Tutorial: Choosing the averaging interval when calculating primary reflection coefficients from well logs
A. T. Walden and J. W. J. Hosken
Journal name: Geophysical Prospecting
Issue: Vol 36, No 8, November 1988 pp. 799 - 824
Info: Article, PDF ( 1.14Mb )
Most seismic data is processed using a sample interval of 4 ms two-way time (twt). The study of the statistical properties of primary reflection coefficients showed that the power spectrum of primaries can change noticeably when the logs are averaged over blocks of 0.5, 1 and 2 ms twt (block-averaging). What is a suitable block-averaging interval for producing broadband synthetics, and in particular how should the power spectrum of primaries be constructed when it is to be used to correct 4 ms sampled deconvolved seismic data for the effects of coloured primary reflectivity?
In this paper we show that for a typical sonic log, a block-averaging interval of 1 ms twt should satisfy some important requirements. Firstly, it is demonstrated that if the reflection coefficients in an interval are not too large the effect of all the reflection impulses can be represented by another much sparser set at intervals of Δt twt, The coefficient amplitudes are given by the differences in the logarithmic acoustic impedances, thus justifying block-averaging. However, a condition for this to hold up to the aliasing (Nyquist) frequency is that Δt takes a maximum value of about 1 ms twt. Secondly, an event on a log should be represented in the seismic data. For this the acoustic impedance contrast must have sufficient lateral extent or continuity. By making some tentative suggestions on the relation between continuity and bed-thickness, a bed-thickness requirement of 0.15 m or more is obtained. Combining this requirement with the maximum number of beds allowable in an interval in order that multiple reflections do not contribute significantly to the reflections in the interval, again suggests a value of about 1 ms for the block-averaging interval.
With this in mind an experiment was performed on three sonic logs. The logs were block-averaged at 1 ms, and primary reflection coefficients calculated. These primaries were then anti-alias filtered and resampled to get a series of primaries at 4 ms, followed by ARMA spectrum fitting. The same logs were also block-averaged at 4 ms directly and primaries computed, followed by ARMA spectrum fitting. In all three cases the first approach gave the ARM A model spectrum with greatest dynamic range, which strongly suggests that direct 4 ms block-averaging introduces significant aliased energy into low frequencies of the primaries spectrum.
The conclusion is that routine computation of broadband synthetics (primaries only or primaries plus multiples) should be carried out using a block-averaging interval of 1 ms twt, followed by anti-alias filtering and thinning to the desired final sample interval. In theory it would be advantageous to go to even finer intervals-say 0.5 ms-but in practice at this level the averaging of slowness imposed by the somic logging tool appears to attenuate high-wave number fluctuations, i.e. it interferes with the‘real’data. The 1ms choice is thus a reasonable compromise which will help minimize non-trivial aliasing effects and should give better matches to the seismic data.