The Buffon's needle problem and the design of a geophysical survey

A logical procedure for designing a geophysical survey when sampling an area with a regular grid can be summarized as follows: model the expected anomaly, estimate the expected noise level, estimate the area of the anomaly above the noise level, choose the spacing, in both the x - and y -directions, of the measurement grid. This last step can be approached according to two main strategies: either when applying the sampling theorem to the shortest dimension of the anomaly or when using a coarser grid, leaving a more complete definition of the anomaly to a later fitting. When following this second option, it can be constructive to estimate the probability of intercepting a given anomaly with a specific segment of profile and a given profile spacing. This latter procedure is analysed by considering a rectangle approximating the plane projection of the anomaly shape and taking into account various ratios between the grid spacing and the rectangle sides. The formulae for estimating the probability of intersecting a given anomaly with a given segment of a given profile spacing are calculated. To demonstrate the accuracy of the results, a Monte-Carlo simulation on a synthetic magnetic map was performed, obtaining, for different ratios between the sides, segment length and profile interval, an agreement better than 0.1% with the analytical formulae.