Extending potential-field data to vertical complex-plane slices provides opportunities, unique to analytic functions, for subsurface imaging. We begin with an existing numerical method for differentiation by integration using Cauchy's integral formula. An example from data over northern Ontario, Canada, illustrates better conditioning compared to derivatives computed using finite differences. Next, these quantities are used for analytic continuation using rational complex-series expansion (Padé approximation). This method provides stable downward continuation to distances that are significantly greater than are achievable by Taylor series and spectral methods. A synthetic test confirms that the rational approximant faithfully reproduces the magnetic response of a thin sheet. A sign reversal of the total field, coincident with the origin of the sheet, features prominently on this example. We also provide synthetic results for sheets with finite depth extent, point sources, thin slabs, and contacts. These also exhibit polarity flips near the source location. Further, these provide a template for interpreting the source geometry using the shape of the approximating function. Additional synthetic tests illustrate an automatic method for locating closely spaced random assemblages of isolated poles. We apply our methods to a deposit-scale heliborne survey over a recently discovered volcanogenic massive sulphide ore deposit in a Canadian greenstone belt. The analytically continued profile over the deposit suggests the presence of a subvertical thin sheet of unknown depth extent. This interpretation coincides with a sulphide ore body that has been previously delimited by several drill holes. Some known prospects near this deposit are also imaged on maps and profiles of the Padé approximant.

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