Kalman Filtering of the GPS Data and NeQuick and NHPC Comparison

Transcription

1 WDS'12 Proceedings of Contributed Papers, Part II, , ISBN MATFYZPRESS Kalman Filtering of the GPS Data and NeQuick and NHPC Comparison Z. Mošna, 1,2 D. Kouba, 1,2 P. Koucká Knížová 2 1 Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. 2 Institute of Atmospheric Physics, Academy of Sciences, Prague, Czech Republic. Abstract. GPS data were used to test the possibility of using Kalman filtering to remove oscillations with periods of the order of seconds to improve fast estimation of the ground positioning and estimate ionospheric variations. Input parameters of the filter were tested to optimise the filter performance for a real-time prediction of TEC (total electron concentration). Additionally, TEC derived using the NHPC software was compared with the NeQuick model. In a chosen dataset, large systematic differences were found during day time while during night the NHPC and NeQuick are in good agreement. Introduction The ionosphere is a part of Earth s atmosphere notably affecting propagation of electromagnetic waves due to a sufficient amount of plasma (typical maxima of electron concentration lay in the range of cm 3 which corresponds to plasma frequencies in the order of MHz). The bottom border can be usually found around 50 km during day time and does not occur below around 90 km during night time. The upper part is called the topside ionosphere and it meets the plasmasphere (by definition in heights where light ions of H + and He + start to dominate over heavier ions, especially O + ). The ground measurements using the ionosondes [e.g., Reinisch et al., 2005] allow us to study the structure of the lower part of the ionosphere up to the maximum of electron concentration (e.g., maximum electron concentrations and heights of present layers) and its variability [Hargreaves, 1992, Davies, 1990, Sauli et al., 2006, Cander, 2009]. Direct observation of upper part of the ionosphere is impossible using the ground ionosondes. It is because vertically transmitted signal from the ionosonde is reflected if its frequency is lower than the maximum plasma frequency. If the transmitted frequency is higher, it passes away and the information about the topside ionosphere is lost as the signal keeps propagating upward. The ionosphere is highly variable with a wide range of short-time to long-time oscillations (e.g., travelling ionospheric disturbances, daily variations, periods connected to planetary waves, 27 day and annual periods, 11 years modulation etc.). The oscillations are caused mainly due to the solar and geomagnetic activity as well as influenced by the neutral atmosphere and other effects. In present paper, we concentrate on ionospheric oscillations with periods of the order of tens seconds and longer. Total electron content (TEC) in the zenith direction is defined as T EC = zenith N e ds, where N e is an electron density and s is a vertical path [Xu, 2003]. Slant TEC is total electron content computed along a slant path with zenith angle > 0. Both parameters are connected via a so-called mapping function [e.g., Hernandez Pajares et al., 2004]. The TEC is a crucial parameter used to compute an ionosphere-induced delay of a signal between the GNSS (Global Navigation Satellite System) transmitters and ground receivers [Brunini and Azpilicueta, 2010, Komjathy, 1997, among others]. The Galileo project is primarily aimed to a positioning similarly to other GNSS systems (GPS, Glonass etc.), however it may bring additional and more precise information about the state of the ionosphere. Usually, two frequencies are normally used to reduce the ionospheric error [Xu, 2003]. A broad band E5 signal is publicly accessible Galileo code and will serve as as an improved positioning system even using one frequency ( snap/publications/nagaraj&dempster2009b.pdf). 210

2 Besides the positioning, a ionospheric measurement may be carried on with a stationary receiver as the time-of-flight of the signal is related strongly to the TEC. This paper consists of two parts. In the first part, we tested signals from the GPS satellites to optimise a Kalman filter for a simple real-time prediction and smoothing of single satellite receiver derived TEC. This is an introductory step to use the Kalman filtering which will be followed by more ambitious goals (e.g., 2D mapping using large set of satellites and receivers). In the second part we compared the NeQuick 2 model [Leitinger et al., 2005, Nava et al., 2008] with ionosonde derived TEC using NHPC software [Huang and Reinisch, 1996]. Both NeQuick and NHPC are advanced and widely used tools for the electron profile estimation. Data TEC data for the Kalman filter come from Bogota GPS observatory (4.6 N, 74 W) and WTZR (Wettzell, Germany, 49 N, 12.5 E), Oct 2003 which are days with varying and moderately disturbed geomagnetic activity (1+ Kp 6, Dst min = 55 nt). Time resolution of the data is one second. TEC values were computed using dual frequency approach [Xu, 2003]. Ionograms with 15 min time resolution were measured at the Pruhonice observatory (50 N, 14.5 E) in geomagnetically quiet days (Kp 2) of Nov 5 6, They were manually processed and the TEC was computed using the NHPC algorithm. NHPC TEC values were compared to the NeQuick model with height ranges 0 1,000 km and 0 20,000 km. Kalman filtering Kalman filter is an algorithm producing estimates of unknown variables and since its development it has been used in a wide number of applications (e.g., navigation, engineering, etc.). Here, we only give a short introduction and refer the reader to the original paper of Kalman [1960] which is rather complex, or to some of introductory papers explaining basic ideas of this tool (e.g., 08.pdf or short but very illustrative papers of Simon [2001] or Maybeck [1979]). Kalman filter usage for purposes of GPS derived TEC estimation or data assimilation is described in e.g. Carrano et al. [2009], Hajj et al. [2004], among others. During the filter process, the estimates (predictions) are computed from indirect, inaccurate and uncertain measurement using recursive steps. Kalman filter is an optimal estimator which means that it minimises the mean error of the estimated parameters. In case of a linear system, the process may be described using the state equation as and the output (measurement) is x k+1 = Ax k + Bu k + w k (1) y k = Cx k + z k, (2) where A,B and C are matrices, k is the time index, x is the state of the system, u is the known input to the system, y is the output and w and z are process and measurement noise, respectively. It is necessary to point that the state of the system x cannot be measured directly due to the noise w and z; we only know the measurement value y k and both x and y are by some extent corrupted by noise [Simon, 2001]. Two criteria should be valid to find the best estimation of ˆx. First, the average value of the estimation is equal to the average value of the process (E ˆx = E x ). Second, we need as small error variance as possible (minimum E( ˆx x 2 )). Using Kalman filtering, we use the assumption that both process noise w and measurement noise z have normal distribution N(0, Q) and N(0, R), respectively, and that there is no correlation between w and z (they are independent). The estimation ˆx k+1 is computed as (Aˆx k +Bu k ) plus the correction term K k (y k+1 C ˆx k ). ˆx k+1 = (Aˆx k + Bu k ) + K k (y k+1 C ˆx k ) (3) 211

3 The Kalman gain ( gain of the correction ) K k is computed as K k = AP k C T (CP k C T + S z ) 1 (a 1 and T superscripts mean matrix inversion and matrix transposition, respectively), where P matrix is called the estimation error covariance. In words, if the measurement noise S z is large, then K k will be small and thus the correction will be small (i.e., we won t give much credibility to the measurement y when computing next ˆx) and vice versa, small measurement noise S z means that the measurement is significant for estimation of next ˆx. Eq. 3 shows an important feature of the Kalman filter. It uses only one previous measurement/estimation to compute next step; thus we avoid using large matrix. This advantage is important especially for the GNSS receivers usage. Electron concentration profile and TEC The NeQuick model estimates slant or vertical TEC on the defined path (e.g. between the satellite and the ground receiver) and it is planned to be used as a real-time model in the Galileo project included in the receivers as a more advanced model to reduce an ionospheric error than a simple parabolic Klobuchar model used in GPS [Klobuchar, 1987]. The TEC estimation will be computed using following parameters: day of year, daytime, path of the signal (position of the satellite and the receiver) and monthly or daily index of solar activity by means of solar flux F10.7 (radio emission with the wavelength 10.7 cm), or SSN (sunspot number) which are supposed to be fully dependent in the model [Leitinger et al., 2005]. The profile of electron concentration is computed in a selected range of heights. The model is divided into two regions: the bottomside and the topside, described by means of semi-epstein layers [Radicella and Leitinger, 2001]. The parameters of the Epstein layers are computed on the basis of the ionosonde parameters. An advantage of NeQuick is its ability to easily accommodate measured values for these parameters [Bidaine and Warnant, 2010]. Topside sounding data were incorporated into a topside electron profile model [Coisson et al., 2006]. The NHPC is a software implemented in the UMLCAR ionosondes and digisondes [Reinisch et al., 2005]. It computes vertical electron concentration between the ground station and height adjusted by the setting of the ionosonde (1,000 km in our case). Electron density profile below the maximum of electron concentration is computed directly using detected reflections of the signal. Upper part between the maximum of electron concentration and upper boundary of the profile is modelled using Chebychev polynomials [e.g., Scotto, C., 2009]. Electron content above this range is neglected. In the section NeQuick/NHPC Comparison we discuss plasmaspheric contribution above this limit to TEC. Results and Discussion Kalman filtering The parameters Q (connected to the process noise) and R (measuring noise) were tested independently for both stations. The Q value was estimated according to the variance of short parts of the signal. Both values were than modified so that the filter removed short-time oscillations and that the phase delay was sufficiently low. Surprisingly, the Kalman filter had relatively low sensitivity to the change of Q and R. Fig. 1 demonstrates difference between various settings of filter parameters Q and R for the WTZR station. Choice of what we want to remove from the signal and what we do not want to remove is relatively arbitrary. However, we suppose that rapid oscillations (usually seen as small amplitude oscillations) are connected with the measurement noise while those with periods of the order of minutes and longer are of ionospheric origin and thus of a high importance for us. Our intention is thus to remove oscillations of the order of seconds but not eliminate those of the order of minutes and higher. Solid line represent a setting with Q=0.1, R=0.01 which corresponds to a low estimation of measurement error. This does not result in a demanded filtering process as the output practically follows the measurement (black dots) 212

4 WTZR 52 Data Q 0.1 R 1 Q 0.1 R 0.1 TEC (TECU) Time (s) Figure 1. Station Wetzer, Germany. Filter optimised to remove micro oscillations (measurement noise) while the periods due to ionospheric variabilities remain significant. Dashed line shows optimal setting of the filter with Q=0.1, R=1. Solid line represents improper filtering with too low estimation of measurement error. NeQuick and NHPC derived TEC values 15 NeQuick h<20000 NeQuick h<1000 NHPC TEC (TECU) Time (UT) Figure 2. Comparison of NeQuick derived TEC (dashed thin line denotes TEC between 0 and 1,000 km, solid thick line shows TEC between 0 and 20,000 km) and NHPC output (dotted line, 0 to 1,000 km) for a day of low geomagnetic activity (Kp 2). Contribution of electron content between 1,000 km and 20,000 km is only a small part of total NeQuick estimation. and the micro oscillations remain unfiltered. Dashed line is computed for Q=0.1, R=1 and we suggest that this setting is in good agreement with our goal to remove fast oscillation and not change longer periods. Small phase shift of about 1 2 s is present, however it is relatively good result in the real-time prediction process. Parameters for the Bogota station were set to Q=0.1, R=0.2 with similarly well removed fast oscillations. NHPC/NeQuick comparison Comparison of NeQuick and NHPC from 5 Nov 2010 is demonstrated in Fig. 2 (both days show very similar results). NHPC outputs are computed for range between ground and 213

5 1,000 km. NeQuick results are for range between ground and 1,000 km and between ground and 20,000 km which is an approximate height of the GPS satellites. All TEC results reflect typical ionospheric behaviour: rapid increase of TEC after sunrise (creation of the E-layer and increase in electron concentration in the F-layer) with a peak around noon and evening decrease with night low values. NHPC digisonde output follows this pattern, however the results are much more irregular, especially during time close to noon. It may be a possible result of wave activity (corresponding to gravity wave domain, e.g., Sauli et al. [2006], Hocke and Schlegel [1996]). Another source of these irregularities may be in the model used for the TEC computation. TEC estimation as an integral value of whole electron density profile is strongly dependent on the quality of the ionograms. Ionogram interpretation is relatively complicated task and changes in critical frequency estimation, thickness of ionospheric layers, topside ionosphere behavior etc. play an important role in the profile of electron concentration [Sauli et al., 2007] which is an input for the TEC estimation. Differences between the NHPC and NeQuick (up to 1,000 km) vary with time. During night, the NHPC shows relatively lower values by approximately 10%. During daytime (between sunrise and sunset) the differences reach values between 20 50% (NHPC lower than NeQuick) most of the time. Exceptionally, we observe close results of both tools (07:00, 08:30, 15:00 UT). Contribution of the plasmaspheric electron content (PEC) above 1,000 km to TEC according to the NeQuick accounts for about 10% during whole day. Lunt et al. [1999] used the SUPIM model to estimate plasmasphere electron content for the GPS purposes. Their results show generally substantial contribution of the plasmasphere to TEC. According to the time and phase of the solar cycle the estimations were ( 10%) with a maximum value of about 50% during night time and solar minimum in the European sector. Balan [2002] derived relative contribution of PEC to TEC with a result of 20 30% during night time and 10% around noon time (with more less constant absolute PEC during whole day). Conclusion We tested the possibility of using of the Kalman filtering in real-time prediction of the TEC using data from two GPS stations. The process (ionospheric variations) noise was described by equal parameters Q, while the parameter R describing the measurement noise was chosen different for both stations. This result may be interpreted so that the fast ionospheric processes are independent on the station location, however the instrument errors (due to the receivers, antenna settings etc.) are different. Additional information is that the filtering has relatively low sensitivity to both Q and R. Comparison of the NeQuick and NHPC TEC estimations shows systematic relative and absolute differences during daytime. Differences during night are very small in absolute values and both tools are in a good agreement. We propose that digisonde measurement with 15 minutes sampling could bring important information that could improve the real time TEC estimation, especially during daytime and during increased geomagnetic activity. Bottom part of the ionosphere below the maximum of electron concentration contains important content of electrons which can be estimated with a good precision. The ionosonde measurement can be useful supplement to the TEC models because the bottom part of the profile is this way computed with a good knowledge of the actual situation. In future work we will extend usage of the Kalman filter for a data from stations close to the Pruhonice ionosonde to improve knowledge of GPS derived TEC distribution and its relation to the ionosonde results. Further, we suggest that more detailed comparison of the NeQuick, NHPC and other models (IRI, POLAN) with a special attention to a distinguishing between bottom and topside part of the profile contributions will be helpful to detect weak and strong points of the models. Acknowledgments. This work was supported by the Grant P209/12/2440 and the project SX 5 ( 214

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