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 Volume 38, Issue 4, 1990
Geophysical Prospecting  Volume 38, Issue 4, 1990
Volume 38, Issue 4, 1990


INVERSION OF REFRACTED ARRIVALS: A FEW PROBLEMS^{1}
By LUIGI ZANZIAbstractThe aim of refracted arrivals inversion is the computation of near‐surface information, i.e. first‐layer thicknesses and refractor velocities, in order to estimate the initial static corrections for the seismic data. The present trend is moving towards totally automatic inversion techniques, which start by picking the first breaks and end by aligning the seismic traces at the datum plane.
Accuracy and computational time savings are necessary requirements. These are not straightforward, because accuracy means noise immunity, which implies the processing of large amounts of data to take advantage of redundancy; moreover, owing to the non‐linearity of the problem, accuracy also means high‐order modelling and, as a consequence, complex algorithms for making the inversion.
The available methods are considered here with respect to the expected accuracy, i.e. to the model they assume. It is shown that the inversion of the refracted arrivals with a linear model leads to an ill‐conditioned problem with the result that complete separation between the weathering thickness and the refractor velocity is not possible. This ambiguity is carefully analysed both in the spatial domain and in the wavenumber domain. An error analysis is then conducted with respect to the models and to the survey configurations that are used.
Tests on synthetic data sets validate the theories and also give an idea of the magnitude of the error. This is largely dependent on the structure; here quantitative analysis is extended up to second derivative effects, whereas up to now seismic literature has only dealt with first derivatives. The topographical conditions which render the traditional techniques incorrect are investigated and predicted by the error equations.
Improved solutions, based on more accurate models, are then considered: the advantages of the Generalized Reciprocal Method are demonstrated by applying the results of the error analysis to it, and the accuracy of the non‐linear methods is discussed with respect to the interpolation technique which they adopt. Finally, a two‐step procedure, consisting of a linear model inversion followed by a local non‐linear correction, is suggested as a good compromise between accuracy and computational speed.



WEIGHTED STACKING OF SEISMIC DATA USING AMPLITUDE‐DECAY RATES AND NOISE AMPLITUDES^{1}
Authors RICHARD G. ANDERSON and GEORGE A. McMECHANAbstractThe signal‐to‐noise (S/N) ratio of seismic reflection data can be significantly enhanced by stacking. However, stacking using the arithmetic mean (straight stacking) does not maximize the S/N ratio of the stack if there are trace‐to‐trace variations in the S/N ratio. In this case, the S/N ratio of the stack is maximized by weighting each trace by its signal amplitude divided by its noise power, provided the noise is stationary. We estimate these optimum weights using two criteria: the amplitude‐decay rate and the measured noise amplitude for each trace. The amplitude‐decay rates are measured relative to the median amplitude‐decay rate as a function of midpoint and offset. The noise amplitudes are measured using the data before the first seismic arrivals or at late record times. The optimum stacking weights are estimated from these two quantities using an empirical equation.
Tests with synthetic data show that, even after noisy‐trace editing, the S/N ratio of the weighted stack can be more than 10 dB greater than the S/N ratio of the straight stack, but only a few decibels more than the S/N ratio of the trace equalized stack. When the S/N ratio is close to 0 dB, a difference of 4 dB is clearly visible to the eye, but a difference of 1 dB or less is not visible. In many cases the S/N ratio of the trace‐equalized stack is only a few decibels less than that of the optimum stack, so there is little to be gained from weighted stacking. However, when noisy‐trace editing is omitted, the S/N ratio of the weighted stack can be more than 10 dB greater than that of the trace‐equalized stack. Tests using field data show that the results from straight stacking, trace‐equalized stacking, and weighted stacking are often indistinguishable, but weighted stacking can yield slight improvements on isolated portions of the data.



Lp‐NORM DECONVOLUTION^{1}
Authors H. W. J. DEBEYE and P. VAN RIELAbstractThe purpose of deconvolution is to retrieve the reflectivity from seismic data. To do this requires an estimate of the seismic wavelet, which in some techniques is estimated simultaneously with the reflectivity, and in others is assumed known. The most popular deconvolution technique is inverse filtering. It has the property that the deconvolved reflectivity is band‐limited. Band‐limitation implies that reflectors are not sharply resolved, which can lead to serious interpretation problems in detailed delineation.
To overcome the adverse effects of band‐limitation, various alternatives for inverse filtering have been proposed. One class of alternatives is Lp‐norm deconvolution, L1norm deconvolution being the best‐known of this class.
We show that for an exact convolutional forward model and statistically independent reflectivity and additive noise, the maximum likelihood estimate of the reflectivity can be obtained by Lp‐norm deconvolution for a range of multivariate probability density functions of the reflectivity and the noise. The L∞‐norm corresponds to a uniform distribution, the L2‐norm to a Gaussian distribution, the L1‐norm to an exponential distribution and the L0‐norm to a variable that is sparsely distributed. For instance, if we assume sparse and spiky reflectivity and Gaussian noise with zero mean, the Lp‐norm deconvolution problem is solved best by minimizing the L0‐norm of the reflectivity and the L2‐norm of the noise. However, the L0‐norm is difficult to implement in an algorithm. From a practical point of view, the frequency‐domain mixed‐norm method that minimizes the L1norm of the reflectivity and the L2‐norm of the noise is the best alternative.
Lp‐norm deconvolution can be stated in both time and frequency‐domain. We show that both approaches are only equivalent for the case when the noise is minimized with the L2‐norm.
Finally, some Lp‐norm deconvolution methods are compared on synthetic and field data. For the practical examples, the wide range of possible Lp‐norm deconvolution methods is narrowed down to three methods with p= 1 and/or 2. Given the assumptions of sparsely distributed reflectivity and Gaussian noise, we conclude that the mixed L1norm (reflectivity) L2‐norm (noise) performs best. However, the problems inherent to single‐trace deconvolution techniques, for example the problem of generating spurious events, remain. For practical application, a greater problem is that only the main, well‐separated events are properly resolved.



THE OPTIMUM GATE LENGTH FOR THE TIME‐VARYING DECONVOLUTION OPERATOR^{1}
Authors VIJAY P. DIMRI and KIRTI SRIVASTAVAAbstractThe time‐varying deconvolution operator designed by dividing time‐varying sequence has been extended to include an optimal division of the input data. A numerical example illustrates that the error energy is less in the case of optimally divided input in comparison with arbitrary division.



3D AND 2½ D MODELLING OF GRAVITY ANOMALIES WITH VARIABLE DENSITY CONTRAST^{1}
Authors D. BHASKARA RAO, M. J. PRAKASH and N. RAMESH BABUAbstractThe decrease of density contrast in sedimentary basins may be approximated by a quadratic function. A sedimentary basin may be viewed as a number of prisms placed in juxtaposition. Equations in closed form for the gravity anomalies of 3D and 2½ D prismatic models are derived. Approximate equations for these models are also derived for rapid calculations. Efficient methods are developed for anomaly calculation by an appropriate use of the exact and approximate equations, and hence, for 3D and 2½ D modelling. The depths to the basement are adjusted iteratively by comparing the calculated anomalies with the observed anomalies. These methods are applied for analysis of the residual anomaly map of the Los Angeles Basin, California.



A PHYSICAL SCALE MODEL STUDY OF THE COMPARATIVE PERFORMANCE OF TWO MODES OF OPERATION FOR FIXED‐LOOP TURAM‐TYPE EM SYSTEMS^{1}
Authors K. DUCKWORTH and C. CUMMINSAbstractThe results of a physical scale model study of the conventional mode of operation of fixed loop electromagnetic systems and an alternative mode called the tx‐parallel mode in which traverses are run parallel to the long axis of the rectangular transmitter loop are presented. The results show that over thick or dipping conductors, the tx‐parallel configuration provides coupling with the target which is comparable with that provided by the conventional configuration. In addition, the tx‐parallel configuration is shown to provide more consistent indications of the direction and magnitude of conductor dip. Over wide conductors, where separate conventional surveys are needed to define the opposite edges of such conductors, it is shown that only a single tx‐parallel survey is needed to locate both edges of the conductor.
The tx‐parallel results were found to allow better resolution of the individual anomalies due to closely spaced parallel conductors.
The tx‐parallel response of identical sheet conductors of opposite dip indicated that the response of the separate sheets could be recognized even when the two sheets were placed at zero separation. This was found to be due in part to spatial displacement of the individual current vortices within each conductor owing to their mutual repulsion.

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