{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:30:24Z","timestamp":1772253024800,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,12,23]],"date-time":"2017-12-23T00:00:00Z","timestamp":1513987200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"COST (European Cooperation in Science and Technology)","award":["Action TU1208 \u201cCivil engineering applications of Ground Penetrating Radar\u201d"],"award-info":[{"award-number":["Action TU1208 \u201cCivil engineering applications of Ground Penetrating Radar\u201d"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>This paper deals with bistatic subsurface probing of a horizontally layered dielectric half-space by means of ultra-wideband electromagnetic waves. In particular, the main objective of this work is to present a new method for the solution of the two-dimensional back-scattering problem arising when a pulsed electromagnetic signal impinges on a non-uniform dielectric half-space; this scenario is of interest for ground penetrating radar (GPR) applications. For the analytical description of the signal generated by the interaction of the emitted pulse with the environment, we developed and implemented a novel time-domain version of the coupled-wave Wentzel-Kramers-Brillouin approximation. We compared our solution with finite-difference time-domain (FDTD) results, achieving a very good agreement. We then applied the proposed technique to two case studies: in particular, our method was employed for the post-processing of experimental radargrams collected on Lake Chebarkul, in Russia, and for the simulation of GPR probing of the Moon surface, to detect smooth gradients of the dielectric permittivity in lunar regolith. The main conclusions resulting from our study are that our semi-analytical method is accurate, radically accelerates calculations compared to simpler mathematical formulations with a mostly numerical nature (such as the FDTD technique), and can be effectively used to aid the interpretation of GPR data. The method is capable to correctly predict the protracted return signals originated by smooth transition layers of the subsurface dielectric medium. The accuracy and numerical efficiency of our computational approach make promising its further development.<\/jats:p>","DOI":"10.3390\/rs10010022","type":"journal-article","created":{"date-parts":[[2017,12,26]],"date-time":"2017-12-26T03:06:38Z","timestamp":1514257598000},"page":"22","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Application of Coupled-Wave Wentzel-Kramers-Brillouin Approximation to Ground Penetrating Radar"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9634-9431","authenticated-orcid":false,"given":"Igor","family":"Prokopovich","sequence":"first","affiliation":[{"name":"Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation (IZMIRAN), 108840 Moscow, Russia"}]},{"given":"Alexei","family":"Popov","sequence":"additional","affiliation":[{"name":"Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation (IZMIRAN), 108840 Moscow, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4764-6338","authenticated-orcid":false,"given":"Lara","family":"Pajewski","sequence":"additional","affiliation":[{"name":"Department of Information Engineering, Electronics and Telecommunications, Sapienza University of Rome, 00185 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9494-9097","authenticated-orcid":false,"given":"Marian","family":"Marciniak","sequence":"additional","affiliation":[{"name":"National Institute of Telecommunications, 04-894 Warsaw, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2017,12,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Benedetto, A., and Pajewski, L. (2015). Civil Engineering Applications of Ground Penetrating Radar, Springer.","DOI":"10.1007\/978-3-319-04813-0"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Persico, R. (2014). Introduction to Ground Penetrating Radar: Inverse Scattering and Data Processing, Wiley-IEEE Press.","DOI":"10.1002\/9781118835647"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1016\/j.conbuildmat.2005.06.007","article-title":"Modelling ground penetrating radar by GprMax","volume":"19","author":"Giannopoulos","year":"2005","journal-title":"Constr. Build. Mater."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Warren, C., Giannopoulos, A., and Giannakis, I. (2015, January 7\u201310). An advanced GPR modelling framework\u2014The next generation of gprMax. Proceedings of the 8th International Workshop on Advanced Ground Penetrating Radar (IWAGPR 2015), Florence, Italy.","DOI":"10.1109\/IWAGPR.2015.7292621"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1109\/JSTARS.2015.2430933","article-title":"Experimental and Modeled Performance of a Ground Penetrating Radar Antenna in Lossy Dielectrics","volume":"9","author":"Warren","year":"2016","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1364\/JOSAA.30.000783","article-title":"Spectral domain method for the electromagnetic scattering by a buried sphere","volume":"30","author":"Frezza","year":"2013","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.jappgeo.2013.01.004","article-title":"Cylindrical-Wave Approach for Electromagnetic Scattering by Subsurface Metallic Targets in a Lossy Medium","volume":"97","author":"Frezza","year":"2013","journal-title":"J. Appl. Geophys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1109\/LGRS.2012.2197172","article-title":"Electromagnetic Scattering by a Metallic Cylinder Buried in a Lossy Medium with the Cylindrical-Wave Approach","volume":"10","author":"Frezza","year":"2013","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Bourlier, C., Le Bastard, C., and Pinel, N. (July, January 30). Full wave PILE method for the electromagnetic scattering from random rough layers. Proceedings of the 15th International Conference on Ground Penetrating Radar (GPR 2014), Brussels, Belgium.","DOI":"10.1109\/ICGPR.2014.6970483"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3","DOI":"10.2495\/CMEM150011","article-title":"Transmitted field in the lossy ground from ground penetrating radar (GPR) dipole antenna","volume":"59","author":"Poljak","year":"2015","journal-title":"WIT Trans. Model. Simul."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Poljak, D., Dori\u0107, V., Birki\u0107, M., El Khamlichi Drissi, K., Lallechere, S., and Pajewski, L. (2017, January 28\u201330). A simple analysis of dipole antenna radiation above a multilayered medium. Proceedings of the 9th International Workshop on Advanced Ground Penetrating Radar (IWAGPR 2017), Edinburgh, UK.","DOI":"10.1109\/IWAGPR.2017.7996037"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1721","DOI":"10.1109\/TAP.2005.846719","article-title":"FDTD simulation of TE and TM plane waves at nonzero incidence in arbitrary layered media","volume":"53","author":"Winton","year":"2005","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"045712","DOI":"10.1088\/1464-4258\/11\/4\/045712","article-title":"Light propagation in 1D inhomogeneous deterministic media: The effect of discontinuities","volume":"11","author":"Diamant","year":"2009","journal-title":"J. Opt. A Pure Appl. Opt."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zeng, Q., and Delisle, G.Y. (2010, January 12\u201316). Transient analysis of electromagnetic wave reflection from a stratified medium. Proceedings of the 2010 Asia-Pacific International Symposium on Electromagnetic Compatibility, Beijing, China.","DOI":"10.1109\/APEMC.2010.5475664"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kaganovsky, Y., and Heyman, E. (2010, January 16\u201319). Pulsed beam propagation in plane stratified media: Asymptotically exact solutions. Proceedings of the 2010 URSI International Symposium on Electromagnetic Theory, Berlin, Germany.","DOI":"10.1109\/URSI-EMTS.2010.5637209"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Diamanti, N., Annan, A.P., and Redman, J.D. (July, January 30). Impact of Gradational Electrical Properties on GPR Detection of Interfaces. Proceedings of the 15th International Conference on Ground Penetrating Radar (GPR 2014), Brussels, Belgium.","DOI":"10.1109\/ICGPR.2014.6970480"},{"key":"ref_17","unstructured":"Connor, K.M., and Dowding, C.H. (1999). GeoMeasurements by Pulsing TDR Cables and Probes, CRC Press."},{"key":"ref_18","unstructured":"Schlaeger, S. (2002). Inversion of TDR Measurements to Reconstruct Spatially Distributed Geophysical Ground Parameter. [Ph.D. Thesis, University of Karlsruhe]."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1073\/pnas.16.10.627","article-title":"Reflection of waves in an inhomogeneous absorbing medium","volume":"16","author":"Epstein","year":"1930","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_20","first-page":"116","article-title":"A discretization model for the solution of Maxwell\u2019s equations for six-component fields","volume":"31","author":"Weiland","year":"1977","journal-title":"Archiv Elektronik Uebertragungstechnik"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/0021-9991(73)90167-8","article-title":"An integro-differential equation technique for the time-domain analysis of thin wire structures. I. the numerical method","volume":"12","author":"Miller","year":"1973","journal-title":"J. Comput. Phys."},{"key":"ref_22","first-page":"525","article-title":"A Green\u2019s function approach to the determination of internal fields","volume":"11","author":"Krueger","year":"1989","journal-title":"Appl. Math. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"L21405","DOI":"10.1029\/2007GL031459","article-title":"Fast evaluation of zero-offset Green\u2019s function for layered media with application to ground-penetrating radar","volume":"34","author":"Lambot","year":"2007","journal-title":"Geophys. Res. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1364\/JOSAA.23.000099","article-title":"Reflectance optimization of inhomogeneous coatings with continuous variation of the complex refractive index","volume":"23","author":"Larruquert","year":"2006","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1088\/1464-4258\/8\/2\/005","article-title":"Reflectivity of a stratified half-space: The limit of weak inhomogeneity and anisotropy","volume":"8","author":"Grafstrom","year":"2006","journal-title":"J. Opt. A Pure Appl. Opt."},{"key":"ref_26","unstructured":"Landau, L.D., and Lifshits, E.M. (1977). Quantum Mechanics: Non-Relativistic Theory, Butterworth-Heinemann."},{"key":"ref_27","unstructured":"Brekhovskikh, L.M. (1980). Waves in Layered Media, Academic Press. [2nd ed.]."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1007\/978-3-642-45895-8_4","article-title":"Propagation of Electromagnetic Waves","volume":"Volume 4\/16","author":"Flugge","year":"1958","journal-title":"Handbuch der Physik\/Encyclopedia of Physics"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0360-1285(98)00017-3","article-title":"Light scattering for particle characterization","volume":"25","author":"Jones","year":"1999","journal-title":"Prog. Energy Comb. Sci."},{"key":"ref_30","unstructured":"Vinogradov, V.A., Kopeikin, V.V., and Popov, A.V. (2004, January 21\u201324). An approximate solution of 1D inverse problem. Proceedings of the 10th International Conference on Ground Penetrating Radar (GPR 2004), Delft, The Netherlands."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1134\/S0016702913070112","article-title":"Ground Penetrating Radar Investigation of the Supposed Fall Site of a Fragment of the Chelabinsk Meteorite in Lake Chebarkul","volume":"51","author":"Kopeikin","year":"2013","journal-title":"Geochem. Int."},{"key":"ref_32","first-page":"69","article-title":"Search for Chelyabinsk Meteorite Fragments in Chebarkul Lake Bottom (GPR and Magnetic Data)","volume":"2017","author":"Buzin","year":"2017","journal-title":"J. Telecommun. Inf. Technol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1016\/0012-821X(75)90146-6","article-title":"Dielectric Properties of the First 100 Meters of the Moon","volume":"24","author":"Olhoeft","year":"1975","journal-title":"Earth Planet. Sci. Lett."},{"key":"ref_34","unstructured":"Fritz, J. (1982). Partial Differential Equations, Springer. [4th ed.]."},{"key":"ref_35","unstructured":"Claerbout, J.F. (1985). Fundamentals of Geophysical Data Processing, Pennwell Books."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Berkut, A.I., Edemsky, D.E., Kopeikin, V.V., Morozov, P.A., Prokopovich, I.V., and Popov, A.V. (2017, January 7\u201310). Deep penetration subsurface radar: Hardware, results, interpretation. Proceedings of the 9th International Workshop on Advanced Ground Penetrating Radar (IWAGPR 2017), Edinburgh, UK.","DOI":"10.1109\/IWAGPR.2017.7996052"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1134\/S1063771016050110","article-title":"Problems of Moving Ultrasound Penetrative Devices in a Dispersion Medium during Drilling of the Moon\u2019s Regolith","volume":"62","author":"Matveev","year":"2016","journal-title":"Acoust. Phys."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/10\/1\/22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:55:20Z","timestamp":1760208920000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/10\/1\/22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,23]]},"references-count":37,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["rs10010022"],"URL":"https:\/\/doi.org\/10.3390\/rs10010022","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints201712.0035.v1","asserted-by":"object"}]},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,12,23]]}}}