ALBERT

Notes: The geophysical data processor today has on offer a great variety of tools for the inversion of seismic reflection data to estimate geological structure. The major subset of these comprises migration procedures, which span a wide range of sophistication and cost in terms both of computation time and manual effort on the part of interpreters and processing staff. The choice of an over-powerful process can be very wasteful, but on the other hand too naive a migration procedure can lead to wrong interpretations which are much more costly still.Complete inversion procedures which aim to delineate all changes in rock densities and elastic properties in the subsurface are still in the imaginative stages of research. Not even the most sophisticated migration procedure in current use with real data, however, provides a complete inversion, but all depend in some measure on prior knowledge of the velocity structure of the section of the earth traversed by the seismic energy. Such knowledge may be very approximate at first, but each inversion should, through the skill of the interpreter, allow him to revise his velocity model and, up to some limit imposed by the quality and ambiguity of the original data, to improve the next inversion. Paradoxically, he can often be helped by using forward modeling procedures to check the implications of his ideas in the data domain, both in deciding how to update the velocity model and in selecting the most appropriate migration process to use next.We review here the currently available toolkit of migration and modeling processes and make suggestions as to how each process can fit into a learning strategy which can improve the interpretation as economically as possible and in as many iterative steps as the complexity of the earth's velocity structure makes necessary. An example is shown of the strategy being used in a complex overthrust region.The authors wish to thank the Chairman and Board of Directors of BP Exploration Co. for permission to publish this paper, and also make acknowledgment to our colleagues whose labours in research and development have made available to our use many of the essential tools required to implement the strategies we describe.

Notes: This paper investigates the form of the nonwhiteness found in the reflection coefficients from a wide variety of rock sequences around the world. In all but one case densities are taken as constant due to the paucity of suitable density data. The reflection sequences are pseudo-white only above a corner frequency, below which their power spectrum falls away according to a power law fβ, where β is between 0.5 and 1.5. This spectrum can be adequately modelled in practice very simply with an ARMA (1, 1) process which acts on a white innovation sequence. The corollary of this is that before wavelet estimation methods are applied (all of which-except those based on synthetic seismograms—presuppose white reflection sequences) or deconvolution filters estimated, seismic traces should be filtered with the inverse of this process.Interestingly, the estimated ARMA processes group themselves into two clearly differentiated categories, having very different indices of predictability (or, strictly, indices of linear determinism). The two categories apparently correlate precisely with two kinds of sedimentation: one which consists largely of sequences of rocks with repeating properties, called “repetitive” in this paper but perhaps loosely describable as “cyclic”, and the other which is randomly bedded with no apparent pattern of components. The former has indices of predictability which are two to four times as great as those of the latter. Another, probably related, property is that β for the repetitive sequences tends to be greater than that for non-repetitive rock columns.The observed power spectra are shown to be consistent with a simple model for the logarithm of acoustic impedance consisting of a mixture of processes where the distribution of (time) scale parameters is reciprocal.Detailed effects of block-averaging and sampling the logs are shown to depend on the type of sequence under examination.

Notes: One of the important properties of a series of primary reflection coefficients is its amplitude distribution. This paper examines the amplitude distribution of primary reflection coefficients generated from a number of block-averaged well logs with block thicknesses corresponding to 1 ms (two-way time). The distribution is always essentially symmetric, but has a sharper central peak and larger tails than a Gaussian distribution. Thus any attempt to estimate phase using the bi-spectrum (third-order spectrum) is unlikely to be successful, since the third-order moment is almost identically zero. Complicated tri-spectrum (fourth-order spectrum) calculations are thus required. Minimum Entropy Deconvolution (MED) schemes should be able to exploit this form of non-Gaussianity. However, both these methods assume a white reflectivity sequence; they would therefore mix up the contributions to the trace's spectral shape that are due to the wavelet and those that are due to non-white reflectivity unless corrections are introduced.A mixture of two Laplace distributions provides a good fit to the empirical amplitude distributions. Such a mixture distribution fits nicely with sedimentological observations, namely that clear distinctions can be made between sedimentary beds and lithological units that comprise one or more such beds with the same basic lithology, and that lithological units can be expected to display larger reflection coefficients at their boundaries than sedimentary beds. The geological processes that engender major lithological changes are not the same as those for truncation of bedding. Analyses of sub-sequences of the reflection series are seen to support this idea. The variation of the mixing proportion parameter allows for scale and shape changes in different segments of the series, and hence provides a more flexible description of the series than the generalized Gaussian distribution which is shown to also provide a good fit to the series.Both the mixture of two Laplace distributions and the generalized Gaussian distribution can be expressed as scale mixtures of the ordinary Gaussian distribution. This result provides a link with the ordinary Gaussian distribution which might have been expected to be the distribution of a natural series such as reflection coefficients. It is also important in the consideration of the solution of MED-type methods. It is shown that real (coloured) primary reflection series do not seem to be obtainable as the deconvolution result from MED-type deconvolution schemes.

Notes: Electronic circuits are described which simulate the effects of galvanometers and geophones. By placing these circuits in the feedback loop of an operational amplifier it is possible to obtain transfer functions which are the inverse of those of the transducers. That is, they will remove the distortion on the signal which was produced by the geophone or the galvanometer.Both a geophone simulator and its inverse filter have been built and used (O'Brien, 1965). The simulator is accurate over at least seven octaves and the inverse filter restores the ground signal to what it would have been if the recording geophone had had a natural frequency √n+1 times that of the actual geophone, where A is the gain of the operational amplifier used in the circuit.

Notes: 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.