The method of detection and localization of configuration defects in geodetic networks by means of Tikhonov regularization

Abstract In adjusted geodetic networks, cases of local configuration defects (defects in the geometric structure of the network due to missing data or errors in point numbering) can be encountered, which lead to the singularity of the normal equation system in the least-squares procedure. Numbering errors in observation sets cause the computer program to define the network geometry incorrectly. Another cause of a defect may be accidental omission of certain data records, causing local indeterminacy or lowering of local reliability rates in a network. Obviously, the problem of a configuration defect may be easily detectable in networks with a small number of points. However, it becomes a real problem in large networks, where manual checking of all data becomes a very expensive task. The paper presents a new strategy for the detection of configuration defects with the use of the Tikhonov regularization method. The method was implemented in 1992 in the GEONET system (www.geonet.net.pl).

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A Combined Use of TSVD and Tikhonov Regularization for Mass Flux Solution in Tibetan Plateau

Remote Sensing ◽

10.3390/rs12122045 ◽

2020 ◽

Vol 12 (12) ◽

pp. 2045

Author(s):

Tianyi Chen ◽

Jürgen Kusche ◽

Yunzhong Shen ◽

Qiujie Chen

Keyword(s):

Tibetan Plateau ◽

Spatial Resolution ◽

Tikhonov Regularization ◽

Mass Flux ◽

Tikhonov Regularization Method ◽

Normal Equation ◽

The Tibetan Plateau ◽

Visual Spatial ◽

Truncated Normal ◽

Value Decomposition

Limited by the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE-FO) measurement principle and sensors, the spatial resolution of mass flux solutions is about 2–3° in mid-latitudes at monthly intervals. To retrieve a mass flux solution in the Tibetan Plateau (TP) with better visual spatial resolution, we combined truncated singular value decomposition (TSVD) and Tikhonov regularization to solve for a mascon modeling. The monthly mass flux parameters resolved at 1° are smoothed to about 2° by truncating the eigen-spectrum of the normal equation (i.e., using the TSVD approach), and then Tikhonov regularization is applied to the truncated normal equation. As a result, the terms beyond the native resolution of GRACE/GRACE-FO data are truncated, and the errors in higher degree and order components are dampened by Tikhonov regularization. In terms of root mean squared errors, the improvements are 27.2% and 12.7% for the combined method over TSVD and Tikhonov regularization, respectively. We confirm a decreasing secular trend with −5.6 ± 4.2 Gt/year for the entire TP and provide maps with 1° resolution from April 2002 to April 2019, generated with the combined TSVD and Tikhonov regularization method.

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A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

Open Mathematics ◽

10.1515/math-2020-0111 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1685-1697

Author(s):

Zhenyu Zhao ◽

Lei You ◽

Zehong Meng

Keyword(s):

Cauchy Problem ◽

Laplace Equation ◽

Tikhonov Regularization ◽

Regularization Method ◽

Regularization Parameter ◽

Solution Process ◽

Tikhonov Regularization Method ◽

Hermite Expansion ◽

The Cauchy Problem ◽

Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

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