Synthetic Seismogram

From the sonic log, assuming constant density, normal incidence, no multiples or reverberations and no absorption, a synthetic seismic trace was created. The reflection coefficients for the trace were simply computed with the recursion:  

$$RC_i=\frac{V_i\rho_i-V_{i-1}\rho_{i-1}}{V_i\rho_i+V_{i-1}\rho_{i-1}}$$ (1)

where RC_i is the reflection coefficient at the ith interface and V and $\rho$ represent velocity and density respectively. The density was assumed to be constant. The depth-to-time conversion was simply done by integrating the vertical travel time  

$$t_i=\sum_{j=1}^{i-1} \frac{2*\Delta z_j}{V_j}$$ (2)

where $\Delta z_j$ represents the thickness of the j-th layer. In this case the layer thickness was kept constant at 10 feet.

Once converted to time, the reflectivity series was interpolated to a constant time-sampling interval using an 8-point sinc interpolator and convolved with a Ricker wavelet of 60 Hz peak frequency. The right panel of Figure 1 shows the resulting synthetic trace with time zero corresponding to the shallowest depth in the log segment (3600 feet).

 

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Figure 1 Left panel: Sub-sampled sonic log used to generate the synthetic trace on the right panel. The zero of the seismic trace corresponds to the initial depth of the log.