Advances in Radio Science

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1 Advances in Radio Science, 3, , 2005 SRef-ID: /ars/ Copernicus GmbH 2005 Advances in Radio Science Recent Applications and Advances of Numerical Modeling and Wavefield Inversion in Nondestructive Testing R Marklein 1, K J Langenberg 1, K Mayer 1, J Miao 1, A Shlivinski 1, A Zimmer 1, W Müller 2, V Schmitz 2, C Kohl 3, and U Mletzko 4 1 Fachgebiet Theoretische Elektrotechnik, Fachbereich Elektrotechnik / Informatik, Universität Kassel (UNIK), Kassel, Germany 2 Fraunhofer-Institut für Zerstörungsfreie Prüfverfahren (IZFP), Universität des Saarlandes, Gebäude 37, Saarbrücken, Germany 3 Bundesanstalt für Materialforschung und Prüfung (BAM), Berlin, Germany 4 Materialprüfanstalt (MPA) Stuttgart, Universität Stuttgart, Stuttgart, Germany Abstract This paper presents recent advances and future challenges of the application of different linear and nonlinear inversion algorithms in acoustics, electromagnetics, and elastodynamics The presented material can be understood as an extension of our previous work on this topic The inversion methods considered in this presentation vary from linear schemes, like the Synthetic Aperture Radar (SAR) applied electromagnetics and the Synthetic Aperture Focussing Technique (SAFT) as its counterpart in ultrasonics, and the linearized Diffraction Tomography (DT), to nonlinear schemes, like the Contrast Source Inversion (CSI) combined with different regularization approaches Inversion results of the above mentioned inversion schemes are presented and compared for instance for time-domain ultrasonic data from the Fraunhofer-Institute for Nondestructive Testing (IZFP, Saarbrücken, Germany) Convenient tools for nondestructive evaluation of solids can be electromagnetic and/or elastodynamic waves; since their governing equations, including acoustics, exhibit strong structural similarities, the same inversion concepts apply In particular, the heuristic SAFT algorithm can be and has been utilized for all kinds of waves, once a scalar approximation can be justified Relating SAFT to inverse scattering in terms of diffraction tomography, it turns out that linearization is the most stringent inherent approximation A comparison of the inversion results using the linear time-domain inversion scheme SAFT and well tested nonlinear frequency-domain inversion schemes demonstrates the considerable potential to extend and improve the ultrasonic imaging technique SAFT while consulting the mathematics of wavefield inversion, yet, in particular if the underlying effort is considered, the relatively simple and effective SAFT algorithm works surprisingly well Since SAFT is a widely accepted imaging tool in ultrasonic NDE it seems worthwhile to check its formal restrictions and as- Correspondence to: R Marklein (marklein@uni-kasselde) sumptions whether they could be overcome and whether they would outperform the standard and original SAFT algorithm 1 Ultrasonic data: linear and nonlinear inversion results Convenient tools for NDE of solids can be electromagnetic and/or ultrasonic waves; since their governing equations exhibit strong structural similarities, the same modeling and inversion concepts apply (Marklein, 2002; Marklein et al, 2001, 2002a,b) In particular, the heuristic synthetic aperture focusing technique (SAFT) algorithm can be and has been utilized for all kinds of waves, once a scalar approximation can be justified Figure 1 shows a typical multi-monostatic (multi-pulse-echo) ultrasonic data set which is processed by the SAFT algorithm Relating SAFT to inverse scattering in terms of diffraction tomography, it turns out that linearization is the most stringent inherent approximation Hence, the results of nonlinear inversion schemes such as Contrast Source Inversion (CSI), where CSI has been applied in remote sensing very successfully (van den Berg et al, 2003; Marklein et al, 2001), are compared to the output of SAFT for a carefully designed ultrasonic experiment (Marklein et al, 2002a,b) Here we restrict ourselves to the scalar case, the extension and application of CSI to the vector case in ultrasonic and electromagnetic NDE is a future challenge First approaches can be found in Abubakar (2000); Pelekanos and Sevroglou (2003) Figure 2 shows the geometries and photographs of the four aluminum samples with two, four, six, and twelve air-filled circular cylinders which are in the experiments embedded in a water tank Multi-monostatic (multipulse-echo) and multi-bistatic (multi-pitch-catch) ultrasonic data sets (Figs 1 and 4a) have been carefully measured The SAFT reconstructions are given in Figs 3 and 4b Nonlinear CSI inversion results presented in Figs 4d f are in general

2 he infrom R) apussing s, and nlinear bined results ed and a from (IZFP, estrucr elasluding me in- FT alwaves, SAFT phy, it ent aping the tested strates ultramathhe un- effec- AFT is seems ptions would elde) Convenient tools for NDE of solids can be electromagnetic and/or ultrasonic waves; since their governing equations exhibit strong structural similarities, the same modeling and inversion concepts apply (Marklein, 2002; Marklein et al, , 2002a,b) In particular, ther heuristic Marklein synthetic al: Numerical aper- Modeling and Wavefield Inversion in Nondestructive Testing Fig 1 Measured ultrasonic (elastodynamic) multi monostatic multi-monostatic (multi-pulse-echo) (multi pulse echo) time domain data set for an aluminum cylinder with six air-filled holes ture focusing technique (SAFT) algorithm can be and has much been better utilized than the forsaft all kinds results of waves, Figure 5once shows a scalar a comparisoproximation of CSI results can befor justified experimental Fig 1and shows synthetic a typical data multi- The ap- synthetic monostatic data (multi-pulse-echo) are obtained withultrasonic a Domaindata Integral set which Equation is (DIE) processed solver byin thesaft frequency algorithm domain Relating It can besaft seen that to due inverse phase scattering errors and in terms noise in of the diffraction measurements tomography, of theit CSI turns the to reconstructions out that linearization of theis synthetic most data stringent are much inherent better approximation Hence, the results of nonlinear inversion schemes such as Contrast Source Inversion (CSI), where CSI has been 2applied Ultrasonic in remote data: sensing inhomogeneous very successfully anisotropic (van den Berg case et al, (InASAFT) 2003; Marklein et al, 2001), are compared to the output of SAFT for a carefully designed ultrasonic experiment Non-destructive (Marklein et al, testing 2002a,b) of defects in nuclear power plant dissimilar pipe weldings play an important role in safety inspections (see Fig 6) Traditionally the imaging of such defects is performed using the SAFT algorithm, however since parts of the dissimilar welded structure are made of an anisotropic material, this algorithm may fail to produce correct results Here we present results of a modified SAFT algorithm that enables a correct imaging of cracks in complex inhomogeneous anisotropic structures by accounting for the true nature of the wave propagation in such structures The algorithm is called inhomogeneous anisotropic SAFT (InASAFT) (Shlivinski et al, 2004a,b) The InASAFT algorithm is shown to yield better results over the SAFT algorithm for complex environments, but the InASAFT suffers from the same difficulties of the SAFT algorithm, ie ghost images and lack of clearly focused images However these artifacts can be identified through numerical modeling of the wave propagation in the structure, eg with the Elastodynamic Finite Integration Technique (EFIT) (Marklein, 2002) Here we consider a realistic dissimilar weld Figure 6 (left) shows a polished cut of a weld where its modeling, based on the design blue-prints, can be seen in Fig 6 (right) Figure 7 (left) shows a synthetic B-scan as a result of the modeling with EFIT (Marklein, 2002), and Fig 7 (right) displays the InASAFT reconstruction The crack inside the weld is reconstructed correctly The experimental B-scan and InASAFT image is given in Fig 8, where the two focused points in the black ellipse correspond to the crack tips 3 Electromagnetic data: Ground Penetrating Radar (GPR) application Figure 9 shows the geometry of a concrete test specimen as a computer model, which was actually realized by the Federal Highway Research Institute in Germany The specimen contains two circular cylindrical tubes of infinite electrical conductivity with a diameter of 100 mm as an appropriate model for tendon ducts The reinforcement is perfectly conducting, it exhibits two different mesh sizes of 75 mm and 50 mm The specimen has the dimensions 2 m 15 m 07 m, and the concrete material is supposed to be homogeneous and lossless with a relative permittivity ε r =2 For the 3-D simulation of electromagnetic wave propagation and scattering, a plane wave impinging perpendicularly to the surface is chosen, which has the base band pulse structure shown in Fig 9 ( 3 db cut-off frequency: 09 GHz); the electric field strength is linearly polarized parallel to the tendon ducts The numerical method to solve Maxwell s equations is based on the Finite Integration Technique, and here we use a commercially available implementation (CST, 2004) Figure 10 exhibits 2-D slices out of the 3-D wave field along the line L1 as indicated in Fig 9, ie a line through the coarser reinforcement Typically, even though the highest frequency wavelength is much larger than the grid spacing, the plane penetrates the reinforcement and leaves it again as a nearly plane wave, nevertheless much weaker in amplitude than before Hence, the scattering by the tendon ducts is also obvious resulting in a 3-D data field when scanned along the surface; these data are 2-D with regard to the scan coordinate, and the third dimension is time Two slices along the lines L1 and L2 of Fig 9 of the data are displayed in Fig 11: The slice along L1 through the coarser grid clearly exhibits the hyperbolic diffraction curves of the tendon ducts, whereas they are not visible along the finer grid This is confirmed applying a linear diffraction tomographic inverse scattering scheme FT- SAFT (Mayer et al, 1990) to the 3-D data: Fig 12 clearly shows images of the tendon ducts only below the coarser grid Thus, simulations of that kind define a limiting grid size for GPR tendon duct imaging, but it turns out that this upper bound cannot yet be met by experiments 4 Concluding remarks and future work We have presented latest results of the numerical modeling and linear and nonlinear inversion of electromagnetic and ultrasonic synthetic as well as experimental data A comparison of the ultrasonic inversion results using the linear time-domain inversion scheme SAFT and well tested nonlinear frequency-domain inversion schemes demonstrates the considerable potential to extend and improve the ultrasonic imaging technique SAFT while consulting the mathematics of wavefield inversion, yet, in particular if the underlying effort is considered, the relatively simple and effective SAFT algorithm works surprisingly well Even the extension of SAFT to the inhomogeneous anisotropic case, which yields

3 R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing 2 R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing Fig 2 Stepwise increase of crack complexity: (top) samples geometries of two, four, six and twelve holes; (bottom) aluminum samples Fig 2 Stepwise increase of crack complexity: (top) samples geometries of two, four, six and twelve holes; (bottom) aluminum samples Fig 2 Stepwise increase of crack complexity: (top) samples geometries of two, four, six and twelve holes; (bottom) aluminum samples Fig 3 Left: SAFT reconstruction of the multi-monostatic (multi-pulse-echo) data set; Right: SAFT reconstruction embedded in the 3-D geometry of the test sample Fig 3 Left: SAFT reconstruction of the multi-monostatic (multi-pulse-echo) data set; Right: SAFT reconstruction embedded in the 3-D geometry of the test sample Fig 3 Left: SAFT reconstruction of the multi-monostatic (multi-pulse-echo) data set; Right: SAFT reconstruction embedded in the 3-D geometry of Here thewe test restrict sample ourselves to the scalar case, the extension and Fig 4a) have been carefully measured The SAFT reconstructions and Here application we restrict of CSI ourselves to the to vector the scalar case case, in ultrasonic the extension and and Fig 4a) are have given been in carefully Fig 3 and measured Fig 4b The Nonlinear SAFT reconstructions inversion results are given presented in Fig in3 Fig and Fig 4d-f are 4b innonlinear general electromagnetic and applicationnde of CSI is ato future the vector challenge casefirst in ultrasonic approaches and CSI the InASAFT, can electromagnetic be found is working (Abubakar, NDEwell is aand future 2000; inchallenge combination Pelekanos First & Sevroglou, with approaches numerical2003) modeling can befig found 2tools, shows in (Abubakar, like the geometries EFIT, 2000; the and Pelekanos effects photographs of& lineariza- Sevroglou, of the son overcome much of CSI better results and thanwhether for experimental SAFT they results would and Figsynthetic outperform 5 shows data a compari- the The standard much formal CSI inversion better restrictions thanresults the SAFT and presented assumptions results in Fig Fig 54d-f whether shows area in compari- they general could be tion, for four 2003) instance aluminum Fig artifacts 2 samples shows in the with the geometries two, reconstruction, four, and six, photographs and can twelve beof interpreted Obviously, like in the ultrasonic case, simulated airfilled fourcircular aluminum cylinders samples which withare two, in the four, experiments six, and twelve embed- air- (DIE) synthetic solver data in the arefrequency obtained with domain a Domain It canintegral be seen that Equation due the synthetic and sonoriginal of CSI dataresults SAFT/FT-SAFT are obtained for experimental with a algorithm Domain and synthetic Integral Equation data The ded filled in acircular water tank cylinders Multi-monostatic which are in(multi-pulse-echo) the experiments embedded in a water (multi-pitch-catch) imaging using FT-SAFT and to(die) phasesolver errorsinand thenoise frequency in thedomain measurements It can be ofseen the CSI thatthe electromagnetic due multi-bistatic data tank Multi-monostatic ultrasonic (multi-pulse-echo) data leadssets to (Fig an intuitive physical understanding of GPR probing of embedded regarding the theoretical formulation of linear wavefield Final conclusions: Not very much remains to be done and 1 reconstructions to phase errorsof and thenoise synthetic in thedata measurements are much better of the CSI the multi-bistatic (multi-pitch-catch) ultrasonic data sets (Fig 1 reconstructions of the synthetic data are much better structures in concrete The still existing discrepancy between inversion, yet improvements with nonlinear inversion can simulation and experiments will be investigated in greater certainly be made Ockham s razor is simple: Will it work in detail in the future, and transmission data will be included practice? Presently, we are working on the development and as well Since SAFT/FT-SAFT are a widely accepted imaging tools in ultrasonic NDE it seems worthwhile to check its ultrasonics as proposed by Abubakar (2000) and Pelekanos implementation of the vector case in electromagnetics and and Sevroglou (2003)

4 170 R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing 3 (a) (b) 10 z in mm Angle in degree (c) Time sample -10 z in mm (d) z in mm (e) (f) z in mm z in mm Fig Fig 4 4 a) (a) Measured ultrasonic multi-bistatic (multi-pitch-catch) time-domain data set for the transmitter at at 0 0; (b) ; b) Time-domain SAFT reconstruction; c) (c) Actual profile; d) (d) Frequency-domainCSI reconstruction using data at six different frequencies, no noa apriori information used; used; e) (e) Frequency-domain CSI reconstruction using single frequency data and anda priori a information, the the contrast contrast is imaginary is imaginary valued; valued; f) (f) Frequency-domain CSI CSI reconstruction using using data data four at four different different frequencies and and a priori a priori information, the the contrast is is imaginary valued 2 Ultrasonic data: inhomogeneous anisotropic case (InASAFT) Non-destructive testing of defects in nuclear power plant disof the dissimilar welded structure are made of an anisotropic material, this algorithm may fail to produce correct results Here we present results of a modified SAFT algorithm

5 R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing Fig 5 Top: CSI reconstructions of experimental data and synthetic data; Bottom: CSI reconstructions of synthetic data which are obtained Fig 5 Top: CSIbyreconstructions an Domain Integralof Equation experimental (DIE) solver data in the and frequency synthetic domain data; Bottom: CSI reconstructions of synthetic data which are obtained by an Domain Integral Equation (DIE) solver in the frequency domain R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing 5 The algorithm is called inhomogeneous anisotropic SAFT (InASAFT) (Shlivinski et al, 2004a,b) The InASAFT algorithmet The specimen has the dimensions 2 m 15 m 07 m, and the concrete material is supposed to be homogeneous is al: shown Numerical to yieldmodeling better results andover Wavefield the SAFT Inversion al- and in Nondestructive lossless with a relative Testing permittivity ε r = 2 For the 3 D R Marklein 5 gorithm for complex environments, but the InASAFT suffers from the same difficulties of the SAFT algorithm, ie ghost images and lack of clearly focused images However these artifacts can be identified through numerical modeling of the wave propagation in the structure, eg with the Elastodynamic Finite Integration Technique (EFIT) (Marklein, 2002) Here we consider a realistic dissimilar weld Fig 6 (left) shows a polished cut of a weld where its modeling, based on the design blue-prints, can be seen in Fig 6 simulation of electromagnetic wave propagation and scattering, a plane wave impinging perpendicularly to the surface is chosen, which has the base band pulse structure shown in Fig 9 ( 3 db cut off frequency: 09 GHz); the electric field strength is linearly polarized parallel to the tendon ducts The numerical method to solve Maxwell s equations is based on the Finite Integration Technique, and here we use a commercially available implementation (CST, 2004) Fig 10 exhibits 2 D slices out of the 3 D wave field along (right) Fig 7 (left) shows a synthetic B-scan as a result of the line L1 as indicated in Fig 9, ie a line through the the modeling with EFIT (Marklein, 2002), and Fig 7 (right) displays the InASAFT reconstruction The crack inside the weld is reconstructed correctly The experimental B-scan and InASAFT image is given in Fig 8, where the two focused points in the black ellipse correspond to the crack tips coarser reinforcement Typically, even though the highest frequency wavelength is much larger than the grid spacing, the plane penetrates the reinforcement and leaves it again as a nearly plane wave, nevertheless much weaker in amplitude than before Hence, the scattering by the tendon ducts is also obvious resulting in a 3 D data field when scanned along the surface; these data are 2 D with regard to the scan coordinate, each each and of of the the the third weld weld dimension parts; parts; Right: Right: is time Blue-print Blue-print Two slices modeling along modeling Fig Fig 6 Left: 6 Left: Polished Polished 3 Electromagnetic cut cut of of the the dissimilar dissimilar data: Ground weld weld Note Note Penetrating the thegrain Radar orientation (GPR) application the lines L1 and L2 of Fig 9 of the data are displayed in Fig 6 Left: Polished cut of the dissimilar weld Note the grain orientation in each of the weld parts; Right: Blue-print modeling Fig 11: The slice along L1 through the coarser grid clearly Fig 9 shows the geometry of a concrete test specimen as a computer model, which was actually realized by the Federal Highway Research Institute in Germany The specimen contains two circular cylindrical tubes of infinite electrical conductivity with a diameter of 100 mm as an appropriate model for tendon ducts The reinforcement is perfectly conducting, it exhibits two different mesh sizes of 75 mm and 50 mm exhibits the hyperbolic diffraction curves of the tendon ducts, whereas they are not visible along the finer grid This is confirmed applying a linear diffraction tomographic inverse scattering scheme FT-SAFT (Mayer et al, 1990) to the 3 D data: Fig 12 clearly shows images of the tendon ducts only below the coarser grid Thus, simulations of that kind define a limiting grid size for GPR tendon duct imaging, but it turns out Fig 7 Fig Left: 7 Left: Synthetic Synthetic B-scan B-scan as aasresult a result of of the the EFIT EFIT modeling; Right: InASAFT reconstruction Fig 7 Left: Synthetic B-scan as a result of the EFIT modeling; Right: InASAFT reconstruction that that this this upper upper bound bound cannot yet yet be be met met by by experiments tion, for instance artifacts in inthethe reconstruction, can can be interpreted Obviously, like likeininthethe ultrasonic case, case, simulated electromagnetic data imaging using using FT-SAFT leads leads to an toin- an in- be intuitive physical understanding of GPR probing of embedded

6 172 R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing 6 R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing 6 R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing Fig 8 Left: Experimental B-scan; Right: InASAFT reconstruction Fig 8 Left: Experimental B-scan; Right: InASAFT reconstruction Fig 8 Left: Experimental B-scan; Right: InASAFT reconstruction Fig 9 Left: Geometry of a concrete test specimen as a computer model; Right: Excitation signal & Sevroglou (2003) General Assembly of the International Union of Radio Science (URSI), CD-ROM, 2002 Fig 9 Left: Geometry of a concrete test specimen as a computer model; Right: Excitation signal Marklein, R, Mayer, K, Hannemann, R, Krylow, T, Balasubramanian, K, Excitation Langenberg, signal K J, and Schmitz, V: Linear and non- Fig 9 Left: Geometry References of a concrete test specimen as a computer model; Right: R Marklein et al: Numerical Modeling and Wavefield Inversion inlinear Nondestructive inversion algorithms Testing applied in nondestructive evaluation, 7 & Abubakar, Sevroglou A: (2003) Three-Dimensional Nonlinear Inversion of Electrical Inverse General Problems, Assembly 18, , of the International 2002 Union of Radio Science Conductivity, Delft University Press, Delft, 2000 Mayer, (URSI), K, Marklein, CD-ROM, R, Langenberg, 2002 K J, and Kreutter, T: CST MICROWAVE STUDIO, Three-dimensional Marklein, R, Mayer, imaging K, system Hannemann, based onr, Fourier Krylow, transform T, Balasubramanian, aperture K, Langenberg, focusing technique, K J, and Ultrasonics, Schmitz, 28, V: , Linear and non- References Marklein, R: The Finite Integration Technique as a General Tool to synthetic Compute Acoustic, Electromagnetic, Elastodynamic, and Coupled Wave A: Fields, Three-Dimensional in Review of Radio Nonlinear Science: Inversion of Electrical URSI, Pelekanos, Inverse G and Problems, Sevroglou, 18, V: , Journal of Computational 2002 and Ap linear inversion algorithms applied in nondestructive evaluation, Abubakar, edited by W R Stone, IEEE Press, Piscataway, , 2002 Conductivity, Delft University Press, Delft, 2000 plied Mathematics, 151, , 2003 Mayer, K, Marklein, R, Langenberg, K J, and Kreutter, T: Marklein, R, Balasubramanian, K, Qing, A, and Langenberg, Shlivinski, A, Langenberg, K J, and Marklein, R: Ultrasonic CST MICROWAVE STUDIO, Three-dimensional imaging system based on Fourier transform K J: Linear and nonlinear iterative scalar inversion of multifrequency multi-bistatic experimental electromagnetic scattering Modelling and Imaging in Dissimilar Welds, Proceedings of Marklein, R: The Finite Integration Technique as a General Tool to synthetic aperture focusing technique, Ultrasonics, 28, , the 30th MPA-Seminar in Conjunction with the 9th German- Compute data, Inverse Acoustic, Problems, Electromagnetic, 17, , 2001 Elastodynamic, and Coupled Wave R, Fields, Balasubramanian, in Review of K, Radio Langenberg, Science: K J, Miao, URSI, J, Shlivinski, Pelekanos, A, Langenberg, G and Sevroglou, K J, Marklein, V: Journal R, Schmitz, of Computational V, Müller, and Ap- Japanese 1990 Seminar, Stuttgart, 2004 Marklein, edited and Sinaga, by W S R M: Stone, Applied IEEELinear Press, and Piscataway, Nonlinear Scalar , Inverse 2002 W, and plied Mletzko, Mathematics, U; Modelling 151, , of Elastic2003 Wave Propagation and Marklein, Scattering: R, ABalasubramanian, Comparative Study, K, Proceedings Qing, A, of and thelangenberg, XXVIIth Shlivinski, A, Langenberg, K J, and Marklein, R: Ultrasonic K J: Linear and nonlinear iterative scalar inversion of multifrequency multi-bistatic experimental electromagnetic scattering the 30th MPA-Seminar in Conjunction with the 9th German- Modelling and Imaging in Dissimilar Welds, Proceedings of data, Inverse Problems, 17, , 2001 Japanese Seminar, Stuttgart, 2004 Marklein, R, Balasubramanian, K, Langenberg, K J, Miao, J, Shlivinski, A, Langenberg, K J, Marklein, R, Schmitz, V, Müller, and Sinaga, S M: Applied Linear and Nonlinear Scalar Inverse W, and Mletzko, U; Modelling of Elastic Wave Propagation and Scattering: A Comparative Study, Proceedings of the XXVIIth Fig 10 2-D slices out of the 3-D wave field along the lines L1 and L2 in Fig 9a Fig 10 2-D slices out of the 3-D wave field along the lines L1 and L2 in Fig 9a

7 R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing 173 Fig 10 2-D slices out of the 3-D wave field along the lines L1 and L2 in Fig 9a Fig 11 Received Fig 11 signals Received along signals lines along L1 lines (left) L1 (left) and L2 and (right) L2 of offig 9 (left) (logarithmic scale: scale: 60 db) 60 db) 8 R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing Imaging of Defects in Dissimilar Welds, Proceedings of the 4th 8 R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing International Conference on NDE in Relation to Structural Integrity for Nuclear and Pressurised Components, London, 2004 van den Berg, P M, Abubakar, A, and Fokkema, J T: Extended Contrast Source Inversion, Radio Science 38, VIC-1 VIC-10, 2003 Fig 12 Linear diffraction tomographic inverse scattering scheme to the 3-D data Fig 12 Linear diffraction tomographic inverse scattering scheme to the 3-D data Fig 12 Linear diffraction tomographic inverse scattering scheme to the 3-D data Fig 13 Three orthogonal slices through the 3-D FT-SAFT image volume of experimental data Fig 13 Three orthogonal slices through the 3-D FT-SAFT image volume of experimental data Fig 13 Three orthogonal slices through the 3-D FT-SAFT image volume of experimental data

8 174 R Marklein et al: Numerical Modeling and Wavefield Inversion in Nondestructive Testing References Abubakar, A: Three-Dimensional Nonlinear Inversion of Electrical Conductivity, Delft University Press, Delft, 2000 CST MICROWAVE STUDIO, Marklein, R: The Finite Integration Technique as a General Tool to Compute Acoustic, Electromagnetic, Elastodynamic, and Coupled Wave Fields, in Review of Radio Science: URSI, edited by: Stone, W R, IEEE Press, Piscataway, , 2002 Marklein, R, Balasubramanian, K, Qing, A, and Langenberg, K J: Linear and nonlinear iterative scalar inversion of multifrequency multi-bistatic experimental electromagnetic scattering data, Inverse Problems, 17, , 2001 Marklein, R, Balasubramanian, K, Langenberg, K J, Miao, J, and Sinaga, S M: Applied Linear and Nonlinear Scalar Inverse Scattering: A Comparative Study, Proceedings of the XXVIIth General Assembly of the International Union of Radio Science (URSI), CD-ROM, 2002 Marklein, R, Mayer, K, Hannemann, R, Krylow, T, Balasubramanian, K, Langenberg, K J, and Schmitz, V: Linear and nonlinear inversion algorithms applied in nondestructive evaluation, Inverse Problems, 18, , 2002 Mayer, K, Marklein, R, Langenberg, K J, and Kreutter, T: Threedimensional imaging system based on Fourier transform synthetic aperture focusing technique, Ultrasonics, 28, , 1990 Pelekanos, G and Sevroglou, V: Journal of Computational and Applied Mathematics, 151, , 2003 Shlivinski, A, Langenberg, K J, and Marklein, R: Ultrasonic Modelling and Imaging in Dissimilar Welds, Proceedings of the 30th MPA-Seminar in Conjunction with the 9th German- Japanese Seminar, Stuttgart, 2004 Shlivinski, A, Langenberg, K J, Marklein, R, Schmitz, V, Müller, W, and Mletzko, U; Modelling of Elastic Wave Propagation and Imaging of Defects in Dissimilar Welds, Proceedings of the 4th International Conference on NDE in Relation to Structural Integrity for Nuclear and Pressurised Components, London, 2004 van den Berg, P M, Abubakar, A, and Fokkema, J T: Extended Contrast Source Inversion, Radio Science 38, VIC-1 VIC-10, 2003