SOURCE ARRAY SCALING FOR WAVELET DECONVOLUTION*

A seismic source array is normally composed of elements spaced at distances less than a wavelength while the overall dimensions of the array are normally of the order of a wavelength. Consequently, unpredictable interaction effects occur between element and the shape of the far field wavelet, which is azimuth-dependent, can only be determined by measurements in the far field. Since such measurements are very often impossible to make, the shape of the wavelet—particularly its phase spectrum—is unknown. A theoretical design method for overcoming this problem is presented using two scaled arrays. The far field source wavelets from the source arrays have the same azimuth dependence at scaled frequencies, and the far field wavelets along any azimuth are related by a simple scaling law. Two independent seismograms are generated by the two scaled arrays for each pair of source-receiver locations, the source wavelets being related by the scaling law. The technique thus permits the far field waveform of an array to be determined in situations where it is impossible to measure it. Furthermore it permits the array design criteria to be changed: instead of sacrificing useful signal energy for the sake of the phase spectrum, the array may be designed to produce a wavelet with desired amplitude characteristics, without much regard for phase.     

ODD-DEPTH STRUCTURES FOR DETERMINISTIC DECONVOLUTION AND SEISMOGRAM TESTING*

In odd-depth structure the two-way traveltime to each boundary is constrained to be an odd integer. The odd-depth property of a model is exposed to possible refutation under a seismogram test. Test function is a simple transformation of a synthetic seismogram. For an odd-depth model the test function has identically the value 1. The testability of a synthetic seismogram over an odd-depth structure provides a method of deterministic deconvolution. There is no need of specialized assumptions, like the minimum-phase property, about the source wavelet. The deconvolution may be performed in the absence of the early segment of a seismogram.     

VIBROSEIS DECONVOLUTION*

Vibroseis deconvolution can be performed either before or after correlation. As regards the deconvolution before correlation, the Vibroseis deconvolution operator can be described as convolution of a spike deconvolution operator with a minimum-phase filter operator with bandpass properties. As regards the deconvolution after correlation, the deconvolution operator can be shown to be the convolution of three operators: spike deconvolution operator and two-fold convolution with a minimum phase operator. Time-varying Vibroseis deconvolution can particularly well be described and performed after correlation.     

DYNAMIC PREDICTIVE DECONVOLUTION*

Dynamic predictive deconvolution makes use of an entire seismic trace including all primary and multiple reflections to yield an approximation to the subsurface structure. We consider plane-wave motion at normal incidence in an horizontally layered system sandwiched between the air and the basement rock. Energy degradation effects are neglected so that the layered system represents a lossless system in which energy is lost only by net transmission downward into the basement or net reflection upward into the air; there is no internal loss of energy by absorption within the layers. The layered system is frequency selective in that the energy from a surface input is divided between that energy which is accepted over time by net transmission downward into the basement and the remaining energy that is rejected over time by net reflection upward into the air. Thus the energy from a downgoing unit spike at the surface as input is divided between the wave transmitted by the layered system into the basement and the wave reflected by the layered system into the air. This reflected wave is the observed seismic trace resulting from the unit spike input. From surface measurements we can compute both the input energy spectrum, which by assumption is unity, and the reflection energy spectrum, which is the energy spectrum of the trace. But, by the conservation of energy, the input energy spectrum is equal to the sum of the reflection energy spectrum and the transmission energy spectrum. Thus we can compute the transmission energy spectrum as the difference of the input energy spectrum and the reflection energy spectrum. Furthermore, we know that the layered system acts as a pure feedback system in producing the transmitted wave, from which it follows that the transmitted wave is minimum-delay. Hence from the computed energy spectrum of the transmitted wave we can compute the prediction-error operator that contracts the transmitted wave to a spike. We also know that the layered system acts as a system with both a feedback component and a feed-forward component in producing the reflected wave, that is, the observed seismic trace. Moreover, this feedback component is identical to the pure feedback system that produces the transmitted wave. Thus, we can deconvolve the observed seismic trace by the prediction-error operator computed above; the result of the deconvolution is the wave-form due to the feedforward component alone. Now the feedforward component represents the wanted dynamic structure of the layered system whereas the feedback component represents the unwanted reverberatory effects of the layered system. Because this deconvolution process yields the wanted dynamic structure and destroys the unwanted reverberatory effects, we call the process dynamic predictive deconvolution. The resulting feedforward waveform in itself represents an approximation to the subsurface structure; a further decomposition yields the reflection coefficients of the interfaces separating the layers. In this work we do not make the assumption as is commonly done that the surface as a perfect reflector; that is, we do not assume that the surface reflection coefficient has magnitude unity.     

ZERO MEMORY NON-LINEAR DECONVOLUTION*

A type of iterative deconvolution that extracts the source waveform and reflectivity from a seismogram through the use of zero memory, non-linear estimators of reflection coefficient amplitnde is developed. Here, we present a theory for iterative deconvolution that is based upon the specification of a stochastic model describing reflectivity. The resulting parametric algorithm deconvolves the seismogram by forcing a filtered version of the seismogram to resemble an estimated reflection coefficient sequence. This latter time series is itself obtained from the filtered seismogram, and so a degree of iteration is required. Algorithms utilizing zero memory non-linearities (ZNLs) converge to a family of processes, which we call Bussgang, of which any colored Gaussian process and any independent process are members. The direction of convergence is controlled by the choice of ZNL used in the algorithm. Synthetic and real data show that, generally, five to ten iterations are required for acceptable deconvolutions.     

AUTOMATIC DECONVOLUTION OF GRAVIMETRIC ANOMALIES*

Existing techniques of deconvolution of gravity anomalies are principally based on upward and downward continuation of measured fields. It can be shown that a unique set of linear filters, depending only on geometrical parameters, relates density distribution at a given depth to gravity measured on the surface. A method to compute the filter coefficients is developed. Very accurate reconstitution of theoretical models of intricate shape, prove the validity of the linear relationship. One of these sets of linear filters is applied to a field case of underground quarries.     

SELF-MATCHING DECONVOLUTION IN THE FREQUENCY DOMAIN *

In a previous paper the author showed how, by computing an inverse filter in the frequency domain, an automatic compromise could be made between the conflicting requirements to spike a wavelet and to keep the attendant noise amplification within bounds. This paper extends the technique to take account of errors in the estimated shape of the wavelet defined to the deconvolution process. The drastic effects which such errors can have if they are ignored are demonstrated. A novel form of filter–called the “self-matching filter”–is defined which allows the user to limit not only the noise amplification but also the sensitivity of the filter to random uncertainties in the estimated wavelet. This is achieved by whitening the spectrum only within automatically selected pass bands whilst suppressing other noise-dominated or uncertainly defined frequency components. Conventional Wiener filtering is shown to be a special case of this more general filter, namely one in which the wavelet uncertainty is completely ignored. The type of phase spectrum which the output pulse should be designed to possess (e.g. zero phase or minimum phase) is briefly discussed.     

ADAPTIVE PREDICTIVE DECONVOLUTION OF SEISMIC DATA*

The Wiener prediction filter has been an effective tool for accomplishing dereverberation when the input data are stationary. For non-stationary data, however, the performance of the Wiener filter is often unsatisfactory. This is not surprising since it is derived under the stationarity assumption. Dereverberation of nonstationary seismic data is here accomplished with a difference equation model having time-varying coefficients. These time-varying coefficients are in turn expanded in terms of orthogonal functions. The kernels of these orthogonal functions are then determined according to the adaptive algorithm of Nagumo and Noda. It is demonstrated that the present adaptive predictive deconvolution method, which combines the time-varying difference equation model with the adaptive method of Nagumo and Noda, is a powerful tool for removing both the long- and short-period reverberations. Several examples using both synthetic and field data illustrate the application of adaptive predictive deconvolution. The results of applying the Wiener prediction filter and the adaptive predictive deconvolution on nonstationary data indicate that the adaptive method is much more effective in removing multiples. Furthermore, the criteria for selecting various input parameters are discussed. It has been found that the output trace from the adaptive predictive deconvolution is rather sensitive to some input parameters, and that the prediction distance is by far the most influential parameter.     

MIGRATION IN TERMS OF SPATIAL DECONVOLUTION*

The relationship between two finite-difference schemes (15° and 40°) and the Kirchhoff summation approach is discussed by using closed form solutions of Claerbout's approximate versions of the wave equation. Forward extrapolation is presented as a spatial convolution procedure for each frequency component. It is shown that downward extrapolation can be considered as a wavelet deconvolution procedure, the spatial wavelet being given by the wave theory. Using this concept, a three-dimensional model for seismic data is proposed. The advantages of downward extrapolation in the space-frequency domain are discussed. Finally, it is derived that spatial sampling imposes an upper limit on the aperture and a lower limit on the extrapolation step.     

A NEW APPROACH TO VIBROSEIS DECONVOLUTION*

A method is proposed to obviate the shortcomings of conventional deconvolution approaches applied to vibroseis data. The vibroseis wavelet reduces the time domain resolution of the earth's impulse response by restricting its passband. The spectrum of the wavelet is assumed to be a “low quefrency”phenomenon, and hence it can be estimated by low cut cepstral filtering. The wavelet's amplitude spectrum can then be removed by spectral division. By using an approach which is consistent with the principle of maximum entropy, the undetermined portions of the seismogram's Fourier transform can be filled in by autoregressive prediction. The process of initially deconvolving in a restricted passband reduces the enhancement of noise contaminated parts of the spectrum, and the spectral extension scheme increases the time domain resolution of the process.     

WAVE SCATTERING DECONVOLUTION BY SEISMIC INVERSION*

We propose a wave scattering approach to the problem of deconvolution by the inversion of the reflection seismogram. Rather than using the least-squares approach, we study the full wave solution of the one-dimensional wave equation for deconvolution. Randomness of the reflectivity is not a necessary assumption in this method. Both the reflectivity and the section multiple train can be predicted from the boundary data (the reflection seismogram). This is in contrast to the usual statistical approach in which reflectivity is unpredictable and random, and the section multiple train is the only predictable component of the seismogram. The proposed scattering approach also differs from Claerbout's method based on the Kunetz equation. The coupled first-order hyperbolic wave equations have been obtained from the equation of motion and the law of elasticity. These equations have been transformed in terms of characteristics. A finite-difference numerical scheme for the downward continuation of the free-surface reflection seismogram has been developed. The discrete causal solutions for forward and inverse problems have been obtained. The computer algorithm recursively solves for the pressure and particle velocity response and the impedance log. The method accomplishes deconvolution and impedance log reconstruction. We have tested the method by computer model experiments and obtained satisfactory results using noise-free synthetic data. Further study is recommended for the method's application to real data.     

MINIMUM ENTROPY DECONVOLUTION WITH AN EXPONENTIAL TRANSFORMATION*

The Minimum Entropy Deconvolution (MED) technique of Wiggins (1977) represents a breakthrough in deconvolution and will undoubtedly find wide application in many fields. MED does not require any phase assumptions about the disturbing function and seeks a deconvolved output which consists of the smallest number of large spikes consistent with the input data. The efficiency of MED is much improved when an exponential transformation is incorporated into the algorithm. This is particularly true when the input traces contain additive noise. In this case the noise suppression characteristics of MED are considerably enhanced by the transformation and the identification of smaller spikes is improved. This paper also presents a kurtosis criterion of simplicity rather than the varimax norm introduced by Wiggins. It appears that for a multiple trace input the kurtosis measure leads to improved results.     

PROTEE A SELF-ADAPTING DECONVOLUTION AND DERINGING PROGRAM*

Deconvolution and deringing are well known subjects and it is not necessary to state again their objectives nor the basical methods used to reach them. Let us just remember that, generally, among many others, the two following assumptions are made for simplification purposes:

• —for deconvolution, it is assumed that the recorded seismic signal is constant, meaning that its shape is the same all along the time interval during which the trace is to be deconvolved;
• —for de-ringing, it is assumed that the ringing period is constant and that the intensity of the ringing phenomenon is independant of the time.

With these two assumptions, a single constant operator can be applied for deconvolving, deringing or both. In most cases, the time variations of the signal or of the ringing are small enough and the error resulting of the application of a constant operator is acceptable. It results into a slight increase of the noise level or into a small residual ringing in the processed trace. When this noise or the residual ringing are too important, the assumption of a constant signal and ringing period must be rejected. This is the case that is examined here according to the following steps:

• —short definition of the problem;
• —fast evaluation of some possible solutions;
• —the selected solution: resulting approximations and how to obviate them, computing method and a remark about the operators;
• —theoretical example: the efficiency of the process used is evaluated on data in which the results aimed at are known; the influence of the selection of numerical values to be assigned to the parameters is examined;
• —real cases: comparison of results obtained with the Protee process and with more conventional processes assuming a time invariance or including a weighted composition of several conventional processes each with a different operator.

     

WAVELET DECONVOLUTION USING A SOURCE SCALING LAW*

We present a new method for the extraction and removal of the source wavelet from the reflection seismogram. In contrast to all other methods currently in use, this one does not demand that there be any mathematically convenient relationship between the phase spectrum of the source wavelet and the phase spectrum of the earth impulse response. Instead, it requires a fundamental change in the field technique such that two different seismograms are now generated from each source-receiver pair: the source and receiver locations stay the same, but the source used to generate one seismogram is a scaled version of the source used to generate the other. A scaling law provides the relationship between the two source signatures and permits the earth impulse response to be extracted from the seismograms without any of the usual assumptions about phase. We derive the scaling law for point sources in an homogeneous isotropic medium. Next, we describe a method for the solution of the set of three simultaneous equations and test it rigorously using a variety of synthetic data and two types of synthetic source waveform: damped sine waves and non-minimum-phase air gun waveforms. Finally we demonstrate that this method is stable in the presence of noise.     

CORRECTING FOR COLOURED PRIMARY REFLECTIVITY IN DECONVOLUTION1

Statistical deconvolution, as it is usually applied on a routine basis, designs an operator from the trace autocorrelation to compress the wavelet which is convolved with the reflectivity sequence. Under the assumption of a white reflectivity sequence (and a minimum-delay wavelet) this simple approach is valid. However, if the reflectivity is distinctly non-white, then the deconvolution will confuse the contributions to the trace spectral shape of the wavelet and reflectivity. Given logs from a nearby well, a simple two-parameter model may be used to describe the power spectral shape of the reflection coefficients derived from the broadband synthetic. This modelling is attractive in that structure in the smoothed spectrum which is consistent with random effects is not built into the model. The two parameters are used to compute simple inverse- and forward-correcting filters, which can be applied before and after the design and implementation of the standard predictive deconvolution operators. For whitening deconvolution, application of the inverse filter prior to deconvolution is unnecessary, provided the minimum-delay version of the forward filter is used. Application of the technique to seismic data shows the correction procedure to be fast and cheap and case histories display subtle, but important, differences between the conventionally deconvolved sections and those produced by incorporating the correction procedure into the processing sequence. It is concluded that, even with a moderate amount of non-whiteness, the corrected section can show appreciably better resolution than the conventionally processed section.     

古地磁数据可靠性的试用判据

古地磁数据的可靠性问题,迄今已成为古地磁学进一步发展的关键所在。早在1964年Irving首次提出了数据可靠性的最低判据。当时绝大部分数据为天然剩磁测试结果（至1963年所搜集的554个数据中,经磁清洗的仅占12％）,因此,该判据强调一致性及野外检验。随着磁清洗技术的不断完善和广泛应用(1964-1970年新增的812个数据     

古地磁数据可靠性的试用判据

古地磁数据的可靠性问题,迄今已成为古地磁学进一步发展的关键所在。早在1964年Irving首次提出了数据可靠性的最低判据。当时绝大部分数据为天然剩磁测试结果（至1963年所搜集的554个数据中,经磁清洗的仅占12％）,因此,该判据强调一致性及野外检验。随着磁清洗技术的不断完善和广泛应用(1964—1970年新增的812个数据     

地震资料自适应时变卡尔曼反褶积

应用卡尔曼滤波的一步预测方法,并根据卡尔曼滤波方程适用于时变系统的特点,给出了利用卡尔曼滤波进行地震资料自适应时变反褶积的方法,文中给出了理论和实际资料处理的例子.