The interpretation of marine magnetic anomalies usually consists either in determining the magnetization distribution assuming the source geometry and magnetization direction or in determining the magnetic layer thickness assuming the magnetization direction and intensity. In this paper, we introduce a new technique that allows modeling of the thickness of the magnetic source layer with very few a priori assumptions about the magnetization: the magnetic layer is assumed to be made of a series of bodies, each having a constant unknown magnetization and an unknown size. This technique is based upon the application of the continuous wavelet transform recently introduced for the interpretation of potential field data as a multipole decomposition. We present applications to synthetic data, to one deep-tow magnetic profile recorded across the Juan de Fuca Ridge (JDF), and to three deep-tow magnetic profiles recorded across the Central Indian Ridge (CIR). Our results confirm that despite significant source thickness variations (100--1200 m across the CIR), measured magnetic anomalies mostly reflect past geomagnetic field intensity fluctuations; however, we show that within the axial region of high magnetization, thickness variations have a significant contribution to short-wavelength variations of deep-tow magnetic signals (>100 nT).