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Geophysical Prospecting: On the form of the space‐time domain reflection function for a simple flat‐lying interface

A description is given of the function R(x, t) forming the kernel of the space‐time domain reflection operator for scalar waves illuminating a single interface in two space dimensions. This reflection function is closely related to migration common‐image gathers and it can be thought of as the set of all plane‐wave reflection coefficients after Fourier synthesis over wavenumber, or slowness, and frequency.R(x, t) is a generalized function with a complicated form at the origin, or around it when smoothing is applied. Much can be inferred from knowledge of the plane‐wave coefficients themselves and a simple scalar‐ or acoustic‐wave case without density variation displays the essential features. The theoretical predictions are visualized by smoothed numerical examples obtained using a discrete Fourier transform method. The space‐time form of the reflection function for a low‐contrast boundary is also considered, as is, briefly, the transmission function.Among the characteristics of R(x, t) are that it is essentially V shaped in the (x, t) plane, with linear arms given by the wave cones for the two media involved. These arms carry  signals, where H(t) is the Heaviside step function. The time dependence along x = 0 is also relatively simple, being like the delta derivative δ′(t). At or around the origin R(x, t) is intricate due to the fine balancing of its plane‐wave components. This intricacy makes the corresponding reflection operator capable of analysing an arbitrary incident wave to give the reflected wave, including such effects as total reflection and head waves and the reflection of short horizontal wavelength evanescent waves due to a point source very close to the interface.The results provide a baseline for understanding common‐image gathers used in migration, including those associated with reverse‐time migration and full‐waveform inversion. Clarity about the space‐time form of the reflection operator will hopefully advance the goal of unifying imaging, amplitude‐versus‐offset analysis and waveform inversion. Application to the time‐domain numerical modelling of waves is also conceivable.

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