Filtering Deterministic Layer Effects in Imaging

Sensor array imaging arises in applications such as nondestructive evaluation of materials with ultrasonic waves, seismic exploration, and radar. The sensors probe a medium with signals and record the resulting echoes, which are then processed to determine the location and reflectivity of remote reflectors. These could be defects in materials such as voids, fault lines or salt bodies in the earth, and cars, buildings, or aircraft in radar applications. Imaging is relatively well understood when the medium through which the signals propagate is smooth, and therefore nonscattering. But in many problems the medium is heterogeneous, with numerous small inhomogeneities that scatter the waves. We refer to the collection of inhomogeneities as clutter, which introduces an uncertainty in imaging because it is unknown and impossible to estimate in detail. We model the clutter as a random process. The array data is measured in one realization of the random medium, and the challenge is to mitigate cumulative clutter scattering so as to obtain robust images that are statistically stable with respect to different realizations of the inhomogeneities. Scatterers that are not buried too deep in clutter can be imaged reliably with the coherent interferometric (CINT) approach. But in heavy clutter the signal-to-noise ratio (SNR) is low and CINT alone does not work. The “signal,” the echoes from the scatterers to be imaged, is overwhelmed by the “noise,” the strong clutter reverberations. There are two existing approaches for imaging at low SNR: The first operates under the premise that data are incoherent so that only the intensity of the scattered field can be used. The unknown coherent scatterers that we want to image are modeled as changes in the coefficients of diffusion or radiative transport equations satisfied by the intensities, and the problem becomes one of parameter estimation. Because the estimation is severely ill-posed, the results have poor resolution, unless very good prior information is available and large arrays are used. The second approach recognizes that if there is some residual coherence in the data, that is, some reliable phase information is available, it is worth trying to extract it and use it with well-posed coherent imaging methods to obtain images with better resolution. This paper takes the latter approach and presents a first attempt at enhancing the SNR of the array data by suppressing medium reverberations. It introduces filters, or annihila∗Published electronically November 8, 2012. This paper originally appeared in Multiscale Modeling & Simulation, Volume 7, Number 3, 2009, pages 1267–1301. http://www.siam.org/journals/sirev/54-4/88097.html †Department of Computational and Applied Mathematics, Rice University, MS 134, Houston, TX 77005-1892 ([email protected]). The work of this author was partially supported by AFSOR grant FA9550-12-1-0117, ONR grants N00014-12-1-0256, N00014-09-1-0290, and N00014-05-1-0699, and NSF grants DMS-0907746, DMS-0934594, and DMS-0604008. ‡Shell Oil Bellaire Technology Center, Houston, TX 77025 ([email protected]). The Ph.D. studies of this author were supported by ONR grant N00014-05-1-0699 and by NSF grants DMS-0604008, DMS-0305056, and DMS-0354658. §Department of Mathematics, Stanford University, Stanford, CA 94305 ([email protected]. edu). The work of this author was partially supported by U.S. Army grant W911NF-07-2-0027-1 and by AFOSR grant FA9550-11-1-0266. ¶Department of Applied Mathematics, University of Crete and IACM/FORTH, GR-71409 Heraklion, Greece ([email protected]). The work of this author was partially supported by European FP7 Marie Curie International Reintegration grant MIRG-CT-2007-203438 and the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 239959. 757 D ow nl oa de d 12 /1 7/ 12 to 1 71 .6 7. 21 6. 22 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp :// w w w .s ia m .o rg /jo ur na ls /o js a. ph p