From well known mathematical theory it can be demonstrated that most contour maps may be considered to be built up by the superposition of a double infinity of elementary undulating surfaces, each of which has the form of a horizontal sinusoidally corrugated sheet, infinite in extent. These elementary surfaces may have all possible wave lengths, orientations, amplitudes, and phases. Several examples are given of simple mosaic-type composite maps built up by combining only two such elementary surfaces in different ways. These resemble geophysical contour maps in many significant respects.Residual maps are often prepared by using a template procedure for computing the residual value at any point as a linear combination of several neighborhood values interpolated from the original map. An expression is derived for the Fourier transform of any residual map prepared in this way. This transform gives the amplitude spectrum of the residual map in terms of the amplitude spectrum of the original map and the geometry of the template pattern. It is applied to the special case of an original map of the two-component mosaic variety mentioned above. The results are presented quantitatively in the form of attenuation, or filter, curves which show the amplitudes of the residual anomalies for various sizes and shapes of original anomalies, and for several different residual templates.The geometrical significance of 'second derivative' maps is discussed, and it is shown that they may be prepared by a process which is a limiting case of applying a residual template pattern of very simple type.Attenuation curves are presented for several kinds of residual templates when applied to an idealized original contour map consisting of a single anomaly of various shapes. These filter curves are very similar to those for original maps of the simple mosaic type. It is concluded that, since most geophysical maps may be considered to be of a kind intermediate between these two extreme types, the attenuation curves given here may be useful for designing residual templates which will have desired selective characteristics.