A framework for solving meta inverse problems: experimental design and application to an acoustic source problem
Tóm tắt
The selection of optimal measurement locations in remote sensing or imaging algorithms is of large practical interest in many applications. The target is usually to choose a measurement setup that best resolves some particular quantity of interest. This work describes a general framework for selecting such an optimal setup within a given set Q of possible setups for the formulation and solution of the meta inverse problem. The work shows that it is crucial to incorporate the basic ingredients which are usually part of the inversion process. In particular, it takes care of the nature and the size of the measurement error, the choice of the regularization scheme which is employed for the inverse problem, and the prior knowledge on solutions. The basic idea of the framework is to minimize the errors associated with the reconstruction of a given quantity of interest. Five functional layers which reflect the structure of the meta inverse problem are introduced. Further, with framework adaption, an iterative algorithm is formulated to solve the meta inverse problem at each iterative step in order to obtain improved reconstructions of the inverse problem. Using the initial reconstructions as input for meta inversion, the framework adaption algorithm does not require prior knowledge of the source distribution. The feasibility of the framework adaption algorithm is illustrated by using it to solve the inverse acoustic source problem.
Tài liệu tham khảo
Athanasiou EN, Tsourlos PI, Papazachos CB, Tsokas GN (2006) Optimizing resistivity array configurations by using a non-homogeneous background model. In: Proceedings, 12th Annual Meeting EAGE-Near Surface Geophysics, Helsinki, Finland
Athanasiou EN, Tsourlos PI, Papazachos CB, Tsokas GN (2009) Optimizing electrical resistivity array configurations by using a method based on the sensitivity matrix. In: Proceedings, 15th Annual Meeting EAGE-Near Surface Geophysics, Dublin, Ireland
Bouttier F, Kelly G (2001) Observing-system experiments in the ECMWF 4D-Var data assimilation system. QJR Meteorol Soc 127:1469–1488
Colton D, Kress R (1992) Inverse acoustic and electromagnetic scattering theory, volume 93 of applied mathematical sciences. Springer, Berlin
Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems, volume 375 of mathematics and its applications. Kluwer Academic Publishers Group, Dordrech
Gautam PK, Biswas A (2016) 2D geo-electrical imaging for shallow depth investigation in Doon Valley Sub-Himalaya, Uttarakhand, India. Model Earth Syst Env 2:175
George NJ, Ekanem AM, Ibanga JI, Udosen NI (2017) Hydrodynamic implication of Aquifer Quality Index (AQI) and flow zone (F21) in ground water abstraction: a case study of coastal hydro-lithofacies in South-eastern Nigeria. J Coast Conserv 21(6):759–776
Groetsch C (1993) Inverse Problems in the mathematical sciences (theory & practice of applied geophysics). Friedrich Vieweg & Sohn, Germany
Hennig T, Weller A, Miller (2008) Object orientated focussing of geoelectrical multielectrode measurements. J Appl Geophys 65:57–64
Kang Y, Lin TW, Chang YJ, Chang YP, Wang CC (2008) Optimal balancing of flexible rotors by minimizing the condition number of influence coefficients. Mech Mach Theory 43(7):891–908
Kirsch A (1996) An introduction to the mathematical theory of inverse problems, volume 120 of applied mathematical sciences. Springer, New York
Kress R (1999) Linear integral equations, volume 82 of applied mathematical sciences. Springer, New York
Lin TW, Kang Y, Wang CC, Chang CW, Chih-Pin C (2005) An investigation in optimal locations by genetic algorithm for balancing flexible rotor-bearing systems. In: Proceedings of the ASME Turbo Expo, vol 4, Reno, Nevada, USA, 6–9 June 2005
Loke MH, Wilkinson PB (2009) Rapid parallel computation of optimized arrays for electrical imaging surveys. In: Proceedings, 15th Annual Meeting EAGE-Near Surface Geophysics, Dublin, Ireland
Loke MH, Alfouzan FA, Nawawi NM (2007) Optimisation of electrode arrays used in 2D resistivity imaging surveys. In: Extended Abstracts, 19th ASEG Geophysical Conference and Exhibition
Olagunju E, Ariyibi E, Awoyemi M, Adebayo A, Dasho O, Adenika C (2017) Application of geochemical and geophysical approach to environmental impact assessment: a case study of Emirin active open dumpsite, Ado-Ekiti Southwestern Nigeria. Model Earth Syst Env. https://doi.org/10.1007/s40808-017-0381-1
Potthast R (2001) Point sources and multi-poles in inverse scattering theory, vol 427 of Chapman & Hall/CRC Research Notes in Mathematics. Chapman & Hall/CRC, Boca Raton
Stummer P, Maurer H, Green A (2004) Experimental design: electrical resistivity data sets that provide optimum subsurface information. Geophysics 69(1):120–139
Tan D, Andersson E, Fisher M, Isaksen L (2007) Observing-system impact assessment using a data assimilation ensemble technique: application to the ADM Aeolus wind profiling mission. QJR Meteorol Soc 133:381–390
Udosen NI, George NJ (2018) Characterization of electrical anisotropy in North Yorkshire, England using square arrays and electrical resistivity tomography. J Geomech Geophys Geo-energy Geo-resour. https://doi.org/10.1007/s40948-018-0087-5
Udosen NI, Potthast RWE (2018) Automated optimization of electrode locations for electrical resistivity tomography. J Model Earth Syst Env. https://doi.org/10.1007/s40808-018-0472-7
Wilkinson PB, Chambers JE, Meldrum P, Ogilvy RD, Caunt S (2006a) Optimization of array configurations and panel combinations for the detection and imaging of abandoned mineshafts using 3D Cross-Hole Electrical Resistivity Tomography. J Environ Eng Geophys 11(3):213–221
Wilkinson PB, Meldrum PI, Chambers JE, Kuras O, Ogilvy RD (2006b) Improved strategies for the automatic selection of optimized sets of electrical resistivity tomography measurement configurations. Geophys J Int 167(3):1119–1126
Wilkinson PB, Ogilvy RD, Chambers JE, Meldrum PI, Kuras O (2007) Array optimisation for multi-channel electrical resistivity tomography instruments. In: Proceedings, 13th Annual Meeting EAGE Near Surface Geophysics, Istanbul, Turkey
Wilkinson PB, Meldrum PI, Chambers JE, Kuras O, Ogilvy RD (2009) Optimised survey design for geoelectrical resistivity tomography. In: Short Abstracts, 10th International Conference on Biomedical Applications of Electrical Impedance Tomography (EIT 2009), Manchester, United Kingdom