Bayesian statistical ionospheric tomography improved by incorporating ionosonde measurements

Norberg, Johannes; Virtanen, Ilkka; Roininen, Lassi; Vierinen, Juha; Orispää, Mikko; Kauristie, Kirsti; Lehtinen, Markku

doi:10.5194/amt-9-1859-2016

Cited by 15 publications

(19 citation statements)

References 11 publications

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“…There are different suggestions to suppress these negative estimates, for instance by the inclusion of pseudo-measurements (see Norberg et al, 2016). Alternatively, one might apply probability density distributions that inherently ensure positive electron density estimates, e.g.…”

Section: Kriging Of the Electron Density With A Spatial Covariance Modelmentioning

confidence: 99%

“…Subsequently the given electron densities are modified according to the measurements without touching the model coefficients itself (see, e.g., Bust et al, 2004;Wang et al, 2004;Wen et al, 2007;Hobiger et al, 2008;D. Minkwitz et al: Gradient-enhanced kriging of the electron density Scherliess et al, 2009;Angling and Jackson-Booth, 2011;Norberg et al, 2015;Gerzen and Minkwitz, 2016).…”

Section: Introductionmentioning

confidence: 99%

“…For that purpose, the definition of spatial and temporal correlations between electron densities at different locations is essential and incorrect specifications degrade the estimates of the ionospheric parameters. Motivated by the already successful application of kriging for the estimation of TEC grids/errors for the different satellite-based augmentation systems (Sparks et al, 2011;Sunda et al, 2013) and the provision of regional and global TEC maps (Orús-Pérez, 2005;Sayin et al, 2008), we advance the 2-D kriging to a 3-D kriging approach introduced in Minkwitz et al (2015). This approach has its origin in the geostatistics (see Wackernagel, 2003;Chilès and Delfiner, 2012), estimates the optimal covariance between two electron densities by the STEC measurements and predicts the electron density based on that.…”

Section: Introductionmentioning

confidence: 99%

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Ionospheric tomography by gradient-enhanced kriging with STEC measurements and ionosonde characteristics

et al. 2016

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Abstract. The estimation of the ionospheric electron density by kriging is based on the optimization of a parametric measurement covariance model. First, the extension of kriging with slant total electron content (STEC) measurements based on a spatial covariance to kriging with a spatialtemporal covariance model, assimilating STEC data of a sliding window, is presented. Secondly, a novel tomography approach by gradient-enhanced kriging (GEK) is developed. Beyond the ingestion of STEC measurements, GEK assimilates ionosonde characteristics, providing peak electron density measurements as well as gradient information. Both approaches deploy the 3-D electron density model NeQuick as a priori information and estimate the covariance parameter vector within a maximum likelihood estimation for the dedicated tomography time stamp. The methods are validated in the European region for two periods covering quiet and active ionospheric conditions. The kriging with spatial and spatial-temporal covariance model is analysed regarding its capability to reproduce STEC, differential STEC and foF2. Therefore, the estimates are compared to the NeQuick model results, the 2-D TEC maps of the International GNSS Service and the DLR's Ionospheric Monitoring and Prediction Center, and in the case of foF2 to two independent ionosonde stations. Moreover, simulated STEC and ionosonde measurements are used to investigate the electron density profiles estimated by the GEK in comparison to a kriging with STEC only. The results indicate a crucial improvement in the initial guess by the developed methods and point out the potential compensation for a bias in the peak height hmF2 by means of GEK.

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Section: Kriging Of the Electron Density With A Spatial Covariance Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ionospheric tomography by gradient-enhanced kriging with STEC measurements and ionosonde characteristics

et al. 2016

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“…The general methods to solve this ill-posed problem are iterative algorithms (such as the Algebraic Reconstruction Technique and the Multiplicative Algebraic Reconstruction Technique) with some prior information and constraints [e.g., Andreeva, 1990;Hobiger et al, 2008;Raymund et al, 1990;Wen et al, 2012], use of basis functions (such as spherical harmonics-generated basis functions and empirical orthonormal basis functions) [e.g., Fremouw et al, 1992;Garcia and Crespon, 2008;Mitchell and Spencer, 2003;Na and Lee, 1990], singular value decomposition algorithms [Hajj et al, 1994], multisource data fusion algorithms [e. g., Alizadeh et al, 2011;Dettmering et al, 2011;Li et al, 2012;Yue et al, 2012], constrained least squares algorithms [Seemala et al, 2014], Bayesian approaches [Markkanen et al, 1995;Norberg et al, 2015Norberg et al, , 2016, artificial neutral networks methods [Ma et al, 2005], data assimilation approaches (three-dimensional variational, four-dimensional variational, and Kalman filter) [e.g., Bust et al, 2004;Hajj et al, 2004;Pi et al, 2003;Scherliess et al, 2004;Schunk et al, 2004;Wang et al, 2004], and regularization methods [Fehmers et al, 1998;Lee et al, 2007;Nygrén et al, 1997]. However, it needs to be noted that ill-posedness is still a crucial problem in ionospheric tomography algorithm [Yao et al, 2015].…”

Section: Introductionmentioning

confidence: 99%

Three‐dimensional ionospheric tomography reconstruction using the model function approach in Tikhonov regularization

et al. 2016

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Ionospheric tomography is based on the observed slant total electron content (sTEC) along different satellite‐receiver rays to reconstruct the three‐dimensional electron density distributions. Due to incomplete measurements provided by the satellite‐receiver geometry, it is a typical ill‐posed problem, and how to overcome the ill‐posedness is still a crucial content of research. In this paper, Tikhonov regularization method is used and the model function approach is applied to determine the optimal regularization parameter. This algorithm not only balances the weights between sTEC observations and background electron density field but also converges globally and rapidly. The background error covariance is given by multiplying background model variance and location‐dependent spatial correlation, and the correlation model is developed by using sample statistics from an ensemble of the International Reference Ionosphere 2012 (IRI2012) model outputs. The Global Navigation Satellite System (GNSS) observations in China are used to present the reconstruction results, and measurements from two ionosondes are used to make independent validations. Both the test cases using artificial sTEC observations and actual GNSS sTEC measurements show that the regularization method can effectively improve the background model outputs.

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“…A comprehensive introduction to statistical inverse problems is provided in [8] and [9]. In ionospheric tomography, the Bayesian inference has been applied in [10]- [13]. The Ionospheric Data Assimilation Three-Dimensional (IDA3D) presented in [14] is based on the Three-Dimensional Variational Data Assimilation Technique (3DVAR) and uses slightly different terminology.…”

mentioning

confidence: 99%

Gaussian Markov Random Field Priors in Ionospheric 3-D Multi-Instrument Tomography

Norberg

Vierinen

Roininen

et al. 2018

IEEE Trans. Geosci. Remote Sensing

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In ionospheric tomography, the atmospheric electron density is reconstructed from different electron density related measurements, most often from ground-based measurements of satellite signals. Typically, ionospheric tomography suffers from two major complications. First, the information provided by measurements is insufficient and additional information is required to obtain a unique solution. Second, with necessary spatial and temporal resolutions, the problem becomes very high dimensional, and hence, computationally infeasible. With Bayesian framework, the required additional information can be given with prior probability distributions. The approach then provides physically quantifiable probabilistic interpretation for all model variables. Here, Gaussian Markov random fields (GMRFs) are used for constructing the prior electron density distribution. The use of GMRF introduces sparsity to the linear system, making the problem computationally feasible. The method is demonstrated over Fennoscandia with measurements from global navigation satellite system (GNSS) and low Earth orbit (LEO) satellite receiver networks, GNSS occultation receivers, LEO satellite Langmuir probes, and ionosonde and incoherent scatter radar measurements.Index Terms-Bayesian, Gaussian Markov random fields (GMRFs), ionospheric tomography, multi-instrument.

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Bayesian statistical ionospheric tomography improved by incorporating ionosonde measurements

Cited by 15 publications

References 11 publications

Ionospheric tomography by gradient-enhanced kriging with STEC measurements and ionosonde characteristics

Ionospheric tomography by gradient-enhanced kriging with STEC measurements and ionosonde characteristics

Three‐dimensional ionospheric tomography reconstruction using the model function approach in Tikhonov regularization

Gaussian Markov Random Field Priors in Ionospheric 3-D Multi-Instrument Tomography

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