The ability of marine vibrators to accurately control the frequency and phase of the emitted signal offers new and interesting possibilities. In terms of deblending, one could, for example, imagine having simultaneously operating vibrators in narrow non-overlapping frequency bands. Deblending, could then be done by applying a simple bandpass filter. In a sensitive survey area, one could imagine that vibrators omit the frequencies used by the local mammal population to communicate, thus reducing the overall environmental impact. In such cases, there is a need to interpolate or fill in the missing frequencies. In seismic processing, interpolating missing frequencies is a new problem, not directly related to the more well studied problem of interpolating missing spatial data. In this work, we present both classical signal processing methodology as well as CNN-based approaches for interpolation of missing frequency bands in seismic reflection data.@en

Waves in space-dependent and in time-dependent materials obey similar wave equations, with interchanged time- and space-coordinates. However, since the causality conditions are the same in both types of material (i.e., without interchangement of time- and space-coordinates), the solutions are dissimilar. We present a systematic treatment of wave propagation and scattering in 1D space-dependent and in 1D time-dependent materials. After formulating unified equations, we discuss Green's functions and simple wave field representations for both types of material. Next we discuss propagation invariants, i.e., quantities that are independent of the space coordinate in a space-dependent material (such as the net power-flux density) or of the time coordinate in a time-dependent material (such as the net field-momentum density). A discussion of general reciprocity theorems leads to the well-known source–receiver reciprocity relation for the Green's function of a space-dependent material and a new source–receiver reciprocity relation for the Green's function of a time-dependent material. A discussion of general wave field representations leads to the well-known expression for Green's function retrieval from the correlation of passive measurements in a space-dependent material and a new expression for Green's function retrieval in a time-dependent material. After an introduction of a matrix–vector wave equation, we discuss propagator matrices for both types of material. Since the initial condition for a propagator matrix in a time-dependent material follows from the boundary condition for a propagator matrix in a space-dependent material by interchanging the time- and space-coordinates, the propagator matrices for both types of material are interrelated in the same way. This also applies to representations and reciprocity theorems involving propagator matrices, and to Marchenko-type focusing functions.

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The ability to extract information from scattered waves is usually limited to singly scattered energy even if multiple scattering might occur in the medium. As a result, the information in arrival times of higher-order scattered events is underexplored. This information is extracted using fingerprinting theory. This theory has never previously been applied successfully to real measurements, particularly when the medium is dispersive. The theory is used to estimate the arrival times and scattering paths of multiply scattered waves in a thin sheet using an automated scheme in a dispersive medium by applying an additional dispersion compensation method. Estimated times and paths are compared with predictions based on a sequence of straight ray paths for each scattering event given the known scatterer locations. Additionally, numerical modelling is performed to verify the interpretations of the compensated data. Since the source also acts as a scatterer in these experiments, initially, the predictions and the numerical results did not conform to the experimental observations. By reformulating the theory and the processing scheme and adding a source scatterer in the modelling, it is shown that predictions of all observed scattering events are possible with both prediction methods, verifying that the methods are both effective and practically achievable. @en

Marchenko redatuming can retrieve the impulse response to a subsurface virtual source from the single-sided surface reflection data with limited knowledge of the medium. We illustrate the concepts and practical aspects of Marchenko redatuming on a simple 1D acoustic lossless medium in which the coupled Marchenko equations are exact. Defined in a truncated version of the actual medium, the Marchenko focusing functions focus the wavefields at the virtual source location and are responsible for the subsequent retrieval of the downgoing and upgoing components of the medium's impulse response. In real seismic exploration, where we have no access to the truncated medium, we solve the coupled Marchenko equations by iterative substitution, relying on the causality relations between the focusing functions and the desired Green's functions along with an initial estimate of the downgoing focusing function. We show that the amplitude accuracy of the initial focusing function influences that of the retrieved Green's functions. During each iteration, propagating an updated focusing function into the actual medium can be approximated by explicit convolution with the broadband reflection seismic data after appropriate processing, which acts as a proxy for the true medium's reflection response.

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