Ultrasonic non destructive testing (NDT) is an efficient method to detect flaws in industrial parts. The detection of flat bottom holes (FBH) is a typical problem, which serves as a reference in the NDT community. It is nevertheless a hard task if the FBH is short because its echo overlaps with the backwall echo. In this paper, we propose to use a sparse deconvolution approach to separate the FBH echo from the backwall echo and hence to detect the FBH with high resolution. From experimental data acquired with a FBH drilled in an aluminum plate, we show that the FBH echo can be modeled as a high-pass filtered version of the incident ultrasonic wave. Therefore, we build a propagation model depending on the instrument response and of a specific attenuation function. A sparse deconvolution technique is then proposed to precisely locate the flaw and the backwall positions. In application to real data, we show that the developed approach is more efficient than conventional techniques such as gates or invariant sparse deconvolution.