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A B S T R A C TIn the past, integral formulations for marine data-driven demultiple methods have been derived from reciprocity theorems. Two fundamental assumptions in these derivations were that the sea-surface is flat and has a known reflection coefficient, often taken to be minus one. In this paper, we show that for dual sensor data these assumptions can be relaxed. The sea-surface has to obey the same conditions as any other reflecting boundary in the subsurface: it must be constant in time but shape and reflection strength can vary in space. For both surface-related multiple elimination, and multiple attenuation by multi-dimensional deconvolution, we derive integral equations that depend only on the measured pressure and particle velocity fields. Finally, we show there is an intimate connection between the integral equations for the methods. I N T R O D U C T I O NIntegral equations for data-driven, multi-dimensional multiple attenuation can be derived from reciprocity theorems. The case of surface-related multiple elimination was derived by Fokkema and van den Berg (1993). Weglein et al. (2003) showed that for surface-related multiples the inverse scattering series approach, developed by Weglein et al. (1997), results in an equivalent integral formulation. Amundsen (2001) performed a similar derivation for multiple attenuation by multi-dimensional deconvolution.In all derivations, the boundary condition was that the seasurface is flat and that the pressure is zero at this surface. Berkhout and Verschuur (1997) derived a method to predict and subtract surface-related multiples through a feedback mechanism in which the sea-surface reflectivity can be parameterized. In practice, the surface-related multiples are subtracted in an adaptive way Verschuur and Berkhout (1997). In this paper, we reformulate surface-related multiple elimination and multi-dimensional deconvolution such that it can

Abstract. One-way wave equations conveniently describe wave propagation in media with discontinuous and/or rapid variations in one direction, but with smooth and slow variations in the complementary transverse directions. In the past, reciprocity theorems have been developed in terms of one-way wave fields. The boundaries of the integration volumes and the variations of the medium parameters must adhere to strict conditions. The variations must have the smoothness required by pseudodifferential operators, while the boundaries have to be flat. To extend the applicability to nonflat boundaries, this paper formulates one-way wave equations and corresponding reciprocity theorems in terms of curvilinear coordinates of the semiorthogonal (SO) type. In SO coordinate systems, one of the covariant basis vectors is orthogonal to the others, which can be nonorthogonal among each other. The same applies to the contravariant basis vectors. Furthermore, the orthogonal directions coincide; that is, the orthogonal co-and contravariant basis vectors coincide. SO coordinates are characterized by a local property of the basis vectors. An extra specification is necessary to make them conform in any way to nonflat boundaries. This can be done in terms of so-called lateral Cartesian (LC) coordinates. Cartesian coordinates are mapped to LC coordinates by applying an invertible transformation to one coordinate while keeping the others the same. LC coordinates are a straightforward means to describe or conform to nonflat boundaries. Applications of the extended reciprocity theorems include removal of multiple reflections, removal of complex propagation effects, wave field extrapolation, and synthesis of unrecorded data.

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