Journal of Applied Geodesy

2013

DOI: 10.1515/jag-2013-0006

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Adjustment of survey networks by topological grid search

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Cited by 4 publications

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“…4 ; Figure 5 c). The centre of gravity of their intersection practically coincides with the Best Linear Unbiased Estimator (BLUE-type estimate) of M. This approach explains the function of the TOPINV algorithm (Saltogianni and Stiros 2012a , b ; Harvey 2013 ). The variance (quality, uncertainty) of the estimator, however depends on the prior information for point M, i.e.…”

Section: Introductionmentioning

confidence: 69%

“…Such integer ambiguities represent a major source of error in GPS positioning (Han and Rizos 1996 ), but also in satellite radar measurements (Usai 2003 ; Kampes and Hanssen 2004 ) etc. As has been shown by Harvey ( 2013 ), certain of these problems can be solved on the basis of the TOPINV.…”

Section: Introductionmentioning

confidence: 99%

“…A topological inversion technique for the solution of redundant systems of non-linear equations with n unknowns has recently been presented by Saltogianni and Stiros ( 2012b ; 2013 ) and was further assessed by Harvey ( 2013 ). This technique, thereafter called TOPINV (from Topological Inversion, or TGS, Topological Grid Search), exploits the power of modern computers and is based on the principle of intersection of geometric loci in the R n space.…”

Section: Introductionmentioning

confidence: 99%

“…It is a technique inspired from the traditional lighthouse navigations, and one of its major advantages is that it does not require any inversion of matrices. For this reason, it was proposed that this method can also be applied for the solution of certain types of underdetermined systems of equations (Harvey 2013 ), the solution (inversion) of which traditionally leads to inversion of singular matrices (Matsu’ura and Hirata 1982 ).…”

Section: Introductionmentioning

confidence: 99%

“…In this article we present a generalization of the idea of Harvey ( 2013 ) that TOPINV (or TGS) can cover a wide range of underdetermined systems of equations observed in engineering and geophysics. We also explain that the only requirement for this method of inversion is the a priori knowledge of the possible range of values for each of the unknown variables.…”

Section: Introductionmentioning

confidence: 99%

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Solution of underdetermined systems of equations with gridded a priori constraints

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2014

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The TOPINV, Topological Inversion algorithm (or TGS, Topological Grid Search) initially developed for the inversion of highly non-linear redundant systems of equations, can solve a wide range of underdetermined systems of non-linear equations. This approach is a generalization of a previous conclusion that this algorithm can be used for the solution of certain integer ambiguity problems in Geodesy.The overall approach is based on additional (a priori) information for the unknown variables. In the past, such information was used either to linearize equations around approximate solutions, or to expand systems of observation equations solved on the basis of generalized inverses. In the proposed algorithm, the a priori additional information is used in a third way, as topological constraints to the unknown n variables, leading to an Rn grid containing an approximation of the real solution.The TOPINV algorithm does not focus on point-solutions, but exploits the structural and topological constraints in each system of underdetermined equations in order to identify an optimal closed space in the Rn containing the real solution. The centre of gravity of the grid points defining this space corresponds to global, minimum-norm solutions. The rationale and validity of the overall approach are demonstrated on the basis of examples and case studies, including fault modelling, in comparison with SVD solutions and true (reference) values, in an accuracy-oriented approach.

“…4 ; Figure 5 c). The centre of gravity of their intersection practically coincides with the Best Linear Unbiased Estimator (BLUE-type estimate) of M. This approach explains the function of the TOPINV algorithm (Saltogianni and Stiros 2012a , b ; Harvey 2013 ). The variance (quality, uncertainty) of the estimator, however depends on the prior information for point M, i.e.…”

Section: Introductionmentioning

confidence: 69%

“…Such integer ambiguities represent a major source of error in GPS positioning (Han and Rizos 1996 ), but also in satellite radar measurements (Usai 2003 ; Kampes and Hanssen 2004 ) etc. As has been shown by Harvey ( 2013 ), certain of these problems can be solved on the basis of the TOPINV.…”

Section: Introductionmentioning

confidence: 99%

“…A topological inversion technique for the solution of redundant systems of non-linear equations with n unknowns has recently been presented by Saltogianni and Stiros ( 2012b ; 2013 ) and was further assessed by Harvey ( 2013 ). This technique, thereafter called TOPINV (from Topological Inversion, or TGS, Topological Grid Search), exploits the power of modern computers and is based on the principle of intersection of geometric loci in the R n space.…”

Section: Introductionmentioning

confidence: 99%

“…It is a technique inspired from the traditional lighthouse navigations, and one of its major advantages is that it does not require any inversion of matrices. For this reason, it was proposed that this method can also be applied for the solution of certain types of underdetermined systems of equations (Harvey 2013 ), the solution (inversion) of which traditionally leads to inversion of singular matrices (Matsu’ura and Hirata 1982 ).…”

Section: Introductionmentioning

confidence: 99%

“…In this article we present a generalization of the idea of Harvey ( 2013 ) that TOPINV (or TGS) can cover a wide range of underdetermined systems of equations observed in engineering and geophysics. We also explain that the only requirement for this method of inversion is the a priori knowledge of the possible range of values for each of the unknown variables.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Solution of underdetermined systems of equations with gridded a priori constraints

¹

,

²

2014

View full text Add to dashboard Cite

The TOPINV, Topological Inversion algorithm (or TGS, Topological Grid Search) initially developed for the inversion of highly non-linear redundant systems of equations, can solve a wide range of underdetermined systems of non-linear equations. This approach is a generalization of a previous conclusion that this algorithm can be used for the solution of certain integer ambiguity problems in Geodesy.The overall approach is based on additional (a priori) information for the unknown variables. In the past, such information was used either to linearize equations around approximate solutions, or to expand systems of observation equations solved on the basis of generalized inverses. In the proposed algorithm, the a priori additional information is used in a third way, as topological constraints to the unknown n variables, leading to an Rn grid containing an approximation of the real solution.The TOPINV algorithm does not focus on point-solutions, but exploits the structural and topological constraints in each system of underdetermined equations in order to identify an optimal closed space in the Rn containing the real solution. The centre of gravity of the grid points defining this space corresponds to global, minimum-norm solutions. The rationale and validity of the overall approach are demonstrated on the basis of examples and case studies, including fault modelling, in comparison with SVD solutions and true (reference) values, in an accuracy-oriented approach.

A new algorithm for modelling simple and double Mogi magma sources in active volcanoes: accuracy, sensitivity, limitations and implications

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2013

Bull Volcanol

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Multivariable inversion using exhaustive grid search and high-performance GPU processing: a new perspective

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et al. 2020

Geophysical Journal International

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SUMMARY Exhaustive searches in regular grids is a traditional and effective method for inversion, that is numerical solution of systems of non-linear equations which cannot be solved using formal algebraic techniques. However, this technique is effective for very few (3–4) variables and is slow. Recently, the first limitation was to a major degree overpassed with the new TOPological INVersion (TOPINV) algorithm which was used for inversion of systems with up to 18, or even more unknown variables. The novelty of this algorithm is that it is not based on the principle of the mean minimum misfit (cost function) between observations and model predictions, used by most inversion techniques. The new algorithm investigates for each gridpoint whether misfits of each observation are within specified uncertainty intervals, and stores clusters of ‘successful’ gridpoints in matrix form. These clusters (ensembles, sets) of gridpoints are tested whether they satisfy certain criteria and are then used to compute one or more optimal statistical solutions. The new algorithm is efficient for highly non-linear problems with high measurement uncertainties (low signal-to-noise ratio, SNR) and poor distribution of observations, that is problems leading to complicated 3-D mean misfit surfaces without dominant peaks, but it is slow when running in common computers. To overcome this limitation, we used GPUs which permit parallel processing in common computers, but faced another computational problem: GPU parallel processing supports only up to three dimensions. To solve this problem, we used CUDA programming and optimized the distribution of the computational load to all GPU cores. This leads up to 100x speedup relative to common CPU processing, as is derived from comparative tests with synthetic data for two typical inversion geophysical problems with up to 18 unknown variables, Mogi magma source modeling and elastic dislocation modeling of seismic faults. This impressive speedup makes the GPU/CUDA implementation of TOPINV practical even for low-latency solution of certain geophysical problems. This speedup in calculations also permitted to investigate the performance of the new algorithm in relation to the density of the adopted grids. We focused on a typical problem of elastic dislocation in unfavorable conditions (poor observations geometry, data with low SNR) and on synthetic observations with noise, so that the difference of each solution from the ‘true’/reference value was known (accuracy-based approach). Application of the algorithm revealed stable, accurate and precise solutions, with quality increasing with the grid density. Solution defects (bias), mainly produced by very coarse grids, can be identified through specific diagnostic criteria, dictating finer search grids.

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