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Nonlinear modulation of gravity-capillary waves travelling principally in one direction at the surface of a three-dimensional fluid leads to the Davey–Stewartson system for the wave amplitude and the induced mean flow. In this paper, we present a rigorous derivation of the system and show that the resulting wavepacket satisfies the water wave equations at leading order with precise bounds for the remainder. Key steps in the analysis are the analyticity of the Dirichlet–Neumann operator with respect to the surface elevation that defines the fluid domain, precise bounds for the Taylor remainders and the description of individual terms in the Taylor series as pseudo-differential operators and their estimates under multiple scale expansions. Résumé La modulation nonlinéaire d’ondes se propageant principalement dans une direction à la surface d’un canal tridimensionnel conduit au système de Davey–Stewartson pour l’amplitude de l’onde et le champ moyen induit. Dans cet article, nous présentons une dérivation rigoureuse du système et nous montrons que l’approximation modulationnelle satisfait les équations des ondes de surface à l’ordre dominant. Les étapes importantes dans l’analyse sont l’analyticité de l’opérateur de Dirichlet–Neumann par rapport à l’interface qui définit le domaine du fluide, des estimations du reste dans le développement de Taylor de l’opérateur, ainsi que la description des différents termes de la série comme opérateurs pseudo-differentiels et leurs développements multi-échelles.

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A B S T R A C TThe stimulation of a geothermal well in Basel, Switzerland produced a distribution of microseismic event locations with an overall alignment in the direction of the maximum horizontal stress. Fault plane solutions of individual larger events indicated movements on fracture planes at an angle to the maximum horizontal stress that could not be reliably interpreted from the event locations. To obtain higher resolution images of the microseismic event locations, events with similar waveforms have been identified by multiplet analysis. A number of receivers were used in the multiplet processing to ensure each multiplet is represented by a unique group of waveforms. The location accuracy within each multiplet has been significantly improved using cross-correlation to refine the shear-wave traveltime picks. The distribution of events within each multiplet can be interpreted as being due to movements on a single fracture or a number of near parallel fractures. It is shown that whilst the overall distribution of events is around the direction of the maximum horizontal stress, the individual multiplets representing fracture planes have a variety of azimuths and dips.

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