earth radius Rxy – Rxy-axis of the coordinate system Perigee – defined as a point on straight line between satellites placed the

closest to the centre of the spherical Earth model at a given moment Occultation Point – a point above which the radio occultation takes place Curvature Center – center of curvature for a given occultation point

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3. Implementation of FSI method based on simulated data (Case 9) 3.1. Description of input data

The bending angle of the ray path can be calculated using one of the methods mentioned

in section 1 operating on the RO data. However, before we proceed with bending angle retrieval, these data must be received by LEO satellite and preprocessed.

In chapter 3 implementation of the FSI method is described. The implemented method is then tested on simulated data (Case 9). All input data are retrieved artificially. Nevertheless, first we proceed with the explanation how the input data are preprocessed in “real life”, to give the reader a concept of the problem.

The system used to process the input data is similar to the simplified scheme of the receiver and is depicted in figure 3.1 [13]. In this thesis all calculations are provided for GPS L1 frequency, thus only processing of L1 signal will be discussed. First, the GPS signal with the L1 carrier modulated by C/A code and navigation message is received by an antenna. This signal is filtered using band-pass filter and amplified before it is down converted using sine wave with the constant frequency close to the L1 carrier e.g. 1577.29 MHz. Next, down converted signal is sampled by analog to digital converter. Sampling is performed at rate 141 Msps in the implementation made by [18]. Such prepared digital signal still contains C/A code, navigation signal, some noise and its carrier is phase-shifted due to the Doppler effect. The next step is to get carrier phase shift φNCO, in which information about bending angle is included.

Whole processing is in digital domain and is performed with a sample rate 28.25 MHz. Based on Doppler phase shift φNCO model, carrier of the received signal is compared (mixed) with two L1 NCO carriers phase-shifted with 90 degrees relative to each other. Both L1 carriers are generated on-board LEO satellite by Numerically Controlled Oscillator guided by Doppler model. On the output of that comparison the in-phase (I) and quadrature (Q) correlator amplitudes are determined. Next, these correlator values are multiplied by C/A code also generated on-board the LEO satellite. This step removes C/A code from the received signal. Then 28250 consecutive digital values of the correlators are summed, so that the output signals with sampling rate 1 kHz are obtained. Finally the output of whole process are three parameters determined: I and Q correlator amplitudes and φNCO defined as model based Doppler phase shift.

In the program all these parameters described above are retrieved artificially. All processed input data are provided by RUAG. The input data retrieval strategy described further is not a subject of the thesis.

To determine artificially retrieved input data values, a set of refractivity reference profiles from European Centre for Medium-Range Weather Forecasts (ECMWF) Integrated Forecast System is selected [17]. Refractivity profiles are selected for points in space and time that are close to real occultation measurement points, from which satellites positions and decoding navigation bit stream can be used. This is due to the fact that navigation bit stream cannot be simulated. Based on selected refractivity profiles, the artificial correlator amplitudes and phase shift are determined using wave propagation software at DMI, known also as a Wave Optics Propagator (WOP). This software, which is a receiver simulation model, allows the

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user to simulate noise, co-channel errors and other receiver errors occurring during reception of the GPS signal [16].

Figure 3.1. Block diagram of GPS signal processing in OL mode

In the provided simulation three artificially retrieved variables are used: φNCO defined as

model based Doppler phase, I as in-phase correlator value, and Q as quadrature correlator value. To check the reliability of the FSI method and efficiency of filtering, a random noise to I and Q correlator amplitudes is introduced. The noise is Gaussian and is added by Matlab’s function “randn()”. Its power is controlled by carrier to noise density ratio (CNR) as follows:

10

2

10

)()(maxCNR

bw

tjQtINoisePower

(3.1)

Where: I(t), Q(t) – [V] values of correlators measurement at time t bw – [Hz] is bandwidth of the signal in this thesis equals 1000 Hz CNR – [dBHz] is carrier to noise density ratio

The CNR is determined by the system link budget and depends on several parameters. One other important parameter is the antenna gain, which is given by the antenna design. Typical values for antenna gain and corresponding CNR values at high altitude are:

Table 3.1. Values for antenna gain Antenna type Antenna gain CNR Low gain antenna 0 dBi 45 dBHz Medium gain antenna 10 dBi 55 dBHz High gain antenna 20 dBi 65 dBHz

For further needs, correlator values with noise are combined into a single complex variable defined as Doppler signal amplitude Anav(t). Anav(t) is formed of I(t) + jQ(t). In figures 3.2, 3.3 and 3.4 |Anav(t)| with different ratios is presented.

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To calculate the bending angle, navigation bit stream is needed. As mentioned the navigation message is contained in the received signal and therefore it must be removed before further processing. This is done by multiplying the Anav values by navigational data bit stream converted into a sequence of (+1) - binary 1 and (-1) - binary 0.

The last input data necessary to calculate bending angle are positions of satellites during occultation. Positions of satellites are described in coordinate system fixed to the GPS satellite and which has the origin in the center of the Earth.

All listed input data are sampled with 1 kHz frequency. The measurement time is over 88.4 s, which gives exactly 88451 samples for each data set.

Figure 3.2. Absolute value of Anav(t) with and without noise (CNR equals 65 dBHz)

The fluctuations in the signal amplitude visible on time from 70 s to around 85 s are caused by strong multipath interference.


0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

Time: t[s]

|Ana

v(t)| [V

]

Anav

& noise

Anav

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

Time: t[s]

|Ana

v(t)| [V

]

Anav

& noise

Anav

25


The scenarios for realistic noise levels are presented on above figures (3.2, 3.3 and 3.4). The highest noise value is for CNR ratio of 45 and the lowest for CNR ratio of 65. The noise visible in figure 3.4 is a significant part of the signal.

3.2. Implementation and filtering 3.2.1. Implementation

In this chapter necessary equations for FSI method implementation are presented. There

are a few steps to follow in order to make the FSI method working. The received signal from the GPS travels through parts of the atmosphere where multipath occurs. The signals are then composed of one or more narrow banded subsignals with varying instantaneous frequency dependant on the multipath. Each of the subsignals corresponds to a single path, each instantaneous frequency will be related to a given ray. It is easy to compute the instantaneous frequency of a single tone as a derivative of the phase, but it is much more complicated if signal are composed of more than one tone. FSI is a simple and efficient way to derive instantaneous frequency of a signal composed of narrow banded subsignals [5]. In FSI the Doppler effect is used to retrieve the information about conditions in the lower troposphere. The Doppler effect on the measured navigation signals is caused mainly by two factors: satellites movement and impact of the atmosphere. The impact of satellites movements can be accounted for by using navigation data. The measured Doppler signal y(t) and its phase φ(t) can be defined as follows:

tti

navNCONCOetAnavty )()()(

(3.2)

)()( tyt (3.3)

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time: t[s]

|Ana

v(t)| [V

]

Anav

& noise

Anav

26

Where: nav - [-] +1 or -1 navigation bits obtained from ground segment Anav(t) - [V] amplitude of the Doppler signal φNCO(t) - [rad] is the NCO phase of the measurement caused by Doppler effect ΩNCO - [rad/s] is the estimated centre angular frequency according to φNCO(t) and is calculated by equation 3.4:

2

)(max

)(min

dt

td

dt

td NCONCO

NCO

(3.4)

Estimated centre angular frequency is defined as mean of maximum and minimum value of Doppler phase derivative in time domain.

All calculations are based on Fast Fourier Transform analysis of the Doppler signal. It has the following form:

)'()'()'( iueUyfftV (3.5) Where: V(ω’) - [V] is the complex valued Fourier transform of the measured Doppler signal ω’ - [rad/s] is the transform angular freq. shifted with the estimated centre angular freq. fft - is the fast Fourier transform algorithm U(ω’) - is the amplitude of the Fourier transform u(ω’) - is the phase of the Fourier transform The sampling frequency is 1 kHz, so the fft domain is in range of ω’ = 2πf, where f = [-500,500] Hz. Now the subtracted frequency ΩNCO is introduced as a shift of the Fourier transform:

)'()'()(

~NCOiu

NCO eUV (3.6)

ω = ω’ + ΩNCO - [rad/s] is the transform angular freq. shifted with the estimated centre angular frequency. The Fourier transform amplitude and phase of the Doppler signal with CNR equals 50 dBHz are presented in figures 3.5 and 3.6.

27

Figure 3.5. Amplitude U(ω) of Fourier transform in whole range of angular frequency

Figure 3.6. Phase of Fourier transform u(ω) in whole range of angular frequency illustrating a problem

with phase unwrapping at an angular frequency of 1.275 rad/s.

Because of the low sampling rate there are problems with phase unwrapping as illustrated in Figure 3.6. To interpolate frequency to a denser grid zero padding was introduced to the time signal. The measurement vector y(t) of 88451 samples was extended to 188451 samples by zeros padding. This solution avoids distortions of the fft phase visible in figure 3.6. The phase u(ω) without problems of phase unwrapping is presented in figure 3.7.

1.25 1.26 1.27 1.28 1.29 1.3 1.31

x 105

-160

-140

-120

-100

-80

-60

-40

Angular frequency: [rad/s]

Am

plitu

de o

f F

our

ier

tran

sfor

m:

U(

)[dB

]

1.24 1.25 1.26 1.27 1.28 1.29 1.3 1.31

x 105

-6

-5

-4

-3

-2

-1

0

1x 10

4


Pha

se o

f Fou

rier tran

sfor

m: u

() [rad

]

28

Figure 3.7. Phase of the Fourier transform u(ω) in whole range of angular frequency

As can be seen in figure 3.5 the signal frequency is in the range of 1.267 - 1.291 rad/s. That is why only for this frequency range the phase of fft is considered. Figures 3.8 and 3.9 present amplitude and phase in considered range of angular frequency.

Figure 3.8. Amplitude of the Fourier transform U(ω) in useful range

Figure 3.9. Phase of the Fourier transform u(ω) in useful range

1.25 1.26 1.27 1.28 1.29 1.3 1.31

x 105

-18

-16

-14

-12

-10

-8

-6

-4

-2

0x 10

4


Phase

of Fou

rier

tra

nsfo

rm: u(

) [rad]

1.27 1.275 1.28 1.285 1.29

x 105

-52

-51

-50

-49

-48

-47

-46

-45

-44


Am

plitu

de o

f F

ourie

r tr

ansf

orm

: U

( )[

dB]

1.27 1.275 1.28 1.285 1.29

x 105

-14

-13

-12

-11

-10

-9

-8

-7

-6

-5x 10

4


Pha

se o

f Four

ier tran

sfor

m: u(

) [rad

]

29

Because the signal is penetrating the atmosphere, its frequency components will be separated and shifted, thus each Doppler frequency will be received by LEO in different time, according to [5]. Under the assumption that each Doppler frequency component is received only once in the occultation, reception time can be determined as follows:

)(

)(u

t (3.7)

t(ω) - [s] is the reception time for the measured frequency component ω ω - [rad/s] is the received angular frequency component Let us now use Equation 3.7 to the simulated data shown in Figure 3.9. The result are shown in Figure 3.10.

Figure 3.10. Reception time t(ω) determined from the Fourier transform phase

The reception time t(ω) is a parameter needed to calculate the atmospheric properties. In the rest of the chapter all calculations are based on t(ω) calculated by Equation 3.7. Because the bending angle α is basically retrieved from FFT phase in angular frequency domain, all used time domain parameters such as satellites positions have to be associated to an angular frequency domain as well. Thus the direct relation between these parameters and angular frequency is developed using t(ω).

The next step is to determine impact parameter a. To do so, the separation angle Θ(t) between LEO and GPS satellites (Figure 2.1) has to be calculated and transformed to the angular frequency ω domain using reception time t(ω). Based on the positions of satellites, the separation angle Θ(t) in the centre of the Earth is calculated, as follows:

)()(

)()(arccos)(

tt

ttt

LEOGPS

LEOGPS

PP

PP (3.8)

PGPS(t) - GPS position vector PLEO(t) - LEO position vector

1.27 1.275 1.28 1.285 1.29

x 105

0

10

20

30

40

50

60

70

80


Rec

eptio

n tim

e: t

( )

[s]

30

Figure 3.11. Value of GPS and LEO satellite separation angle Θ(t) in time t

When the Θ(t) is known, the aim is to calculate ( ) which is the separation angle in angular frequency domain. By using the reception time t(ω) in the frequency domain it is possible to map Θ(t) from the time domain to ( ) in a frequency domain. Mapping is done by interpolation Θ(t) to a continuous function and then by finding the appropriate values of Θ for t(ω). Thanks to that we can retrieve every value of Θ for t(ω). Then the

~ as a

function of angular frequency ω is obtained. The Θ angle approximately changes linearly in time, but in order to increase the accuracy, Θ was approximated to a third order polynomial where coefficient were determined by using “polyfit()” Matlab’s function. Function approximated by this method has the following form:

1.754 + 4-6.042E + 09-4.607E - 11--3.391E)( 23 tttt (3.9)

Figure 3.12. Θ(t) in discrete time domain and its approximated continuous function

0 10 20 30 40 50 60 70 80 901.75

1.76

1.77

1.78

1.79

1.8

1.81

1.82

Time: t[s]

Sat

ellit

es s

epar

atio

n an

gle:

[rad

]

0 10 20 30 40 50 60 70 80 901.75

1.76

1.77

1.78

1.79

1.8

1.81

1.82

Separ

atio

n a

ngle

: (t)[ra

d]

time: t[s]

(t) in discrete domain

(t) approximated to continuous function

31

Figure 3.13. Θ(t) in discrete time domain and its approximated continuous function, over a short time

period

To show the accuracy of the approximation the root mean square error (RMS) was calculated and is equal to 7.92E-7 rad, which means that it is very small and can be neglected. Response of the approximated function described by the equation 3.9 and discrete values of Θ(t) are shown in figures 3.12 and 3.13.

Figure 3.14. Calculated ( ) in frequency domain

Similar solution was applied to calculate the position of LEO satellite for arbitrary time t(ω). The position of the GPS satellite did not require any estimation while all calculations are provided for coordinate system where the satellite does not change its position during occultation.

40.835 40.84 40.845 40.85 40.855 40.86 40.865 40.87 40.875

1.779

1.779

1.779

1.779

1.7791

Separ

atio

n an

gle

: (t)[ra

d]

time: t[s]

(t) in discrete domain

(t) approximated to continuous function

1.27 1.275 1.28 1.285 1.29

x 105

1.75

1.76

1.77

1.78

1.79

1.8

1.81

1.82


Sat

ellit

es s

epar

atio

n an

gle:

[r

ad]

32

The next step is to determine the impact parameter a(ω) and that is according to [2]:

)(

)(~

)(~)(

1)(

)(~

)(~)(

1)(

)(~ 22

dt

da

r

a

dt

rd

r

a

dt

rdk

LEO

LEO

GNSS

GPS

(3.10)

2

k (3.11)

k - wave number of L1 carrier, where λ equals the length of the wave In the equation there are distances to the two satellites from the Earth centre. First the derivatives of the distances between the Earth centre and satellites have to be determined based on the positions of the satellites in angular frequency domain. Then for a known angular frequency ω, it is possible to find the impact parameter values that are true for Equation 3.10. Equation 3.10 is solved numerically by a Newton-Raphson method. This method finds such a(ω) for which the error estimation between calculated and desired ω is less than 0.0001. When it is greater the difference between both ω is added to the a(ω) in next iteration and calculation proceeds. As an initial condition, a simplified definition for impact parameter is used. If the ideal circular satellites’ orbits would be considered, then derivatives of radii obtain zero values and derivative of Θ will be constant. This enables to determine direct equation for the initial condition of the impact parameter which has following form:

)(

)(~)(

dt

dk

a

(3.12)

Using the initial condition the solution to Equation 3.10 is found and the corresponding a(ω). Finally having impact parameter a(ω) bending angle α can be determined according to equation 3.13:

)(~)(

arcsin)(~

)(arcsin)(

~)(

GPSLEO r

a

r

aa (3.13)

LEOr~ (ω) - [m] value at time t(ω) of the distance between LEO satellite and Earth centre

GPSr~ (ω) - [m] value at time t(ω) of the distance between GPS satellite and Earth centre

A flow chart (Figure 3.15) showing connections between variables and above equations

can be very useful in facilitation the understanding of the dependencies in the FSI method. It can be seen that bending angle is a function of the position of the satellites, the angle between them and the impact parameter.

33

Figure 3.15. Flow chart of all variables and equations used in FSI method implementation

3.2.2. Filtering

In order to remove the noise from data i.e. on Anav(t)), φNCO(t)), and on u(ω) a zero-phase

low-pass filter “filtfilt” from Matlabs library was used. Because of high noise levels in input signals, a number of filters was proposed and tested. This is done in order to find the one that provides best results. Tested filters is found e.g. in [6] and they are: Bessel filter - IIR type filter, its frequency response is flat in stopband and in passband. Bessel filter has the widest transition zone. In Matlab the filter is described like: [B,A] = BESSELF(N,Wo), where B are the coefficients in the numerator and A are the coefficients in the denumerator, N is the filter order and Wo is the frequency up to which irrespective of the signal frequency its delay remains roughly constant. Butterworth filter – IIR type filter, as well as the Bessel filter, its frequency response is flat in the stopband and in the passband. In Matlab the filter formula is: [B,A] = BUTTER(N,Wn), where B are the coefficients in the numerator and A are the coefficients in the denumerator, N is the filter order and Wn is the cutoff frequency and must be in the range of 0<Wn<1 (1 is a half of a sample rate). Chebyshev I filter – IIR type filter, contains ripples in the passband. In Matlab the filter is written as: [B,A] = CHEBY1(N,R,Wp), where B are the coefficients in the numerator and A are the coefficients in the denominator, N is the filter order, R is peak-to-peak ripple in the passband and Wp is the normalized passband edge frequency.

34

Chebyshev II filter – IIR type filter, contains ripples in the stopband. In Matlab the function is: [B,A] = CHEBY2(N,R,Wst), where B are the coefficients in the numerator and A are the coefficients in the denumerator, N is the filter order, R stopband ripple and Wst is a normalized stopband edge frequency. Moving average filter – FIR type filter, it is operating on a set of input data and averaging its value to produce output value. In Matlab the filter is described as: [B,A] = ones(N,1)/N, where B are the coefficients in the numerator and N is the filter length and denominator equals A = 1.

3.2.2.1. Amplitude of the Doppler signal (Anav(t))

To find the proper values of the filters settings, a couple of tests were performed. Because of random character of the noise, a mean of the tests was calculated. At first, a single filter was tested using “for” loop in Matlab, where its parameters values were increasing. Results of bending angle errors for noise level 40 dBHz for Butterworth filter are shown in Table 3.2. To speed up the calculations, the mean was calculated from errors in 10 sets of noise. Table 3.2. RMS bending angle errors for Butterworth filter. Each value*10E-3 [rad] N Wn 0.003 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027 0.030 0.033 0.036 0.039

1 2.048 2.025 2.021 2.018 2.018 2.017 2.024 2.045 2.097 2.140 2.227 2.240 2.322

2 2.032 2.018 2.017 2.016 2.016 2.016 2.016 2.045 2.047 2.045 2.076 2.174 2.174

3 2.024 2.017 2.017 2.016 2.016 2.016 2.016 2.046 2.046 2.046 2.044 2.063 2.169

4 2.021 2.017 2.017 2.016 2.016 2.016 2.016 2.046 2.046 2.017 2.025 2.050 2.164

5 2.019 2.017 2.016 2.016 2.016 2.016 2.016 2.046 2.030 2.017 2.025 2.050 2.175

6 2.018 2.018 2.016 2.016 2.016 2.016 2.018 2.046 2.018 2.018 2.024 2.050 2.167

As can be seen in above table, the best results (marked in dark blue) are achieved for Wn in the range (0.006 to 0.021). After that analysis, tests with higher accuracy were performed as follows, i.e. the range of Wn is set to values of 0.006 to 0.022, N in the range of 2 to 6. The mean was calculated from 40 error values obtained from 40 noise sets. The best result is accepted as filter settings. After all tests for Anav data, the following filters are chosen: Table 3.3. Filter settings used in further calculations Filter Settings Bandwidth [Hz] Bessel [N,Wo] N=2 - Wo=38.340 0 – 4.75 Butterworth [N,Wn] N=4 - Wn=0.010 0 – 5.00 ChebyshevI [N,R,Wp] N=6 R=0.5 Wp=0.009 0 – 4.76 ChebyshevII [N,R,Wst] N=5 R=20.0 Wp=0.011 0 – 4.64 Moving average [N] N=86 - - 0 – 5.14 As can be seen in the above table, all filters have similar bandwidth in the range of 0 – 5 Hz. The Anav(t) is a low frequency signal, as can be seen in figure 3.16.

35

Figure 3.16. Amplitude of the Doppler signal in frequency domain

To receive meaningful results for each filter, we tested them under exactly the same conditions, by using 150 sets of noise generated by Matlab and saved previously to a file and calculate the mean of error values and its standard deviations for each program run. The tests were performed for 4 different noise values: 65, 55, 45 dBHz – noise levels in realistic scenario and 20 dBHz, to show the effectiveness of the filter. The test results are shown below in Table 3.4: Table 3.4. RMS bending angle errors [rad] for different noise levels applied to Anav(t)

Filter Carrier to noise density ratio [dBHz] / Standard deviation σ

65 σ 55 σ 45 σ 20 σ

No filter 2.2457

E-5 2.3668

E-6 4.4927

E-5 1.1685

E-5 5.8543

E-4 1.0947

E-4 0.0217

2.5490E-4

Bessel 2.0090

E-5 1.1280

E-8 2.0090

E-5 3.5683

E-8 2.0117

E-5 1.1309

E-7 9.3201

E-5 5.0726

E-5

Butterworth 2.0060

E-5 1.1412

E-8 2.0061

E-5 3.6113

E-8 2.0089

E-5 1.1502

E-7 1.2104

E-4 5.0872

E-5

Chebyshev I 1.9948

E-5 1.1994

E-8 1.9949

E-5 3.8054

E-8 1.9990

E-5 1.5180

E-7 1.3937

E-4 5.0535

E-5

Chebyshev II 2.0058

E-5 1.1429

E-8 2.0059

E-5 3.6180

E-8 2.0086

E-5 1.1482

E-7 1.1376

E-4 5.6065

E-5

Moving average 2.0087

E-5 1.1296

E-8 2.0087

E-5 3.5734

E-8 2.0114

E-5 1.1342

E-7 1.0086

E-4 5.4598

E-5

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

frequency: f/fs[-]

Do

pp

ler

sig

na

l am

plit

ud

e in

fre

q d

om

ain

A na

v(f)[

dB

]

36

Results marked on blue are the best for each noise value and results marked on red are the worst respectively. As can be seen on Table 3.3, the best results for noise values in range 45 – 65 dBHz achieved by Chebyshev I filter, and the worst results are obtained by Bessel filter. For noise level 20 dBHz the roles reversed. The best result is derived by a Bessel filter and the worst by Chebyshev I. For further calculations Chebyshev I is chosen as default filter for Anav(t) data. Error values in row “No filter” explain that the use of the filter is necessary in order to calculate correct bending angle profile.

3.2.2.2. FFT phase

FFT phase filtering was tested in the same way as Anav(t) data. At first filter settings were found based on 10 noise sets. Settings for filters found in the tests are shown below in Table 3.5:

Table 3.5. Filter settings used in further calculations Filter Settings Bandwidth [Hz] Butterworth [N,Wn] N=1 - Wn=0.003 0 – 1.50 ChebyshevI [N,R,Wp] N=1 R=0.5 Wp=0.001 0 – 1.40 ChebyshevII [N,R,Wst] N=2 R=20.0 Wp=0.009 0 – 1.90 Moving average [N] N=274 - - 0 – 1.60 After the selection of the bests filter settings, the tests based on 150 noise sets were performed for different noise values. The mean RMS error from 150 runs of the program are shown in Table 3.6: Table 3.6. RMS bending angle errors [rad] for filtering phase of the FFT

Filter Carrier to noise density ratio [dBHz] / Standard deviation σ [rad]

65 σ 55 σ 45 σ 20 σ

No filter 9.0813

E-5 6.5031

E-6 8.6735

E-5 7.7316

E-6 8.8037

E-5 7.5591

E-6 6.8142

E-4 2.8433

E-4

Butterworth 2.3601

E-5 1.3903

E-6 2.2671

E-5 2.1452

E-6 2.1121

E-5 1.5120

E-6 1.3835

E-4 4.8764

E-5

Chebyshev I 2.4560

E-5 1.6784

E-6 2.3370

E-5 2.5453

E-6 2.1486

E-5 1.8166

E-6 1.3582

E-4 4.8356

E-5

Chebyshev II 2.3437

E-5 1.1593

E-6 2.2790

E-5 1.2934

E-6 2.2319

E-5 8.6881

E-7 1.5497

E-4 5.3371

E-5


E-5 1.2205

E-8 1.9941

E-5 3.8652

E-8 1.9984

E-5 1.5083

E-7 1.4043

E-4 5.1228

E-5 On red are marked the worst results and on blue the best results, respectively. The best efficiency has moving average filter and it is chosen as a default u(ω) variable filter. The “No filter” row shows the necessity of the filter usage.

37

3.2.2.3. Phase φNCO

Phase φNCO is a variable generated in the oscillator based on the Doppler model. The measuring system, oscillator and other devices used in the processing can introduce noise. For phase φNCO only moving average filter was tested. The method in filter’s settings selection is exactly the same as mentioned above. The results for different noise levels are presented below: Table 3.7. RMS bending angle errors [rad] for filtering the phase φNCO


65 σ 55 σ 45 σ 20 σ

No filter 1.3904

E-4 7.6342

E-9 1.3543

E-4 9.8194

E-6 1.2315

E-4 2.1859

E-5 6.1684

E-4 1.7752

E-4


E-5 1.2182

E-8 1.9835

E-5 3.8523

E-8 1.9868

E-5 1.2231

E-7 1.3932

E-4 5.1317

E-5 The radio occultation signal phase φNCO was filtered by function “filtfilt” in Matlab as

well. A simple filter which averages 308 consecutive samples was used (bandwidth 0 – 1.44 Hz). The filter is necessary because the bending angle error values calculated without the filter are an order higher that with the filter.

3.2.2.4. Bending angle α(ω)

In this case only moving average filter was tested. The best results were obtained for filter length of 8 samples. The results for different noise values are shown below: Table 3.8. RMS bending angle errors [rad] for filtering bending angle


65 σ 55 σ 45 σ 20 σ

No filter 1.9834

E-5 1.2182

E-8 1.9835

E-5 3.8523

E-8 1.9868

E-5 1.2231

E-7 1.3932

E-4 5.1317

E-5


E-5 1.2178

E-8 1.9835

E-5 3.8509

E-8 1.9869

E-5 1.2223

E-7 1.3931

E-4 5.1319

E-5

Results without filter are almost the same as results with filter, therefore the use of the filter is not necessary as it does not change the final results. In further calculations the filter for bending angle α was not used.

3.3. Data analysis and methodology

To evaluate accuracy of the FSI method and filtering, root mean square error is calculated between reference and calculated bending angle. Because calculated height h is not a continuous function and not given in the same points as the reference chart it was necessary to interpolate calculated bending angle in points of reference domain. Note that the height is used as vertical coordinate.

38

3.4. Test results and conclusion The simulated data below comes from reference profile number 9 as defined in the EUMETSAT study [21]. Below are presented 4 figures showing bending angle calculated for 3 different CNR noise levels 65, 55, 45 dBHz and a figure presenting distribution of the bending angle error in terms of height for those 3 noise levels.

Figure 3.17. Bending angle at different heights for CNR-65 dBHz


2 4 6 8 10 12 14 16

x 10-3

2

4

6

8

10

12

14

16

18

20

Bending angle: [rad]

Hei

ght:

h[km

]

Calculated bending angleReference bending angle

2 4 6 8 10 12 14 16

x 10-3

2

4

6

8

10

12

14

16

18

20


Hei

ght:

h[km

]


39


As illustrated in Figures 3.17, 3.18, 3.19 the method implemented in this work can calculate bending angle very well for a moderate noise level. For realistic (45 – 65 dBHz) noise scenarios, the shape is practically the same as reference bending angle. The difference between retrieved and reference bending angle in terms of height for different C/No ratios is depicted in figure 3.20

Figure 3.20 Difference between calculated and reference bending angle profiles in function of height for

different CNR ratios

2 4 6 8 10 12 14 16

x 10-3

2

4

6

8

10

12

14

16

18

20


Hei

ght:

h[km

]


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 10-4

2

4

6

8

10

12

14

16

18

20

Error of Bending Angle [rad]

Hei

ght:

h[km

]

CNR = 45 dBHzCNR = 55 dBHzCNR = 65 dBHz

40

The highest noise values are present in the height of about 6 to 9 km. The RMS results for each test are presented in Table 3.9. Table 3.9. RMS bending angle errors for different level of noises after filter application

Carrier to noise density ratio [dBHz] 65 55 45 RMS [rad] 2.0379E-5 2.0367E-5 2.0802E-5

The filters are working with very high efficiency, the percentage difference between error values for CNR 45 and 65 dBHz is in the range of 2%, while the percentage difference between error values for noise ratio 55 and 65 dBHz can be neglected (0.06%). Filters are chosen correctly to retrieve bending angle profiles as close to the reference profiles as possible.

41

4. Implementation of FSI method for more demanding cases of input data (Case 42)

4.1. Presentation of input data

4.1.1. Similarities and differences between the two input data sets

More demanding case of input occultation data (Case 42) are affected by many unpredictable factors with very random character and that is why it is far more complicated to process them compared to previous data set presented in section 3. Case 42 is important to analyze to ensure that the algorithms work in the real case. The main differences and resemblances between data for Case 42 and Case 9 are showed below:

- Both data are based on L1 frequency. - The influence of multipath is visible on both data sets. - The noise level in Case 42 is similar to the noise in Case 9 described in chapter 3

with CNR of 55 dBHz. The differences:

- More demanding input data set is longer (200 s) than Case 9 (88 s). - The noise in all simulated RO signals are not colored - Gaussian with zero mean,

while in Case 42 it is more realistic. The noise sources can be as follows: instrumental noise, ionospheric noise and noise due to small-scale turbulence. Ionospheric noise is the most significant part of the noise, while small-scale turbulence is the most important for absorption obtaining from radio occultations [7].

- More complex input data consists of many data gaps, especially in the first 90 s of measurement (rising). Data gaps are causing information loss and to restore the information, the signal requires filtration. However that does not guaranty that the correct result is obtained.

- The Anav(t) amplitude level in Case 9 is in (0 - 2)*106 range, while in more demanding data set (Case 42) it is scaled to the lower range of (0 - 6)*10-8.

4.1.2. Case 42

The algorithms used in Case 42 is an extended version of the programs used in section 3

on Case 9. That means that all FSI algorithms remain unchanged, i.e. calculations of waves’ reception time, determination satellites’ positions in arbitrary time or finding impact parameter by iterative method. For details see [4]. The main difference is that Case 42 uses sophisticated and more complex methods to achieve satisfying results.

Case 42 contains two sets of data. First set contains measurements of the occultation signals and satellites positions when LEO satellite is rising as it is depicted on the Figure 4.1. In the second set, measurements are obtained for the setting LEO satellite as presented in Figure 4.2. The setting occultation data is created from the rising occultation data by reversing the order of the samples. Hence, the expected bending angle is the same.

42

Figure 4.1. Positions of the satellites when for rising LEO satellite. (Blue circle indicates GPS satellite,

brown circle LEO satellite)

Figure 4.2. Positions of the satellites for setting LEO satellite. (Blue circle indicates GPS satellite, brown

circle LEO satellite)

43

Presented on the Figures 4.1 and 4.2, red circle indicate the point on the Earth surface above which the bending angle is calculated. Trajectories of the LEO satellites are marked by brown lines. Small circles at the end of the trajectories indicate positions of the LEO satellites at the end of the measurements. Positions of GPS satellites are represented by small blue circles on the left. There are no trajectories of the GPS satellites, because they do not change their positions in presented coordinate system. The blue dashed line determines the Earth’s rotation axis which coincides with z-axis of this coordinate system.

The data sets are similar. They are received and preprocessed in the same way, using open loop mode described in details in chapter 3.1. The data sets contain similar types of noise and they are carrying the same information about bending angle, thus they are processed the same way. Due to these similarities, only the rising LEO satellite will be described in details.

First of all the more demanding case of input data are very noisy and additionally contain set of gaps. Fortunately most of the gaps appear when the tracking of Doppler phase is performed, and where the GPS signal is not received yet. They result from the simulated receiver’s search for the signal. In case of the rising LEO, the gaps visible in figure 4.3 are present in the part of the signal when only thermal noise appears (first 90 s). In case of the setting LEO shown in figure 4.7, great majority of gaps appear in the last 140 seconds of the data set, but it is still possible that these gaps appear when signal from GPS is received, thus those signals, especially its phase require strong filtering. Signal of the rising LEO is presented in figure 4.3. For such a case the gaps in correlator values signal Anav are replaced by samples with values equal 0 [8].

Figure 4.3. Magnitude of the Anav in time domain for rising LEO satellite

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time: t[s]

|Ana

v| [V

]

44

Figure 4.4. Angle of the Anav in time domain for rising LEO satellite

In case of the rising LEO satellite, illustrated in Figure 4.4, influence of multipath is visible in around 100 – 120 s.

Figure 4.5. Magnitude of the Anav in time domain for setting LEO satellite

Figure 4.6. Angle of the Anav in time domain for setting LEO satellite

0 20 40 60 80 100 120 140 160 180 200-800

-600

-400

-200

0

200

400

600

Time: t[s]

(A

nav)

[rad

]

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time: t[s]

|Ana

v| [V

]

0 50 100 150 200 2500

200

400

600

800

1000

1200

1400

1600

Time: t[s]

(A

nav)

[rad

]

45

As can be seen in Figure 4.6, for setting LEO satellite, the provided measurement time is longer and is 250 s. The multipath region is visible between 80 and 100 s for the setting satellite.

The phases of the occultation data φNCO also have to be fixed and filtered. Due to presence of the gaps, the phases φNCO lost their continuity and took sawtooth shape over the gaps (see Figures 4.7 and 4.8)

Figure 4.7. Phase φNCO of occultation signal for rising LEO satellite

Figure 4.8. Phase φNCO of occultation signal for setting LEO satellite

Satellites positions were obtained by ground segment thus do not require any filtering.

Same as in the case of the simulated data, positions are described in coordinate system fixed to the GPS satellite.

46

4.2. Implementation of FSI method on Case 42

In this chapter, Case 42 improvements of the program are presented. The program used to process data from Case 9 is not able to perform the calculations correctly for Case 42. The problems that appear are considerable enough in order to make α profile retrieval more difficult. There is a need to implement many changes to the program to make it a proper working tool. The program’s processing steps with a short explanation are presented in figure 4.9.

Figure 4.9. Processing steps in the program

Preprocessing the input data

Satellites’ corrections

Radio Holographic filtering

FFT of the Doppler signal

Calculation of the reception time

Determination of satellites' possitions

Calculation and analysis of impact parameter

Bending angle retrieval

Calculation of impact parameter based on the Doppler signal components' frequency and satellites possitions

Analysis and eventual correction of impact parameter profile

Determination of the satellites possitions for every signal component based on the reception time

Determination of Doppler signal components reception time based on the FFT phase

Calculating FFT of such processed Doppler signal

Combining the correlator values and Doppler phase shift into Doppler signal and its filtration using RH filter to remove noise

Transformation of satellites' posstions due to Earth oblateness

Filling data gaps in measured corellators val. using samples equal to "0"

Fixing the discontinuity and filtering of the Doppler phase measurement

Calculation of BA based on impact parameter profile and satellites' possitions for every frequency components of Doppler signal

47

4.2.1. Fixing the discontinuity of φNCO

Due to gaps in signal during reception, gaps in correlator signal also appears. Since the phase φNCO is retrieved from the same noisy signal, lost samples also will affect into φNCO measurements. At the moment of signal loss, NCO resets the counter and starts calculating the phase from the beginning. This causes discontinuity in the φNCO measurements, which must be resolved.

A solution proposed in this master thesis is based on the derivative of the φNCO analysis and removal of higher values with opposite sign. Removed values of derivative are replaced by samples calculated from the average of the adjoining samples. Afterwards the phase derivative is also filtered using moving average filter with length 30, then it is integrated. Result of filtering for rising LEO satellite is presented in figure 4.10.

Figure 4.10. Phase φNCO of occultation signal after filtering for rising LEO satellite

4.2.2. Radio holographic filtering (RHF)

Due to sufficient filtering of the Doppler signal y(t), it was not necessary to introduce an advanced filtering method in Case 9, but in Case 42 filters accepted in the previous case were not able to perform the filtering well. That is why the RHF is used. The idea of the radio holographic filter is taken from [9], written by M. E. Gorbunov and K. B. Lauritsen. They investigate several scenarios of calculation the bending angle for noisy and non-noisy occultation data. It appears that using Fourier Integral Operators (FIO) such as CT or FSI, a strongly smoothed amplitude of the radio occultation signal in time domain does not cause a constant value of the FIO amplitude in multipath region. A major role in the FSI plays the phase of occultation signal, not the amplitude. Moreover they observe that the high resolution bending angle can be obtained even when the amplitude of the RO signal is not used. This implies that the most crucial factor is the radio occultation signal phase, thus RHF is focusing mostly on this parameter.

There are two approaches where noise filtering from RO signal is carried out. First, provide filtering based on Doppler signal y(t) (eq. 3.2), and secondly provide it on FFT of the

48

Doppler signal )(~ V (eq. 3.5). Both approaches are based on the same principles. In this

master thesis RHF on Doppler signal y(t) is used, thus only this will be explained. RHF can be divided into five steps. First, the Doppler signal phase φ(t) (eq.3.3) is

smoothed to get a reference phase shape φm(t). Then this reference phase is subtracted from original signal y(t). This step ensures that the spectrum will be narrow-banded and thus the multipath structure would not be disturbed. Next, the FFT of modified signal ym(t) is calculated and multiplied by FFT of the window. The length and shape of the window determines attenuation level. Due to presence of the real noise the best results of filtering are expected for Gaussian window. Then the inverse FFT is calculated and reference phase φm which was initially subtracted is added back. General scheme of filtering is presented in figure 4.11 [9].

Figure 4.11. Scheme of RHF

1. Subtracting the reference phase ))(()()( ti

mmetyty

2. Fast Fourier Transform )()( tyFY mm

3. Gaussian filtering

)()()( tWFtyFY Gaussmfilt

m

4. Inverse Fast Fourier Transform

)()( 1 filtm

filtm YFty

5. Adding the reference phase ))(()()( tifilt

mfilt metyty

49

Figure 4.12. Filtered and original value of the Doppler signal phase for rising LEO satellite

Figure 4.13. Filtered and original value of the Doppler signal phase for rising LEO satellite over a short

time interval

50

Figure 4.14. Filtered and original magnitude of the Doppler signal for rising LEO satellite

As it is presented on the Figures 4.13 and 4.14 radio holographic filter attenuates additional noise. The phase of Doppler signal preserve its shape and therefore it becomes more smooth. The noise from the magnitude was also removed. The value of the Doppler signal magnitude was normalized before filtering.

To obtain such a results, smoothing window of size 2 s (2000 samples) was used to retrieve the reference phase model φm(t). Width of the Gaussian window was set to 0.2 s (200 samples), and standard deviation of the window σw = 0.4. Settings were chosen empirically.

After RH filtering the signal is Fourier Transformed and the phase of the signal in frequency domain still may contain small variations. Due to high sensitivity of the FSI method on these undesirable variations, they are removed using smoothed window with length 1600 samples.

4.2.3. Corrections of satellite's orbits (CSO) due to Earth's oblateness

The next step of improving the FSI method is consideration of the Earth’s oblateness and its impact on the results. In the simulations presented in section 3 this factor could be neglected due to its very small influence, but in Case 42 it may be significant.

To introduce correction to the coordinate system, position of curvature centre has to be determined. To achieve that, set of transformations and calculations is necessary. First step is to transform the satellites positions to J2000 coordinate system.

J2000 is a Cartesian coordinate system fixed to the stars in which the origin is at the centre of the Earth. Z-axis coincidences with Earth rotation axis and X and Y axes span the equatorial plane. In comparison to coordinate system WGS84 (in which the satellites are described), in the J2000 system X and Y axes do not change their orientation when Earth rotates. Since both systems have common Z-axis, the transformation is based on the angle shift γ of the X and Y axes along the equator [10]. It follows that the value of this transformation angle γ is closely related with time when the satellites positions was registered in time due to the satellite movement and Earth rotation. The details and whole algorithm of calculating the γ is clearly described in [10] thus it will not be quoted here.

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

Time: t[s]

Mag

nitu

de: |

y|

|y||y

filtered|

51

When the angle γ is obtained then the transformation is executed by simple matrix multiplication:

E

E

E

z

y

x

z

y

x

100

0cossin

0sincos

2000

2000

2000

(4.1)

Where: x2000,y2000,z2000 - coordinates in J2000 coordinate system fixed to the stars xE,yE,zE - coordinates in Cartesian coordinate system fixed to the GPS satellite (WGS84) γ - transformation angle

In the rest of the report all positions relates to the J2000 coordinate system. When the local curvature centre is calculated (see appendix A), the satellites orbits are corrected by subtracting the local curvature centre position from J2000 coordinates of the satellites positions.

cocGPS_J2000GPS PPP (4.2)

cocLEO_J2000LEO PPP (4.3) 4.2.4. Horizontal gradients correction (HGC)

Horizontal gradients are causing variations of the value of impact parameter a(ω) along a

signal ray. In the occurrence of strong horizontal gradients dependent of height, a bending angle profile may become a multi-valued function. The FSI method bases on the fact that the impact parameter is a unique coordinate in the ray span. It is true for ideally spherical atmosphere, but not always in line with horizontal gradients. This is causing a lot of problems in properly retrieving the bending angle profile. The easiest way to explain this phenomenon is to describe that situation on a simplified scheme:

Figure 4.15. Horizontal gradients

As can be seen in figure 4.15, horizontal gradients are marked on yellow. The signal bends through the atmosphere, and for a certain distance its way is not changing the altitude from the Earth’s surface. Bending angle is dependent of impact parameter, and as can be seen on the above figure, impact parameter for the yellow section is not changing. The solution for

52

that can be introduction of a new coordinate instead of impact parameter, such as the ray manifold can have a unique projection on the suitable axis. The problem is that ray manifold has a specific structure, which is dependent on unfamiliar horizontal gradients. This is causing the impossibility to choose a universal coordinate that can reveal multipath. The solution for that can be a different approach - estimate bending angle errors [11]. This method is useless in our case because it solves horizontal gradients in bending angle, while we still have incorrect impact parameter profile. The impact parameter affected by the horizontal gradients is shown in figure 4.16:

Figure 4.16. Impact parameter a(ω) affected by horizontal gradients. At 3.9 105 the effect of horizontal

gradient can be seen

The peak visible at around 390000 sample is a direct effect of the fluctuations of the frequency in the signal components and thereby horizontal gradients. This is causing problem in the plotting the bending angle profile, because the same height is available for different bending angle values. In this master thesis, based on the principles of FSI method, we propose simple method to identify the range where the horizontal gradients affect on results.

Our method identifies the impact of the horizontal gradients based on analysis of the impact parameter and basic assumptions of the FSI. According to our method there are three conditions which if are not fulfilled indicate on the horizontal gradients impact:

- unambiguity of Doppler signal frequency components - impact parameter is decreasing (for rising LEO) and increasing (for setting LEO)

function in the whole range - absolute value of the impact parameter derivative is less than 400 meters (small

fluctuations) Firstly, we proceed with the explanation of the unambiguity of Doppler signal frequency

components. Full Spectrum Inversion method, as it was mentioned before, relies on the assumption that each frequency component of the occultation signal occurs once, so that unambiguous relation of signal component frequency and its reception time to the LEO satellite occurs [5]. According the Jensen, this relation can be determined as follows:

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

6.38

6.4

6.42

6.44

6.46

6.48x 10

6

no of a sample

impa

ct p

aram

eter

[m]

53

qqqq

q

qqqq tt

d

dt

d

dt

dt

dtt

d

d

d

du

)( (4.4)

Where: u – Fourier Transform phase ω – frequency of the signal component, Fourier Transform domain φq – phase of the signal component q in time domain tq – reception time of the component q

General in parametric form:

d

du

dt

tdtt ,

)(),( (4.5)

According to relations listed above, if the derivative of the Doppler signal phase φ(t) at the reception time t is equal to the signal component frequency ω, then the assumption is fulfilled and FSI method works properly. Unfortunately, this relation in a presence of strong noise, e.g. horizontal gradients can be disturbed and those frequency components for which this equation will not be true, should be treated with uncertainty.

Figure 4.17. Values of the signal components frequencies determined as Fourier Transform domain and

value of the signal component phase derivative

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

-1.3

-1.29

-1.28

-1.27

-1.26

-1.25

x 105

no of the signal component

freq

uenc

y of

the

sign

al c

ompo

nent

[rad

/s]

derivative of the signal component phaseangular frequency of the signal component

54

Figure 4.18. Values of the signal components frequencies determined as Fourier Transform domain and

value of the signal component phase derivative in short time period

As it is depicted in figures 4.17 and 4.18, the values of the frequencies for signal y(t) components from number 387000 to 390700 are different. It is assumed that if that difference exceeds 1.5 rad/s, then for those samples the assumption of the unambiguity is not fulfilled and resultant reception time and impact parameter is defined as erroneous.

Second condition is a consequence of the unambiguity of the Doppler signal. If impact parameter has some local extremes, there will be detected by first condition. However, to evaluate globally the impact parameter, it is necessary to introduce the criterion which will analyze the value of the sample with respect to all previous samples.

Additionally to prevent rapid fluctuations and dynamic changes of the impact parameter values, third condition is checked.

Based on these three conditions the program returns “1” for samples where horizontal gradients may occur (and thus where impact parameter may be erroneous) and “0” for samples where horizontal gradients impact is not detected. Results of the detection the horizontal gradients impact for rising LEO are presented in figure 4.19.

3.875 3.88 3.885 3.89 3.895 3.9 3.905

x 105

-1.3

-1.29

-1.28

-1.27

-1.26

-1.25

x 105

no of the signal component

freq

uenc

y of

the

sign

al c

ompo

nent

[rad

/s]

derivative of the signal component phaseangular frequency of the signal component

55

Figure 4.19. Function which shows whether all conditions are fulfilled “0”, or not “1”

When the presence of the erroneous impact parameter is determined, then a Matlab's function "linspace" is applied in those places. This function creates a linear function from one point to another point, where the ranges of “linspace” function are ones.

Figure 4.20. Impact parameter with substituted parts

As can be seen in figure 4.20, this solution is not perfect, there still exist some distortions in the impact parameter profile, especially visible around 40000th sample, but such an impact parameter exhibit much better performance in final results.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

0

0.2

0.4

0.6

0.8

1

no of sample

valu

e

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

6.38

6.4

6.42

6.44

6.46

6.48x 10

6

no of sample

impa

ct p

aram

eter

[m]

56

4.3. Results In figure 4.21. a flow chart presenting all equations and improvements to the program is shown.

Figure 4.21. Flow chart of all variables and equations in FSI method implementation for Case 42

In this section results of bending angle profiles for Case 42 are shown. Each data set

(rising LEO and setting LEO) is presented in 4 ways depending on the use of RHF (section 4.2.2) and CSO (section 4.2.3):

a) Bending angle without RHF and without CSO b) Bending angle with RHF and without CSO c) Bending angle with RHF and with CSO d) Bending angle with RHF, CSO and HGC

Also bending angle with RHF, CSO and HGC are shown for: e) Case 9 f) Case 5

The derived programs are also tested for the Case 9 given in section 3 and the Case 42.

57

a) Bending angle without RHF and without CSO:

Figure 4.22. Bending angle vs. height based on data from rising LEO satellite (Case 42). This case is

without RHF and CSO

Figure 4.23. Bending angle vs. height based on data from setting LEO satellite (Case 42). This case is without RHF and CSO

Figures 4.22 and 4.23 present the bending angle profiles obtained using smoothing window on the Fourier Transformed signal’s phase. The length of the window equals 1600 samples it was used instead of RHF to compare the results with the smoothing window. In this case no CSO was introduced.

0 0.01 0.02 0.03 0.04 0.05 0.062

4

6

8

10

12

14

16

18

20


Hei

ght:

h[km

]

reference bending anglecalculated bending angle

0 0.01 0.02 0.03 0.04 0.05 0.062

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ght:

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]


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b) Bending angle with RHF and without CSO:

Figure 4.24. Bending angle vs. height based on data from rising LEO satellite (Case 42). This case is with

RHF and without CSO

Figure 4.25. Bending angle vs. height based on data from setting LEO satellite (Case 42). This case is with

RHF and without CSO

In figures 4.24 and 4.25 bending angle profiles with RHF are depicted. Due to horizontal gradients the bending angle is a multivalued function of impact parameter. Notice that in spite of the similar conditions in which the data was obtained, the results from setting and

0 0.01 0.02 0.03 0.04 0.05 0.062

4

6

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Hei

ght:

h[km

]


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ght:

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]


59

rising LEO are not coherent as maybe expected. It is due to noisy environment and influence of the gaps which cause loss of the data.

c) Bending angle with RHF and with CSO:


RHF and CSO

Figure 4.27. Bending angle vs. height based on data from setting LEO satellite (Case 42). This case is with RHF and CSO

0 0.01 0.02 0.03 0.04 0.05 0.062

4

6

8

10

12

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18

20


Hei

ght:

h[km

]


0 0.01 0.02 0.03 0.04 0.05 0.062

4

6

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10

12

14

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18

20


Hei

ght:

h[km

]


60

In figures 4.26 and 4.27 the bending angle using RH filtering and satellite corrections is presented. The CSO introduced due to Earth oblateness slightly improve results. The Root Mean Square error is smaller than error for bending angle calculated without CSO.

d) Bending angle with RHF, CSO and HGC:


RHF, CSO and HGC

Figure 4.29. Bending angle vs. height based on data from setting LEO satellite (Case 42). This case is with RHF, CSO and HGC

0 0.01 0.02 0.03 0.04 0.05 0.062

4

6

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12

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18

20


Hei

ght:

h[km

]


0 0.01 0.02 0.03 0.04 0.05 0.062

4

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Hei

ght:

h[km

]


61

As can be seen in figures 4.28 and 4.29, bending angle profile is almost properly positioned, only a small error of height profile causes a slight shift downward in rising LEO data, and upward in setting LEO data. Table 4.1. RMS error results based on the following solutions

Solution RMS error rising

LEO [rad] RMS error setting

LEO [rad] a) Bending angle without RHF and CSO 2.0630E-3 2.3015E-3 b) Bending angle with RHF and without CSO 3.5536E-3 5.1089E-3c) Bending angle with RHF and CSO 3.3119E-3 4.5810E-3 d) Bending angle with RHF, CSO and HGC 2.3940E-3 2.0904E-3 As can be seen in Table 4.1, the results for rising LEO seem to be the best for case where no filtration and no CSO or HGC were applied. However, looking at the graphs it is clear that maximum value of the calculated α is much smaller than reference one. These results are far away from desired profile. Using RHF we are able to reconstruct high values of the BA. Thanks to CSO we get closer to the reference shape. The only problem is the proper determination the impact parameter distorted by horizontal gradients. Even small change in the value of impact parameter causes high values of the mean square error. As it was questioned in RQ2, taking into consideration the Earth’s oblateness to satellites orbits gives some benefits to the final bending angle profile. The MSE error is lower in case c) than in case b).

e) Case 9: In figure 4.30 the α profile for Case 42 with noise level of 45 dBHz is presented.

Figure 4.30. Bending angle vs. height based on simulated input data (Case 9)

2 4 6 8 10 12 14 16

x 10-3

2

4

6

8

10

12

14

16

18

20


Hei

ght:

h[km

]


62

The RMS error for the α profile calculated by the final program (3,2363E-5) is higher than with the first program (2.0802E-5), nevertheless shape is preserved with high accuracy even for high noise value.

f) Case 5 In figure 4.31 the α profile for more demanding Case 5 is presented.

Figure 4.31. Bending angle vs. height based on another set of more demanding case of input data (Case 5)

The program calculated the profile correctly, but it is shifted upwards due to incorrect calculated impact parameter. RMS error for the calculated profile is equal to 2,8184E-3.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.0352

4

6

8

10

12

14

16

18

20


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ght:

h[km

]


63

5. Summary and conclusions

In order to predict the weather with high accuracy there has to be measurement made all over the planet. Therefore mankind has to develop good methods to perform that. One of these methods is the radio occultation technique. It allows measurements remotely, using a GNSS satellite as a signal source and a LEO satellite as a receiver with the measurement instruments. Radio occultation also opens new possibilities to meteorological research due to the high coverage incorporating also the big oceans. The large coverage also implies more data to analyze than before and by that better weather forecasting and better opportunity to model the climate for climate change monitoring. It will as well provide more research and by that a larger understanding about Earth´s atmosphere. Important weather and climate properties of the atmosphere are given by the bending angle of the ray path between the satellites. There are two main methods to process RO data in order to obtain bending angle. A traditional one based on geometrical optics, and the second one, due to multipath of the GPS signal in the atmosphere, based on physical optics. One of the physical optics based method is Full Spectrum Inversion, which is the basis of this master thesis. FSI method, as the name can suggest, relies on the Fourier transform. The frequency analysis allows determination of the reception times for different frequency components of the ray. The associated time is then used to estimate the bending angle. The main goal of this master thesis was to implement FSI method in the existing Matlab program delivered by the RUAG company. At first, a set of simulated RO data (Case 9) was processed. These data had an adjustable artificial noise power, and the FSI algorithm had no problems in retrieving bending angle profile. The next sets of data were more complex occultation data with much more noise sources attached, one was for setting and one for rising LEO satellite. These noise sources could be from the measurement instrument, small scale turbulence in the atmosphere and the greatest source - the ionosphere. A substantial use of the filters was added to remove the noise from the input data. In Case 9 disturbances were removed by FIR filters. Chebyshev I had the best result in comparison to other filters, like Butterworth, Bessel and moving average. The data set contained even more complex occultation data (Case 42) was more difficult to process. Beyond the noise, the signal was distorted by data gaps and horizontal gradients. Bending angle profile obtained from the more demanding case of input data was far from the reference one. To improve result, other solutions were applied. One of them was RHF. This method operates on the signal phase in the following order: calculation the reference phase using smoothing window, subtracting the reference phase from Doppler signal, calculating FFT, apply Gaussian filtering to this signal and calculate inverse FFT from filtered signal and the resulted reference phase is added back to the filtered signal. The next method to improve bending angle profiles was corrections of satellite's orbits due to Earth oblateness. This was done by finding the local centre of curvature to the planet. The last considered problem was the horizontal gradients, which breaks the Jensen's assumption that the derivative of the phase at the reception time is equal to the signal component frequency. It causes serious implications, especially in the impact parameter profile, considered as a bending angle domain which can not have the same values of height for different samples. The problem was partially solved by estimation of the improper part of impact parameter. Retrieved bending angle profile for simulated RO data worked almost perfect, while for the more demanding case there are still issues that need further

64

investigation. The most important for further research is to find a proper profile of the impact parameter and an improved filter in order to have better bending angle shape.

Answering to first research question, the impact of noise on the final bending angle profile varies with height, as can be seen in Figure 3.20. The error has the greatest value around 9,6 and 3 km above the earth surface. In Case 42, where more sources of noise was affecting the RO input data, it is very difficult or impossible to detect what influence each kind of noise has on the bending angle profile. The noise can be treated as a sum of its components. The biggest problem in Case 42 is to retrieve a proper shape of a peak visible at around 3.5 km of height. Because of the noise, it is difficult to reproduce rapid value change in the bending angle profile.

Answering to the second research question, as the difference between equatorial radius and polar radius is about 21 km, this should not lead to big error value in the bending angle profile, but to retrieve possibly the best results, it is recommended to implement satellites corrections. In our case, satellites corrections gave better results, the error value is lower about 7% for setting LEO and 9% for rising LEO.

Summarizing, the program presented in this master thesis is able to calculate the bending angles for nearly all input data sets. However, it requires additional tuning in order to reduce the error caused by incorrect height distorted by horizontal gradients impact.

65

6. Future work

a) Case 9 Results of bending angle obtained from Case 9 data are good, very close to reference

model. Therefore there is not many things too improve. The improvements can be: The optimization of program code for shorter processing time, better efficiency

and less computational power. Improving results, especially at the end of bending angle profile (the spit). Corrections of satellites’ orbits.

b) Case 42 The bending angle profiles obtained for more complex RO data are not perfect. There are

still many options to improve the results: Improving the program code for better efficiency and shorter processing time. Implement a good solution for horizontal gradients. The best way can be to find a

universal coordinate instead of impact parameter for which ray manifold has a unique projection on the axis. It is very difficult because we do not know the horizontal gradients. The other method is to estimate errors caused by horizontal gradients.

Improve filtering. Dealing with data gaps. Data gaps are causing significant errors in bending angle

retrievals. Thus it is necessary to deal with it. Gaps can be removed by interpolation or by strong filtration.

Implementation of ionospheric corrections. Ionospheric propagation is causing a nonlinear refraction effects on GPS signal. This is causing an error in the bending angle profile.

Collecting input data from many sources. It can be done by using for example 2 LEO satellites at almost same position as a tandem mission. Each of them will collect data from the same GPS satellite, in almost the same time. That creates a possibility to merge data into one data set, which can allow to eliminate data gaps, influence of the noise, increase the accuracy, but slow down a little processing time.

66

Appendix A

The perigee point for the lowest part of occultation has to be determined (Figure A.1). Perigee is calculated as follows [4]:

2

GPSLEO

GPSGPSLEOGPSLEOGPSper

PP

PPPPPPP

(A.1)

Pper – is the perigee positions for the given satellites positions PGPS – is the GPS satellite positions described in J2000 coordinate system PLEO – is the LEO satellite positions described in J2000 coordinate system

Figure A.1. Occultation geometry

Note that due to bending of the ray path the perigee point may occur under the Earth

surface. However, the most important is the perigee point for the lowest part of the occultation, i.e. placed the closest the Earth centre during whole occultation. In figure A.2 this point is indicated by red point. So from the set of perigee points vectors it is necessary to find one with the smallest length. It is so important, because relative to this point the occultation point is determined in further calculations.

67

Figure A.2. Position of the occultation point and perigee point for the lowest part of occultation

Based on the satellites’ and perigee point positions it is possible to calculate the position of the occultation point which will indicate the location of local centre of curvature. Because of the oblate shape of the Earth, determination of the occultation point can be simplified to

two dimensions [Rxy,z], where . As it is presented in figure 2.2, Rxy and z are coordinates of the cross section along meridian under which the perigee point is identified. For such a model the Earth surface cross section is defined by geometrical equation of the ellipse:

12

2

2

2

n

z

m

Rxy (A.2)

m – major axis of the Earth cross-section, equal: Requator = 6378137 [m] n – minor axis of the Earth cross-section, equal: Requator * (1-fe) = 6356752 [m] where: fe – is flattening factor of the Earth, equal: 0.00335281 Now, if occultation point is the closest point to the perigee, it can be found at the intersection of the ellipse curve and its normal which runs through the perigee point. To determine the linear equation of the normal line, firstly we have to find the slope of the tangent line in the

68

occultation point. Tangent to any curve is defined as the line having slope equal to derivative of this curve. Derivative of the ellipse equals:

2

2

2

2

1n

z

m

Rxy (A.3)

dzn

zdR

m

Rxy

xy

22

20

2 (A.4)

zm

Rn

dR

dz xy

xy2

2

(A.5)

So slope of the tangent line in occultation point takes form:

occ

occxy

zm

Rnslope

2

_2

tan_ (A.6)

Knowing that normal line is perpendicular to the tangent line, slope of normal line in occultation point is equal:

occxy

occ

Rn

zm

slopenormslope

_2

2

tan_

1_ (A.7)

To determine the full formula of the linear equation of the normal: z = slope*Rxy+c, it is necessary to determine constant c. For occultation point linear equation has to be true, so:

cRRn

zmz occxy

occxy

occocc _

_2

2

(A.8)

Then:

2

2

2

2

1n

mz

n

zmzc occ

occocc (A.9)

So for arbitrary occultation point [Rxy_occ, zocc] on the ellipse, the equation of the normal to this ellipse in that point has following form:

2

2

_2

2

1n

mzR

Rn

zmz occxy

occxy

occ

(A.10)

If this normal line runs through the known perigee point [Rxy_per, zper], the coordinates of the occultation point [Rxy_occ, zocc] can be found from following set of equations:

69

1

1

2

2

2

2_

2

2

__

2

2

n

z

m

R

n

mzR

Rn

zmz

occoccxy

occperxyoccxy

occper

(A.11)

Where: 22

_ occoccoccxy yxR

22_ perperperxy yxR

perperper zyx ,, – [m] coordinates of the perigee point position for the lowest part of occultation

occoccocc zyx ,, – [m] coordinates of the occultation point position m – [m] major axis of the Earth cross-section, equal: 6378137 m n – [m] minor axis of the Earth cross-section, equal: 6356752 m Because of the high level of complexity, this set of equations is solved by iterative method similar to this which calculates impact parameter. The initial condition is set to [Rxy_occ zocc] =[Rxy_per, zper]. When coordinates of the occultation point [Rxy_occ, zocc] are determined, the coordinates xocc, yocc and occultation point position Pocc are calculated from:

cos_ occxyocc Rx (A.12)

sin_ occxyocc Ry (A.13)

],,[ occoccocc zyxoccP (A.14) χ – is the longitude of the perigee point, equals: arctan2(yper,xper)

When the position of the occultation point is found, the last parameter necessary to determine local curvature centre position is radius Rc of this curvature. Radius is calculated using geodetic coordinates of the perigee position, according [10]. The longitude, latitude and other geodetic coordinates are presented in figure A.3 and are calculated as follows:

),(2arctan perper xy (A.15)

22_ perperperxy xyR (A.16)

nR

mz

xy

perarctan (A.17)

3

3

cos

sinarctan

meR

nez

xy

per (A.18)

xper, yper, zper – [m] are the Cartesian coordinates of the perigee point [m] χ – [rad] is the longitude of the perigee point Rxy_per – [m] is cylindrical radius of perigee point [m] ψ – [rad] is eccentric anomaly

70

ζ – [rad] is a latitude of Perigee point, where e is earth elipticity, equals:

2)1(1 ef

arctan2() – is the arctan() function providing the ±π radian output defined as:

arctan2(y, x) = arctan for x > 0, and extended to the four quadrants for x ≤ 0

Figure A.3. Perigee position in geodetic coordinate system

Then the parallel and meridian radii of curvature are equal:

2

322

2

sin1

1

e

emrcm

(A.19)

2

122 sin1 e

mrcp

(A.20)

rcm – [m] is the principle radius of curvature along the meridian rcp – [m] is the principle radius of curvature perpendicular to the meridian

71

The angle between the occultation plane and the local meridian is calculated from:

per

per

GPSLEO

GPSLEO

P

P

PP

PP

per

per

z

zarccos (A.21)

Then the Earth local radius of curvature is defined as:

rcprcm

Rc 22 sincos

1

(A.22)

Finally the position of the local curvature centre is obtained from [4]:

ecnR ˆ occcoc PP (A.23)Where en is Earth normal unit vector:

242_

4

2_

2_

2 ,sin,cosˆ

occoccxy

occoccxyoccxye

zmRn

zmRnRnn

(A.24)

72

Reference list [1] S. Syndergaard. (2005). Introduction to GPS Radio Occultation [Online]. Available FTP: www.cosmic.ucar.edu Directory: summercamp_2005/presentations File: Syndergaard_Stig_20050530.pdf [2] RUAG Space AB. Internet: http://www.ruag.com/en/Space/Space_Home/RUAG_Space_Sweden, [Oct. 15, 2012]. [3] G. Blewitt. (1997). Basics of the GPS Technique: Observation Equations [Online]. Available FTP: www.nbmg.unr.edu Directory: staff/pdfs File: Blewitt%20Basics%20of%20gps.pdf [4] M. Bonnedal, “RADIO OCCULTATION (RO) LEVEL 1B PRODUCTS ATBD SCARO-15”, RUAG Company, 2012. [5] A. S. Jensen, M. S. Lohmann, H.-H. Benzon, and A. S. Nielsen, “Full Spectrum Inversion of radio occultation signals,” Radio Science, vol. 38, no. 3, 2003. [6] M. Dahl S. Johansson and M. Winberg, Signal Processing. Collection of Formulas, Department of Telecommunications and Signal Processing, Ronneby, Sweden, October 2005. [7] M. E. Gorbunov, K. B. Lauritsen, A. Rhodin, M. Tomassini, and L. Kornblueh, 2006: Radio holographic filtering, error estimation, and quality control of radio occultation data. J. Geophys. Res., 111, D10105, doi:10.1029/2005JD006427. [8] RUAG Space AB, Final Report, March 2012. [9] M.E. Gorbunov and K.B. Lauritsen, "Radio Holographic Filtering of Noisy Radio Occultations", DMI, Scientific Report 05-06, DMI, Copenhagen, Denmark, 2005. http://www.dmi.dk/dmi/sr05-06.pdf [10] H.-H. Benzon, A. S. Nielsen, and L. Olsen, “An atmospheric wave optics propagator - theory and application,” DMI, Scientific Report 03-01, DMI, Copenhagen, Denmark, 2003. http://www.dmi.dk/dmi/sr03-01.pdf [11] M.E. Gorbunov and K.B. Lauritsen, "Error Estimate of Bending Angles in the Presence of Strong Horizontal Gradients", OPAC-3 Proceedings, Graz, Austria, September 2007. [12] W.G. Melbourne, Radio Occultations Using Earth Satellites: A Wave Theory Treatment, ser. Deep Space Communications and Navigation Series, Jet Propulsion Laboratory, California Institute of Technology, pp. 1-4, 2004. [13] J. Christensen. (2005). GRAS Open Look Tracking. [Online]. Available FTP: http://www.romsaf.org Directory: Workshops/golw File: 05_christensen.pdf [14] von Engeln, A., Andres, Y., Marquardt, C., and Sancho, F.: GRAS Radio Occultation on-board of Metop, Adv. Space Res., 47(2), 336–347, doi:10.1016/j.asr.2010.07.028, 2010. [15] Wikipedia. Internet: http://en.wikipedia.org/wiki/Earth, [Feb. 1, 2013] [16] A. Carlström. (2013, Jan. 24). "RE: Input data - correlator values and phiNCO" Personal e-mail. [17] EUMETSAT Study: Optimisation of Tracking strategies for Radio Occultation, EUM/PEPS/SOW/10/0024 [18] A. Carlström. (2013, Feb. 14). "RE:RE:RE: Input data - correlator values and phiNCO" Personal e-mail.

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[19] Jensen, A. S., M. S. Lohmann, A. S. Nielsen, and H. Benzon (2004), Geometrical optics phase matching of radio occultation signals, Radio Science, vol. 39, RS3009, doi:10.1029/2003RS002899. [20] Gorbunov, M. E. and K. B. Lauritsen, “Analysis of wave fields by Fourier integral operators and their application for radio occultations,” Radio Science, vol. 39, no. 4, 2004. [21] Optimisation of Tracking Strategies for Radio Occultation, EUMETSAT study, http://www.eumetsat.int/Home/Main/Satellites/EPS-SG/Resources/index.htm?l=en [22] Gorbunov, M. E., 2002a: Canonical transform method for processing radio occultation data in the lower troposphere, Radio Sci., 37(5), 1076, doi: 1029/2000RS002592. [23] Pelliccia, F. Pacifici, F. Bonafoni, S. Basili, P. Pierdicca, N. Ciotti, P. Emery, W.J., "Neural Networks for Arctic Atmosphere Sounding From Radio Occultation Data," Geoscience and Remote Sensing, IEEE Transactions on , vol.49, no.12, pp.4846,4855, Dec. 2011, doi: 10.1109/TGRS.2011.2153859 [24] Gorbunov, M. E., and K. B. Lauritsen (2002), Canonical trans-form methods for radio occultation data, Sci. Rep. 02-10,Dan. Meteorol. Inst., Copenhagen. (Available as http://www.dmi.dk/dmi/Sr02-10.pdf) [25] Gorbunov ME, Benzon HH, Jensen AS, Lohmann MS, Nielsen AS (2004) Comparative Analysis of Radio Occultation Processing Approaches Based on Fourier Integral Operators. Radio Science 39(6), RS6004, doi:10.1029/2003RS002916 [26] Sokolovskiy, S., Rocken, C., Schreiner, W., and Hunt, D.: On the uncertainty of radio occultation inversions in the lower troposphere, J. Geophys. Res., 115, D22111, http://dx.doi.org/10.1029/2010JD014058doi:10.1029/2010JD014058, 2010

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