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Fast determination of geometric matrix in ionosphere tomographic inversion with unevenly spaced curvilinear voxels

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Abstract

An efficient tomographic inversion model is very beneficial to image the ionosphere in real time. Geometric matrix determination, one of the indispensable procedures of voxel-based tomographic inversion, is currently very inefficient. We developed a very fast algorithm named splitting-surface intersection and voxel tracing (SIVT) to compute the geometric matrix. Instead of applying an intersecting voxel test and several intersection computations on a large number of voxels, SIVT calculates only the intersections of a ray with the splitting surfaces of a voxel model and determines the intersecting voxels by direct voxel tracing. Theoretically, the SIVT approach costs O(n log n) time (where n refers to the number of splitting surfaces (n)) compared to O(m) for the auxiliary plane (AP) approach (where m refers to the number of voxels and is always larger than n by several orders of magnitude). Our tests show that the SIVT approach outperforms the AP approach by approximately 100–3000 times, and the superiority of the SIVT approach increases exponentially as m or n grows.

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Data availability

The GNSS observation data are available at http://www.epncb.oma.be/, and the source code of the SIVT approach is available at https://github.com/yujieqing/Geometric-Matrix-For-Ionospheric-Tomogrphy.

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Acknowledgments

This work was jointly supported by the National Key Research and Development Program of China (2018YFB0505304), the National Natural Science Foundation of China (41771416) and the National Key Research and Development Program of China (2018YFC1503505).

Author information

Authors and Affiliations

  1. School of Environmental Science and Spatial Informatics, China University of Mining and Technology, Xuzhou, China

    Jieqing Yu

  2. Colorado Center for Astrodynamics Research, University of Colorado Boulder, Boulder, Colorado, USA

    Jieqing Yu, Zhe Yang & Brian Breitsch

  3. College of Surveying and Geo-Informatics, Tongji University, Shanghai, China

    Zhe Yang

  4. School of Geosciences and Info-Physics, Central South University, Changsha, China

    Lixin Wu

Authors
  1. Jieqing Yu
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  2. Zhe Yang
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  3. Brian Breitsch
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  4. Lixin Wu
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Correspondence to Jieqing Yu.

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Appendix: Pseudocodes for the SIVT approach implementation

Appendix: Pseudocodes for the SIVT approach implementation

Geometric matrix determination by the SIVT approach is mainly composed of two processes, valid intersection computation and sorting, and segment length computation and affiliation determination.

The first process outputs the sorted valid intersections according to their t-values. Given a valid ray and a curvilinear voxel model, it can be implemented by the following pseudocode.

figure e

The second process outputs the segments of a ray and their affiliations. Given a valid ray, a voxel model, and the valid intersections associated with the valid ray, the second process can be implemented by the following pseudocode.

figure f

The fourth step of the algorithm II computes the orientation vector of a ray. Given a valid ray, a voxel model and the first and the last valid intersections associated with the ray, the orientation vector of the valid ray can be implemented by the following pseudocode.

figure g

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Yu, J., Yang, Z., Breitsch, B. et al. Fast determination of geometric matrix in ionosphere tomographic inversion with unevenly spaced curvilinear voxels. GPS Solut 26, 27 (2022). https://doi.org/10.1007/s10291-021-01211-1

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