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Marine seismic source arrays are directional. Source directivity is used to attenuate coherent noise, but primary reflected data may be degraded. Source directivity is ignored in a standard processing sequence, so directional source signature deconvolution may be required. In the frequency-wavenumber (f-k) directional deconvolution method, a filter is calculated from far-field source signatures and is applied to the f-k transform of common-receiver gathers. Reflections on common-receiver gathers are often spatially aliased, and this causes practical problems with the technique. Directional deconvolution may also be performed in combination with prestack migration because the prestack Kirchhoff summation migration operator is a function of source take-off angle. The constant-offset section is deconvolved separately with a full range of filters for source signatures radiated in different directions; then the migration summation operator sums across the deconvolved sections, selecting the section which has been deconvolved for the correct source signature at each point. Physical model data, which were acquired over simple models using a directional source, are used to evaluate directional deconvolution assuming constant velocity. Reflector continuity and resolution are improved by using directional deconvolution. Directional deconvolution combined with prestack migration is extended to media in which the velocity varies with depth, and is applied to two datasets from the Southern North Sea. The second dataset, which has shallow steeply dipping reflectors, is improved by using directional deconvolution. Directional deconvolution may be combined with a Kirchhoff migration technique which assumes a linear velocity-depth model. Results are superior to conventional Kirchhoff migration because ray bending is honoured. Directional deconvolution cannot synthesise fully point-source equivalent data from data acquired with a source array without excessive noise amplification. Source arrays with a short in-line dimension should be used where possible. For data which have been acquired with a long source array, directional deconvolution is desirable.