A double-adaptive adjustment algorithm (DAAA) is proposed to reconstruct three-dimensional ionospheric electron density (IED) distribution. In the DAAA method, the relaxation factor of the multiplicative algebraic reconstruction technique (MART) is first adaptively adjusted by introducing adaptive MART (AMART). To avoid the voxels without any rays traversing them becoming dependent on the initial IED values, smoothing constraints are generally imposed on the adaptive multiplicative algebraic reconstruction technique (AMART). In general, the elements of the smoothing matrices are invariant in the iterative process. They affect the accuracy and efficiency of the IED inversion. To overcome the above limitation, the adaptive adjustments of the constrained matrix elements are subsequently carried out. Both numerical simulation and actual global navigation satellite system (GNSS) experimental results validate that the accuracy and efficiency of ionospheric tomography have been improved by the DAAA method. Finally, the new algorithm is applied to reconstruct the three-dimensional structure of the ionosphere during different geomagnetic activities. The comparisons show that the vertical profiles of the DAAA method are in agreement with those recorded from the ionosonde, and the inverted vertical total electron content (VTEC) of the DAAA method also agrees with the ionospheric products of center for orbit determination in Europe (CODE) during geomagnetic quiet and geomagnetic storms. The comparisons confirm the reliability and superiority of the DAAA method.
Ill-posedness of GNSS-based ionospheric tomography affects the stability and the accuracy of the inversion results. Truncated singular value decomposition (TSVD) is a common algorithm of ionospheric tomography reconstruction. However, the TSVD method usually has low inversion accuracy and reconstruction efficiency. To resolve the above problem, a truncated mapping singular value decomposition (TMSVD) algorithm is presented to improve the reconstructed accuracy and computational efficiency. To authenticate the effectiveness and the advantages of the TMSVD algorithm, a numerical test scheme is devised. Finally, ionospheric temporal–spatial variations of the selected reconstructed region are studied using the GNSS observations under different geomagnetic conditions. The reconstructed results of TMSVD can accurately reflect semiannual anomalies, diurnal variations, and geomagnetic storm effects. In contrast with the ionosonde data, it is found that the reconstructed profiles of the TMSVD method are more consistent with than those of the IRI 2016. The study suggests that TMSVD is an efficient algorithm for the tomographic reconstruction of ionospheric electron density (IED).
Tomographic inversion of the ionosphere is a rank-deficient problem. To overcome the above problem, an algebraic reconstruction technique (ART) based on adaptive horizontal constraint and empirical orthogonal function (ARTHCEOF) is proposed. The new algorithm avoids the difficulty of vertically constrained matrix construction and resolves the description of the ionospheric vertical structure by using EOF. To confirm the feasibility and validate the ascendancy of the ARTHCEOF, three algorithms are first tested by using the emulated scheme. The test results show that the ARTHCEOF surpasses the ART and the ART based on the horizontal constraint (ARTHC) in both the inversion accuracy and computational efficiency. Finally, the ARTHCEOF method is applied to invert electron density values using the GNSS measurements during different geomagnetic activities. The tomographic images validate that the ARTHCEOF can reflect ionospheric daily changes in the European region. The altitudinal profiles in a fixed location are illustrated according to the inversion results of ARTHCEOF. Compared with the profiles recorded by the ionosonde station, the altitudinal profiles of ARTHCEOF have a good consistency. In the meantime, the VTEC values are inverted using the CIT results. The differential VTEC values are calculated by means of the inverted VTEC values and ionospheric products of CODE. The differential results further identify the dependability of ARTHCEOF.
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