  The inverse problem we consider is to reconstruct the location and shape of
buried obstacles in the lower half-space of an unbounded two-layered medium in
two dimensions from phaseless far-field data. A main difficulty of this problem
is that the translation invariance property of the modulus of the far field
pattern is unavoidable, which is similar to the homogenous background medium
case. Based on the idea of using superpositions of two plane waves with
different directions as the incident fields, we first develop a direct imaging
method to locate the position of small anomalies and give a theoretical
analysis of the algorithm. Then a recursive Newton-type iteration algorithm in
frequencies is proposed to reconstruct extended obstacles. Finally, numerical
experiments are presented to illustrate the feasibility of our algorithms.
