The local ionospheric modeling by integration ground GPS observations and satellite altimetry data ANNALS OF GEOPHYSICS, 59, 6, 2016, A0654; doi:10.4401/ag-6810 A0654 The local ionospheric modeling by integration ground GPS observations and satellite altimetry data Mohammad Ali Sharifi1,2, Saeed Farzaneh1,* 1 School of Surveying and Geospatial Engineering, College of Engineering, University of Tehran, Iran 2 Research Institute of Geoinformation Technology (RIGT), College of Engineering, University of Tehran, Iran ABSTRACT The free electrons in the ionosphere have a strong impact on the propaga- tion of radio waves. When the signals pass through the ionosphere, both their group and phase velocity are disturbed. Several space geodetic tech- niques such as satellite altimetry, low Earth orbit (LEO) satellite and very long baseline interferometry (VLBI) can be used to model the total electron content. At present, the classical input data for development of ionos- pheric models are based on dual-frequency GPS observations, However, a major problem with this observation type is the nonuniform distribution of the terrestrial GPS reference stations with large gaps notably over the sea surface and ocean where only some single stations are located on is- lands, leading to lower the precision of the model over these areas. In these regions the dual-frequency satellite altimeters provide precise information about the parameters of the ionosphere. Combination of GPS and satel- lite altimetry observations allows making best use of the advantages of their different spatial and temporal distributions. In this study, the local ionosphere modeling was done by the combination of space geodetic ob- servations using spherical Slepian function. The combination of the data from ground GPS observations over the western part of the USA and the altimetry mission Jason-2 was performed on the normal equation level in the least-square procedure and a least-square variance component esti- mation (LS-VCE) was applied to take into account the different accuracy levels of the observations. The integrated ionosphere model is more accu- rate and more reliable than the results derived from the ground GPS ob- servations over the oceans. 1. Introduction The Earth’s ionosphere is a layer of the atmos- phere at an altitude of about 60 km to 2000 km above the Earth’s surface, which contains enough electrons and ions to effectively interact with the electromagnetic fields. This layer plays an important role in wireless communication due to its influence on radio propaga- tion [Blaunstein and Christodoulou 2007]. Ionospheric information such as the vertical total electron content (VTEC) can be extracted from different space geodetic techniques: global navigation satellite system, satellite altimetry missions such as TOPEX/Poseidon and Jason, very long baseline interferometry (VLBI) and low Earth orbit satellites (LEOs). Each technique has its specific characteristics influencing the derived ionosphere pa- rameters so it can be assumed that the combined model of the ionosphere can make the best use of the advan- tages of each particular space geodetic technique and improve the reliability and the accuracy of the VTEC determination. The classical input data for the develop- ment of the local ionosphere map is obtained from ground-based GPS observations, however the distribu- tion of the permanent GPS stations is not globally uni- form. By combination of the GPS observations with the other data it is intended to increase the local iono- sphere maps precision. In the field of combining the different space-geo- detic observations for the ionospheric modeling, several studies have been done, Todorova et al. [2007] developed the global models of the ionosphere by integration of GNSS and satellite altimetry data. Then Alizadeh et al. [2011] enhanced the model accuracy by adding radio occultation measurements (RO) from space-based GPS. For regional modeling Schmidt et al. [2008] combined GNSS, RO and satellite altimetry data and in Dettmer- ing et al. [2011] four observation types (terrestrial GPS, space-based GPS, altimetry, and VLBI) are combined into one regional three-dimensional VTEC model. In this study, terrestrial GPS and altimetry were in- tegrated into one regional three-dimensional VTEC model. In this procedure, all observations were combined in one joint adjustment taking into account the system- atic offsets between the observation groups. Consider- ing the different accuracy levels of the input data in the Article history Received June 16, 2015; accepted October 4, 2016. Subject classification: Slepian function, LS-VCE, Satellite altimetry, GPS, Local ionospheric modeling. stochastic model plays a crucial part in this procedure. In this study, the different variance factors will be esti- mated within a least-square variance component estima- tion (LS-VCE). The remainder of this paper is organized as follows. Firstly, the description of the measurements used as input data will be introduced (Section 2). Then, the general model approach and the concept for data combination will be discussed (Section 3). Finally, the VTEC model results will be presented (Section 4). 2. Input data The present study is based on the data from ground- based GPS and satellite altimetry observations. Both techniques allow the observation and modelling of the ionosphere, but each of them has its specific characteris- tics which influence the derived ionosphere parameters. The satellite altimetry data help to compensate the in- sufficient GPS coverage of the oceans. The 24h obser- vations of 30 stations belong to the International GNSS Service (IGS) network (http://sopac.ucsd.edu/) and the sampling rate of the measurements considered is 30 s. The VTEC values from the satellite altimetry Jason-2 considered in this study were obtained from the dual-frequency ionosphere delay measurements in the Geophysical Data Record (GDR). Jason-2 has taken over and continued Topex/Poseidon and Jason-1 missions since June 2008 in the frame of a cooperation among the Centre national d’études spatiales (CNES), the Euro- pean Organisation for the Exploitation of Meteorological Satellites (EUMETSAT), the National Aeronautics and Space Administration (NASA) and National Oceanic and Atmospheric Administration (NOAA). It is equipped with the Poseidon-3 dual-frequency radar altimeter at 14.6 Ghz (Ku-band) and 5.3 GHz (C-band). 2.1. TEC from ground GPS In this study a method of measuring TEC directly from the differential code delay and carrier phase meas- urement on both the L1 and L2 frequencies is used. For this purpose, the geometry-free linear combination of pseudo range and carrier phase measurements (also termed ionospheric observable) which is formed by subtracting simultaneous pseudorange or carrier phase observations is used [Ciraolo et al. 2007]. For pseudor- ange and carrier phase measurements the ionospheric observable can be obtained by Equations (1) and (2) re- spectively: where P1, P2, U1 and U1 are the code and carrier phase pseudo ranges on the L1 and L2 signals with f1= 1575.42 MHz and f2 = 1227.60 MHz frequencies, re- spectively. I1 and I2 are the ionospheric refraction de- lays at L1 and L2, respectively. br = c (xrp1− x r p2) and Br = c (T rL1− T r L2) are the code and phase inter-frequency bi- ases (IFBs) for the receiver, bs = c (xsp1− x s p2) and Bs = c (TsL1− T s L2) are the code and phase differential inter-fre- quency biases for the satellite, fp and fL are the effects of multipath and measurement noise on the pseudo- range and carrier phase, respectively, mi is the wavelength of the Li carrier phase, and Ni is the integer carrier phase ambiguity. Slant total electron content (STEC) can be computed using Equations (1) or (2). Although carrier phase measurements show low noise behavior in comparison with pseudo range measurements it is not favored here for calculating STEC due to the esti- mation of the ambiguity term in the preprocessing. This requires the unknown ambiguities to be estimated in a pre-processing step. In order to benefit from the ambiguity-independent estimates of STECs derived from the code pseudo-ranges and the high precision of the carrier phase measurement, the pseudo-range ionospheric observables are smoothed using the “car- rier to code leveling process” method [Ciraolo et al. 2007, Nohutcu et al. 2010]. By combining Equations (1) and (2) for simultaneous observations, the following equation can be obtained: (3) Because the noise and multipath term for the car- rier-phase observation is much lower than that for the pseudo-range observation, it can be neglected [Hofmann- Wellenhof et al. 2008]. The interval at which no cycle- slip error has occurred, leading to a constant phase ambiguity, is regarded as the continuous observational arc [Hofmann-Wellenhof et al. 2008]. For cycle slip de- tection several testing quantities have been proposed based on various combinations of GPS observations [Seeber 2003, Hofmann-Wellenhof et al. 2008]. Herein, the observation file of each station has been processed individually and a single receiver test, i.e. the combina- tion of a phase and a code range is applied for cycle slip detection [Hofmann-Wellenhof et al. 2008]. The aver- age of the geometry-free pseudo-range and carrier phase is computed for any satellite and receiver in the continuous arc [Gao et al. 1994]:P4 P1 P2 I I c c p s p s p r p r p P4 P1 P2 1 2 1 2 1 2 x x x x f = - = - + - + + - +R R W W I I N N2 c T T c T TL r L r L s L s L 4 1 2 2 1 1 1 2 N2 1 2 1 2 m m f U U U= - = - + - + - + - +Q QV V P4 N2 Br Bs br bs N p P4 4 2 N21 1m m f U+ = + + + + + + - P4 n P4 N N2 Br Bs br bs 1 arc i n i arc p arc P4 4 P4 4 1 1 1 2 N2m m f U U+ = + = - + + + + + + = Q V| SHARIFI AND FARZANEH 2 (1) (2) (4) 3 where n is the number of measurements in the contin- uous arc. By subtracting Equation (4) from Equation (2) the ambiguity term is removed: (5) P ~ 4 is the pseudo-range ionospheric observable smoothed by the carrier-phase ionospheric one. It will minimize the effect of multipath error so it can be neg- lected [Nohutcu et al. 2010]. Inserting ionospheric de- lays into Equation (5), STEC can be obtained from the smoothed ionospheric observable in TEC units (TECU = 1016 el/m2) by: (6) Since in this study the ionosphere is modeled as a thin single spherical layer (Figure 1) the STEC values have to be converted into the height-independent VTEC by introducing a mapping function [Schaer 1999]: (7) where R is the mean Earth radius, z and z´ are the zenith angles of the satellite at the user position and the ionos- pheric pierce point, and H is the mean altitude which approximately corresponds to the altitude of the max- imum electron density and its height can vary between 250 and 500 km depending on latitude, season, solar and geomagnetic activity conditions [Misra and Enge 2003, Seeber 2003, Hofmann-Wellenhof et al. 2008]. 2.2. TEC from satellite altimetry The advent of the new satellite altimetry technique and the use of several different frequencies allow for es- timating the ionospheric electron content. Satellite radar altimeters directly measure in nadir direction. They per- manently transmit signals to Earth and receive the echo from the surface. The range between the satellite and the sea surface is determined by measuring the satellite-to- surface round-trip time of a radar pulse. Therefore, the VTEC can easily be extracted from the dual-frequency altimetry observations by converting the ionospheric correction as follows [Dumont et al. 2008]: (8) where dR is the Ku-band ionospheric range correction in meter from the Geophysical Data Record (GDR) products (iono_corr_alt_ku), and f is the Ku-band radar frequency in Hz. It is noteworthy that the TEC could then be converted to the C-band ionosphere range correction using the same formula above, and the C-band frequency of 5.3 GHz. This approxima- tion is valid for frequencies higher than 1 GHz [Lan- gley 1996]. As the 1 Hz observations are quite noisy, the smoothing and thinning of the data is advanta- geous. For this purpose, a simple median filter with a length of 20 s has been applied on the altimeter ionos- pheric range correction [Picot et al. 2008, Dettmering et al. 2011]. Figure 1 shows the sample filtered and unfiltered altimeter ionospheric range correction on the Ku-band for the Jason-2 satellite, cycle 201 and pass 137. Because of the lower orbit altitude of the altime- try satellites, the measured VTEC are expected to be lower than the values obtained from GPS. However, several studies have shown that the TEC values ob- tained by the satellite altimetry are higher by about 3- 4 TECU [Chelton et al. 2001, Brunini et al. 2005]. Thus, in the combination stage it can be assumed that the al- timetry measurements have biases, this has the effect of the plasmaspheric component (i.e. the VTEC be- tween ~1000 and ~20,000 km height above the Earth surface [Alizadeh et al. 2011]). In this study a constant daily systematic bias for the Jason-2 satellite was esti- mated in the combination procedure. 3. Spherical Slepian functions Spherical harmonic basis can effectively be used to represent the target function as long as the modeled area covers the whole sphere and the data is distributed regularly [Chambodut et al. 2005, Mautz et al. 2005]. This can be limited by data gaps, the very high degrees with small-value coefficients, and different survey data sampling density. This function is prone to error for problems in which the data measured over the part of the Earth, as the northern hemisphere or a spherical cap, since it is no longer orthogonal over the partial area P4 P4 I I br bs arc p arc L P4 P4 4 4 1 2. f f U U= + - - + + + + - u . STETEC P4 f 2 f 1 f 2 br bs f 1 40 3Lp f arc fQ 22 f 12 2f2 2 P4 f 1 f 2 pf f= - - - + - uR fQW VA D with cos sin sinMF VTETEC STETEC z z R H R z 1 E E= = = +l l VTETEC dR 40= - . f 3 2 # LOCAL IONOSPHERE MAPS OF VTEC Figure 1. Sample filtered and unfiltered altimeter ionospheric range correction on Ku-band for the Jason-2 satellite, cycle 201 and pass 137. [Beggan et al. 2013, Sharifi and Farzaneh 2016]. There- fore, the construction of a local spherical harmonic basis that is orthogonal over the studied area of the sphere is required for the study of the local modeling. In order to localize these functions into a region of in- terest R (target region), the optimization of a local en- ergy criterion can be utilized. This will give a new set of functions in the sense of Slepian [1983]. They are band-limited to a maximum spherical harmonic degree L and, at the same time, are spatially concentrated in- side a target region. In other words, Slepian functions are a particular linear combination of the spherical har- monics. Unlike the spherical harmonics they can be sorted according to their energy concentration inside the target region [Simons et al. 2006]. The spherical Slepian function can be presented as a band-limited spherical harmonic expansion: where Ylm(r) is a real spherical harmonics of degree l and order m, r is the location of a point on the surface of a unit sphere X and glm is defined as: To maximize the spatial concentration of a band- limited function g(r) within a region R, the ratio of the norms should be maximized as: where 0 ≤ m ≤ 1 is a measure of spatial concentration. The maximization of this concentration criterion can be achieved in the spectral domain by solving the alge- braic eigenvalue problem [Simons et al. 2006]: Dg = mg (12) where the elements of (L + 1)2 × (L + 1)2 localizing ker- nel D: [Equation (13) at page bottom] are given by: and g is the (L + 1)2 dimensional vector that represents a Slepian eigenfunction expressed in the spherical har- monics, i.e: g = (g00 ... glm ... gLL) T (15) This ‘localization’ matrix is symmetric and the sub- space of maximum energy is readily obtained by eigen- value decomposition. When the signal g (r) is local, it can be approximated using the Slepian expansion trun- cated at the Shannon number N [Percival and Walden 1993]: where A is the area of the region as a fraction of the full sphere. The data can be approximated, yet with very good reconstruction properties within the region by [Simons 2010]: where g (r̂) and dn are the spherical Slepian function and unknown coefficient, respectively. The combination of different space-geodetic obser- vations for the regional ionosphere modeling introduces g r l m m l L 0 = =-= Q V || g Yl rlm m Ylm Q V| g g r Ylm = X Q V# dYl rYlm XQ V maxax g g R # 2 2 m= X # X # R # d d 2 X X X XQ Q V V dD Yl Yl,lm l m YlmR # Yl mX X X=l l l lQ QV VR# N ( ) n n L 1 1 2 m= = = + | L A1 4 2V r= +Q 2V d r d gn n N 1 . = uQ V | g rn uQ V SHARIFI AND FARZANEH 4 (9) (10) (11) (14) (16) (17) (13) 5 the VTEC of two or more different space-geodetic tech- niques into one adjustment computation. It is obvious that the combined model is usually of higher quality because it is based on a large amount of observation data and takes advantage of the particular qualities of the input data. The least-square adjustment (Gauss-Markov model) is applied on each set of different observations and then the individual normal equations are combined by adding the relevant N-matrices: where Ai and Pi are the design matrix for each observa- tion and the weighted matrix for each observation which are uncorrelated and assumed to be known, respectively. The combination algorithm implies that the individual solutions are statistically independent populations, and that each of them provides an individual covariance ma- trix. Due to the various observation instruments and the functional and stochastic modeling the covariance ma- trices are scaled by individual factors before being in- troduced into the combination process. In other words, In comparison with satellite altimetry, the GNSS tech- nique has more measurements, hence in order to in- crease the effect of the satellite altimetry observations in the combined model, up-weighting of the related data is needed. On the other hand, due to the higher noise of the satellite altimetry measurements, a lower weight would be implemented [Alizadeh 2013]. The be- havior of relative weighting is similar to scaling factor when a combination of different techniques are used. These factors (vi) are estimated within a least-square vari- ance component estimation (LS-VCE) [Teunissen and Amiri-Simkooei 2008]. Its principal idea is to transform the Gauss-Markov model into the model of condition equations [Koch 1997 (p. 239), Bahr et al. 2007]. 4. Results and discussion The present study is based on the data from ground GPS and satellite altimetry observations. In order to solve STEC from ground GPS observations, the re- ceiver IFBs were calculated using the Bernese GPS soft- ware v 5.0 and the IFB values for the satellite were obtained from the Center for Orbit Determination in Europe (CODE). The STEC and VTEC values for each observation were computed as described in Section 2. The altitude for the single-layer model was set to 400 km for the calculation of VTEC and an elevation cut- off angle of degree was used. The precise orbit files, which are provided by several IGS agencies, were in- terpolated to determine satellite positions. These TEC measurements contain ionospheric electron density in- formation around the region above the GPS network and they are used as input data for the ionospheric modeling. Figure 2 shows the distribution of the Jason- 2 observation and the GPS reference stations for Au- gust 25, 2012. For the evaluation of the presented method, first the ionospheric map was developed by means of only the ground GPS observation. For this purpose two cases were considered, first the 2D analyses were per- formed, assuming that the ionosphere is static for the modeling period by neglecting the relatively small tem- poral variations in the ionosphere. The root-mean- square-error value for the residuals is 0.70 TECU for the reference solution. The VTEC map of the two-di- mensional modeling for the mid-point of the modeling period (07:45:00) UT is plotted in Figure 3. The VTEC N A PG A A PA A GPS GPS T PGPS GPS N Altimetryry T PAltimetryry Altimetryry N A Altimetryry 2 2 1 1 GPS N Altltimetryry v v = + + -W -WR R -W -W 14 24 34414444444444444444444444444444444444444444444442442444444444444444444444444444444444444444444 344443 14 24 344144444444444444444444444444424424444444444444444444444444344 LOCAL IONOSPHERE MAPS OF VTEC (18) Figure 2: Distribution of Jason-2 observation and GPS reference stations for August 25, 2012. Figure 3. 2-D VTEC model for 25.08.2012, 07:45:00 (UT) RMSE = 0.70 TECU. maps shown in Figure 3 has some artifacts on the lower left corner. These artifacts may be caused by the nature of the basis functions (definition of the boundaries), as- sociated with lack of data in this region. Second, the three-dimensional approaches based on a system of three-dimensional base functions de- fined as the tensor product of spherical Slepian func- tion for longitude and latitude in an Earth-fixed reference frame, as well as polynomial B-spline func- tions for the time [Sharifi and Farzaneh 2013]. Figure 4 shows the GPS-only VTEC map at 07:45:00 UT of day 238, 2012. The RMSE value for the residuals is 0.10 TECU for the reference solution and the optimum level of B-Spline. As can be seen, by being away from the Slepian functions domain, the results become unrealis- tic, especially over the ocean. The comparison between Figures 4 and 3 depicts the 3D modeling, thereby lo- calizing the temporal variations in the ionosphere more appropriately, because in 2D modeling, it is assumed that the ionosphere has been frozen in the sun-fixed ref- erence frame. Hence, such ionospheric modeling does not capture the small-scale and high-frequency iono- sphere disturbance. Although the temporal variations have been considered in 3D modeling, but it still has low precision since there are no observations over the sea surfaces. To compensate for this deficiency, the GPS and satellite altimetry have been combined. As mentioned before, the satellite altimetry mis- sions provide precise information about the ionosphere over the ocean, and in the view of the fact that not many GPS stations are available there, their observations pro- vide less accurate ionosphere maps over these regions. The combination is made by applying a least-square ad- justment on each set of observations and combining the normal equations. In this procedure, a constant daily bias for the Jason-2 satellite was inserted as an additional un- known parameter in the observation equation and was estimated along with the other unknown parameters. In this study the estimated bias was −3.1 TECU. Figure 5 illustrates the combined VTEC map at 07:45:00. It can be inferred that combining the altime- try data with the GPS observations influences the map mainly over the regions where the satellite altimetry provides information. The RMSE value for the residu- als was 0.08 TECU. To evaluate the approach and its accuracy, some of the altimetry observations were set apart from the combination procedure and were not involved in the modeling process. The number of excluded observa- tions is selected in such a way that the accuracy of the developed model would not reduce by the confidence interval of (1 −a = 95%) [Alizadeh 2013]. After the de- velopment of the combined model, the VTEC values at the footprints of the raw altimetry observations, formerly eliminated from the combination, were derived from the combined model. The differences between the combined model VTEC values and the raw altimetry data were utilized for the quantification of the combined model. Hence for the altimetry satellite measurements the dif- ference has been formed according to Alizadeh [2013]: (19) where VTECCLIM is the combined local ionospheric model (CLIM), VTECGDR is the raw altimetry VTEC, which is not included in CLIM, {alt, malt are the geo- graphic latitude and longitude of the raw altimetry ob- servation and t is the time of the altimetry observation. Figure 6 (the red plot) presents the average of the dif- ferences in all raw observations. As stated before a daily constant offset for the altimetry measurements has been , , , , , , VTETEC t VTETEC t VTETEC t alt alt CLCLIMIM alt alt alt altGDR { m { m { m D = - - Q Q QV V V SHARIFI AND FARZANEH 6 Figure 4. 3-D VTEC model result for August 25, 2012, 70:45:00 (UT) for j = 4, RMSE = 0.10 TECU. Figure 5. 3-D VTEC model of GPS and satellite altimetry combined, for August 25, 2012, 70:45:00 (UT) for j = 4, RMSE = 0.08 TECU. 7 considered within the combination procedure and is es- timated along with the unknown parameters in the least- square procedure. This negative mean bias indicates a systematic under-estimation of VTEC delivered by Jason-2 compared to the values delivered by GPS. The blue plot in Figure 6 depicts the mean VTEC biases after removing this offset. As can be seen, the new biases vary between [−1.5, 1.5] TECU with a mean value of 0.06 TECU. This proves a good agreement of the com- bined model with the raw altimetry observations. 5. Summary and conclusion In this study, the local ionosphere modeling has been done by the combination of space geodetic ob- servations. 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