Ask geoscientists, drillers, etc., and they will tell you, unsurprisingly, that the subsurface is modelled and drilled in Depth. However, whether the seismic is (two-way) Time or Depth indexed, seismic inversion products (impedances) are ubiquitously derived in Time, as convolution of an earth model with an appropriate wavelet (an essential step in any inversion) must be done in that domain where the wavelet can usually be assumed stationary. Put a different way, convolution is not easily or naturally represented in Depth, as the effective wavelet shortens or lengthens with varying subsurface velocity, one of the very quantities the seismic inversion attempts to determine. So in the case of Depth indexed seismic, first a Depth to Time conversion must take place. After the inversion is performed in Time, the Time indexed results are usually Time to Depth converted, for use in e.g. geomodelling workflows. Note that the various domain conversions are often concealed from the user. Whilst this approach is awkward (two domain conversions for Depth indexed seismic, a natural product of PSDM or FWI processing), it is so far acceptable for straightforward seismic inversions.

There are however a number of reasons why a new approach in Depth is required. Firstly, facies (or rock-type) based seismic inversion systems have become increasingly de rigeur (Kemper and Gunning, 2014), meaning that not only impedances are derived, but also facies. Whilst subsequent Time to Depth conversion of the impedances (continuous quantities) is feasible, such a domain conversion of discrete facies is not possible without strong aliasing effects. Secondly, for 4D inversions, the natural domain is Depth (we exclude in this discussion cases with significant compaction), as in Time the baseline and monitor surveys again do not align since production (and injection) will have altered the velocity field. Of course to date most 4D analyses are qualitative (e.g. inspection of the quadrature-phase of the seismic difference) but a good quality facies based 4D inversion in Depth should make the analysis more quantitative.

Recasting the convolution operation to Depth is nontrivial. Singh (2012) has attempted this using a stretching technique: for Depth indexed seismic, the Depth axis is stretched and squeezed so that the resultant velocity is constant, and wavelet convolution can then take place safely in this pseudo Depth domain. The convolutional results are then transformed back to the original Depth axis. This is essentially the same as performing a Depth to Time conversion and post inversion a Time to Depth conversion as described earlier. Another approach that gained traction over the last couple of years is the use of Point Spread Functions (PSFs) (Lecomte et. al., 2015). In practice, PSFs are often difficult to obtain, and the 3D character of the operator makes them CPU intensive for inversion schemes. When they are available, we prefer to use them in a lateral deconvolution preprocessing step prior to inversion (Zabihi Naeini, 2018).

In this paper we introduce a new, practical approach to directly obtaining Depth indexed seismic inversion products, both impedances and facies, independent of whether the seismic is Time or Depth indexed. The new method is a modification of the facies-based inversion system of Gunning and Sams (2018). The model is represented in Depth, so no lossy Time to Depth model remapping is required. The Depth model representation has the considerable benefit of allowing regular or irregular gridding, e.g. corner point grids, with stratigraphic alignment in Depth, which marries well with the discrete facies model.