This paper shows that Betti's reciprocity theorem gives an integral equation procedure to eliminate from the physical multicomponent-source, multicomponent-receiver seismic measurements the effect of the physical source radiation pattern and the response of the physical overburden (that is, the medium above the receiver plane). The physical multicomponent sources are assumed to be orthogonally aligned anywhere above the multicomponentreceiver depth level. Other than the position of the sources, no source characteristics are required. The method, denoted the Betti designature/elastic demultiple, has the following additional characteristics: it preserves primary amplitudes while eliminating all waves scattered from the overburden; it requires no knowledge of the medium below the receiver level; it requires no knowledge of the medium above the receiver level; it requires information only of the local density and elastic wave propagation velocities at the receiver level to decompose the physical seismic measurements into upgoing and downgoing waves.Following the Betti designature/elastic demultiple step is an elastic wavefield decomposition step that decomposes the data into PP-, PS-, SP-, and SS-wave responses that would be recorded from idealized compressional-wave and shear-wave sources and receivers. The combined elastic wavefield decomposition on the source and receiver side gives data equivalent to data from a hypothetical survey with overburden absent, with single-component compressional and shear-wave sources, and single-component compressional and shear-wave receivers.When the medium is horizontally layered, the Betti designature/elastic demultiple scheme followed by the elastic source-receiver decomposition scheme greatly simplifies and is conveniently implemented as deterministic multidimensional deconvolution and elastic source-receiver wavefield decomposition of common-source gathers (or common-receiver gathers when source array variations are negligible).Betti designature/elastic demultiple followed by sourcereceiver wavefield decomposition applies to three different seismic experiments: a 9-component (9C) land seismic experiment, a 12-component (12C) ocean-bottom seismic experiment, and an 18-component (18C) borehole seismic experiment. For the land and ocean-bottom seismic experiments, an additional geophone should be deployed below the zero-offset geophone to predict the source-induced vertical traction vector at the source location.A numerical example for the 12C ocean-bottom seismic experiment over a horizontally layered medium validates the Betti designature/elastic demultiple scheme.
A B S T R A C TThis paper presents the theory to eliminate from the recorded multi-component source, multi-component receiver marine electromagnetic measurements the effect of the physical source radiation pattern and the scattering response of the waterlayer. The multi-component sources are assumed to be orthogonally aligned above the receivers at the seabottom. Other than the position of the sources, no source characteristics are required. The integral equation method, which for short is denoted by Lorentz water-layer elimination, follows from Lorentz' reciprocity theorem. It requires information only of the electromagnetic parameters at the receiver level to decompose the electromagnetic measurements into upgoing and downgoing constituents. Lorentz water-layer elimination replaces the water layer with a homogeneous half-space with properties equal to those of the sea-bed. The source is redatumed to the receiver depth.When the subsurface is arbitrary anisotropic but horizontally layered, the Lorentz water-layer elimination scheme greatly simplifies and can be implemented as deterministic multi-component source, multi-component receiver multidimensional deconvolution of common source gathers. The Lorentz deconvolved data can be further decomposed into scattering responses that would be recorded from idealized transverse electric and transverse magnetic mode sources and receivers. This combined electromagnetic field decomposition on the source and receiver side gives data equivalent to data from a hypothetical survey with the water-layer absent, with idealized single component transverse electric and transverse magnetic mode sources and idealized single component transverse electric and transverse magnetic mode receivers.When the subsurface is isotropic or transverse isotropic and horizontally layered, the Lorentz deconvolution decouples into pure transverse electric and transverse magnetic mode data processing problems, where a scalar field formulation of the multidimensional Lorentz deconvolution is sufficient. In this case single-component source data are sufficient to eliminate the water-layer effect.We demonstrate the Lorentz deconvolution by using numerically modeled data over a simple isotropic layered model illustrating controlled-source electromagnetic hydrocarbon exploration. In shallow water there is a decrease in controlled-source electromagnetic sensitivity to thin resistors at depth. The Lorentz deconvolution scheme is *
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