In finite-difference methods a seismic source can be implemented using either initial wavefield values or body forces. However, body forces can only be specified at finite-difference nodes, and, if using initial values, a source cannot be located close to a reflecting boundary or interface in the model. Hence, difficulties can exist with these schemes when the region surrounding a source is heterogeneous or when a source either is positioned between nodes or is arbitrarily close to a free surface.A completely general solution to these problems can be obtained by using Kaiser windowed sinc functions to define a small region around the true source location that contains several nodal body forces. Both monopole and dipole point sources can be defined, enabling many source types to be implemented in either acoustic or elastic media. Such a function can also be used to arbitrarily locate receivers. If the number of finite-difference nodes per wavelength is four or more (and with a source region half-width of only four nodes) this scheme results in insignificant phase errors and in amplitude errors of no more than 0.1%. Numerical examples for sources located less than one node from either a free surface or an image source demonstrate that the scheme can be used successfully for any surface-source or multisource configuration.

Prestack seismic reflection data contain amplitudes, traveltimes, and moveout information; waveform inversion of such data has the potential to estimate attenuation levels, reflector depths and geometry, and background velocities. However, when inverting reflection data, strong nonlinearities can cause reflectors to be incorrectly imaged and can prevent background velocities from being updated. To successfully recover background velocities, previous authors have resorted to nonlinear, global search inversion techniques.We propose a two-step inversion procedure using local descent methods in which we perform alternate inversions for the reflectors and the background velocities. For our reflector inversion we exploit the efficiency of the back-propagation method when inverting for a large parameter set. For our background velocity inversion we use Newton inverse methods. During the background velocity inversions it is crucial to adaptively depth-stretch the model to preserve the vertical traveltimes. This reduces nonlinearity by largely decoupling the effects of the background velocities and reflectors on the data. Nonlinearity is further reduced by choosing to invert for slownesses and by inverting for a sparse parameter set which is partially defined using geological reflector picks.Applying our approach to shallow seismic data from the North Sea collected over a gas-sand deposit, we demonstrate that the proposed method is able to estimate both the geometry and internal velocity of a significant velocity structure not present in the initial model. Over successive iterations, the use of adaptive depth stretching corrects the pull-down of the base of the gas sand. Vertical background velocity gradients are also resolved. For an insignificant extra cost the acoustic attenuation parameter Q is included in the inversion scheme. The final attenuation tomogram contains realistic values of Q for the expected lithologies and for the effect of partial fluid saturation associated with a shallow bright spot. The attenuation image may also indicate the presence of fracturing.

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