During a recent marine magnetic survey of the Juan de Fuca Rise, two magnetometers were towed near the seafloor, one about 300 m above the other. To understand how to interpret the records, we investigate a simple statistical model: two magnetometers moving on parallel paths above a statistically stationary source, with known spectrum. Magnetometers on paths normal to perfectly lineated magnetic anomalies will measure signals that have unit coherence at all wavelengths. Departure of the system from this ideal state can be diagnosed by a lower coherence, and something about the across-track structure can be learned from the shape of the coherence spectrum. We calculate the power and cross spectra of the profile signals in terms of the two-dimensional power spectrum of the field just above the source region; hence we obtain the coherence and phase spectra. For the special case of a white source spectrum we find surprisingly high coherences. A set of inequalities between the spectral estimates is derived and can be used to check the consistency of the measured signals with the model assumptions. The theory is applied to a magnetic traverse of the Juan de Fuca Rise when two near-bottom magnetometers were deployed. The key results are these: in the wavelength range above about 1 km the observed coherency is substantially higher than that from the disordered field model, consistent with the highly lineated structures observed at the surface over all ocean ridge systems. On scales between 500 m and 1 km the coherence falls to levels indistinguishable from those given by an isotropic flat spectrum, implying that on these scales there is little or no across-track lineation. This finding means that the resolution of paleomagnetic field behavior based on seafloor data in this area is no better than 36,000 years.¿ 1997 American Geophysical Union