Line-source simulation for shallow-seismic data. Part 1: theoretical background

Equivalent line-source seismograms can be obtained from shallow seismic field recordings by (1) convolving the waveforms with sqrt(1/t), (2) applying a sqrt(1/t) time-domain taper, where t is traveltime and (3) scaling the waveform with r_offset*sqrt(2), where roffset is source-to-receiver offset. We require such a procedure when applying algorithms of 2-D adjoint full-waveform inversion (FWI) to shallow-seismic data. Although derived from solutions for acoustic waves in homogeneous full space this simple procedure performs surprisingly well when applied to vertical and radial components of shallow-seismic recordings from hammer blows or explosions. This is the case even in the near field of the force, although the procedure is derived from a far-field approximation. Similar approximative procedures recommended in literature are optimized for reflected waves and do not convert the amplitudes of all shallow seismic wavefield constituents equally well. We demonstrate the suitability of the proposed method for the viscoelastic case by numerical examples as well as analytical considerations. In contrast to the proposed single-trace procedure, integral-transform approaches are exact for all viscoelastic wavefield constituents of the near- and far-field in unknown 1-D-heterogeneous structure. ... mehrUnfortunately, integral-transform approaches suffer from artefacts in applications to data sampled on 2-D structures. Here, we use the Fourier–Bessel integral transformation as a reference in 1-D heterogeneous structures. We unroll the wave-theoretical background of both approaches in order to demonstrate, why the simplistic single-trace simulation approach derived from the asymptotic acoustic case can perform so well when applied to the shallow elastic wavefield. Further we give recommendations for practical implementation and application to field data of the proposed simulation method and compare to the results of alternative conversion rules. The performance of the conversion procedure to data recorded on 2-D heterogeneous structures is presented in a companion study by FWI reconstruction tests.