Invariant properties and rotation transformations of the GPR scattering matrix Almendra Villela ⁎, José M. Romo División de Ciencias de la Tierra, Centro de Investigación Científica y de Educación Superior de Ensenada, Carretera Ensenada-Tijuana #3918, zona Playitas, Ensenada, Baja California 22860, Mexico abstract article info Article history: Received 13 July 2012 Accepted 1 January 2013 Available online 11 January 2013 Keywords: GPR scattering matrix Multiple polarizations Invariant properties GPR matrix transformation We analyze the properties of the scattering matrix associated with the incident and scattered electric fields used in GPR. The elements of the scattering matrix provide information produced by different polarizations of the incident wave field. Rotationally invariant quantities such as trace, determinant and Frobenius norm lead to images that combine the information contained in the four elements of the scattering matrix in a mathematically simple and sound manner. The invariant quantities remove the directional properties implic- it in the dipolar field used in GPR allowing the application of standard processing techniques designed for scalar fields, such as those used in seismic data processing. We illustrate the non-directional properties of the invariants using a 3D simulation of the wavefield produced by a point scatterer. The estimation of the azimuth angle of elongated targets is also explored using rotation transformations that maximize alternative- ly the co-polarized or the cross-polarized responses. The angle estimation is essentially an unstable process, particularly if low amplitudes or noisy data are involved. We apply the Frobenius norm ‖S‖ F as a criterion for selection of the best amplitudes to use for a more stable and significant angle estimation. The performance of our formulation was tested with synthetic data produced by a 3D model of an air-filled metal pipe buried in a homogeneous halfspace. The images resulting from the invariants show a clear diffraction hyperbola suitable for a scalar wavefield migration, while the azimuth of the pipe is neatly resolved for amplitudes selected with ‖S‖ F ≥0.4. A field experiment conducted above an aqueduct pipe illustrates the proposed methods with real data. The images obtained from the invariants are better than those from the individual elements of the scattering matrix. The azimuth estimated using our formulation is in agreement with the probable orientation of the aqueduct. Finally, a field experiment above a buried air-filled barrel shows that combining the information in the way proposed in this work may lead to an improved image of the subsurface target, the cost to pay is the lost of directional informa- tion contained in the scattering matrix. In general, we claim that the methods proposed in this work can be useful to analyze the information acquired by multicomponent GPR surveys using standard scalar wavefield algorithms. © 2013 Elsevier B.V. All rights reserved. 1. Introduction The vector nature of the electromagnetic fields generated and mea- sured by ground penetrating radars (GPR) prompts the study of the response of buried targets to different field polarizations (Roberts, 1994; Roberts and Daniels, 1996). A number of experiments have been attempted to exploit the use of multicomponent measurements for different subsurface targets. It has been shown that using different polarizations can help to better define the size, shape and orientation of the target. For example, Guy et al. (1999) found that non-planar or rough objects are better imaged using a receiver antenna perpendicular to the polarization of the source (cross-polarization); Radzevicius and Daniels (2000) studied the backscattered fields from cylinders with different polarization properties; they found that high impedance dielectric pipes are best imaged with the long axis of the dipoles orient- ed orthogonal to the long axis of the pipes, while using antennas orient- ed parallel to the long axis of the pipes produces better images for low impedance metallic pipes. In recent years, a number of migration algo- rithms have been proposed to properly account for the vector nature of GPR wave propagation as well as for dipolar antenna radiation pat- terns and coupling factors (Lambot et al., 2004; Streich et al., 2007; van der Kruk et al., 2003). Orlando and Slob (2009), using 2.0 GHz multicomponent data to detect cracks in a historical building, found that vector migration images have better resolution than images obtained with standard 2D scalar migration. On the other hand, it is common that algorithms designed for elastic waves processing (seismic reflection data) be used for GPR data pro- cessing. Lehmann et al. (2000) pointed out the convenience of using the sum of copolarized fields to obtain non-directional wavefields suit- able for processing with standard scalar wavefield algorithms. In this work we explore the rotation-invariant properties of the scat- tering matrix and their ability to transform the characteristic directional Journal of Applied Geophysics 90 (2013) 71–81 ⁎ Corresponding author at: CICESE/Earth Sciences, P.O. Box 434843, San Diego, CA, 92143-4843, United States. Tel.: +52 646 175 0500; fax: +52 646 175 0567. E-mail addresses: avillela@cicese.mx (A. Villela), jromo@cicese.mx (J.M. Romo). 0926-9851/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jappgeo.2013.01.001 Contents lists available at SciVerse ScienceDirect Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo