Abstract In this chapter, spectra of functions with a single variable will be studied, for example those that are calculated when seeking the frequency content of an individual seismic trace. These few pages should be regarded only as an introduction to spectra. The special properties of spectra encountered in seismic work will be described in the chapter on signal and noise. Furthermore, the theorems which are given in the present introduction will be supplemented by others which may be stated only after spectral distributions and convolution are described. At this time, a more systematic account of spectral properties will be given.

Abstract Until now, only functions of a single variable have been considered. It is clear that the physical reasons which prompted us to introduce the concept of the single-dimension spectrum are also valid for two, three, four… dimensions. Let us first look at the generalization in two dimensions. It is known that the spectrum of a linear source distribution G(x) is the representation of a flux g(α) in a plane passing through the line (L) in question (Figure 2) according to the formula If a plane distribution of sources is now considered, the flux that it generates is once again the sum of thecontributions of the source elments. For a point located a great distance in the direction defined by the angles α 1 and α 2 (in a rectangular coordinate system O x 1 x 2 x 3 )(Figure 13), the flux g (α 1 , α 2 ) is connected to the source distribution G (x 1 ,x 2 )i n the O x 1 x 2 plane by the relation: Here we have employed g (α 1 , α 2 ) and g(k l , k 2 ) indiscriminately in order to make the physical correspondence between G and g understood. Mathematically, the relation g↔G concerns g(k 1 , k 2 ) and G(x 1 , x 2 ). Nothing is easier than to form spectra, or at least the square of their modulus, by an optical method whihc has previously been advocated in the study of single-dimention spectra. It suffices that a monochromatic point source be situated at some distance and be viewed through a transparent film photo of a grating or a seismic section (with about twenty traces per millimeter). It will be adequate to adjust to infinity

Abstract The modulus M is equal to T │sinc (f T )│. To calculate the proposed integral with a view to attempting synthesis of the square wave, it is necessary to decompose the frequency domain into elemental domains where the sinc function retains the same sign. Furthermore we shall make use of the parity of M :

Abstract Information gathered in the course of seismic exploration and stored on magnetic tape is made up of a mixture of signal and noise in varying proportions. The aim of every geophysicist is to reconstitute records with an increased signal-to-noise ration to simply interpretation. During replay of the magnetic tape, selection is made of the useful signal from other less useful signals and from the noise. This selection is evaluated in the frequency domain. Recordings carried out in marine seismic work illustrate this phenomenon; in order for them to be usable, the spurious effect arising from reasonance of the water layer must be removed. For this purpose the geophusicist resorts to two main procedures at the analog playback center: electrical filtering in the frequency domain and delay-line filering in the time domain. Unlike the former which allow him only passbands, the delay-line filters provide broader possibilities. This kind of equipment can, in addition to the passbands, achieve band-reject filters of different wideth and slopes going from suppression of a fixed frequency, as for example the 60-hz industrial power, to attenuation of a wide frequency band as is the case in deconvolution procedures. Narrow rejection bands can still be regularly spaced over the spectrum allowing for removal of multiples that may exist in land or marine seismic.

Abstract An optical correlator was devised and perfected in 1962 at the French Petroleum Institute. Its use by geophysicists is so firmly established and it performs such importand daily service that it would actually be difficult to consider the processing of seismic information, particularly, marine work, without this device and its performance and then proceed to review its main applications. Linear filters have been used very extensively in geophysics for several years. Authocorrelations, crosscorrelations, convolutions, and deconvolutions 4 are daily operations in seisic data processing. Therefore the IFP uses a simple device which performs the first two operations in an inexpensive and rapid manner. This device is the optical correlator for seismic analysis (CTS 1). We shall consider the application of correlations only within the framework of reflection seismology, which is by far the most used geophysical method throughout the world. Beyound the previous oconsiderations, there is another reason which makes correlation indispensable to the geophysicist. These are the problems posed by marine seismic work. The search for petroleum is turning more and more towards exploration of the continental shelf, and in this ev\nvironment geophysics has a chosen place; surface geologic studies here are almost impossible here. A land crew produces only 3 or 4 km of profile per day, while at sea 60 or 80 km are realized during the same time. But if, we have just said, the implementation of shots and profiles is relatively easy, by contrast the water layer

Abstract Recordings made in land shooting, in the course of pertroleum exlporation by the seismic method, display the existence of vibration phenomena which takes place in three-dimensional space. The very manner of recording and display, made in the form of seismic sections, express these phenomena as thought they had taken place in the vertical plane containing the line connecting shotpoint and seismometers, thus restoring the diagram to two dimensions. Even with this simplification, the theoretical description of the phenomena remains extremely complex and has not yet received a simple general solutuon; up to now this inverese filtering has been obtained only in the case of a single-dimension display. After the outline of this solution, it will be seen how it facilitates presentation of linear and nonlinear filter forms whose application effectively enhance useful information and tend to discard progressively all that be described here that have been found useful without aiming to establish a complete catalog on what has been published on the subject. Likewise, the bibliography is not intended to be complete, but should provide a general insight into the problems posed by the inverse filter.

Abstract Record sections obtained from a marine seismic survey are sometimes rendered unusable by disturbances of periodic nature due to the water layer. The latter is, in fact, bounded by two interfaces with strong reflection coefficients, the air-water interface with coefficient –1, the water-substratum interface with a coefficient value that can reach 0.8. The seismic energy produced in this layer or arriving there from below is successively reflected by the two interfaces with a time interval equal to twice the thickness of the water layer traversed at the velocity of sound in water, namely 1500 m/sec. In deep water, Z > 30 m, this is expressed by multiples or reverberations whose separations in time represent the forward and return travel path in the water. This kind of section is shown in Figure 60a. The water depth is 45 m. To the left of SP 1706, the intensity of the reverberations completely obscures the reflecting horizons. In shallow water, Z ≪ 30 m, the reverberations being as close together as possible give rise to apparent sinusoidal phasing, a phenomenon often termed “singing”. Figure 61a is an illustration of this. The water depth is less than 10 m and the dominant frequency occurs toward 40 hz. First we will describe the method that we have applied in order to make these sections useful. Next we will set forth some results based on experience, limited to the analog filtering of reverberations due solely to the presence of a water layer above a sedimentary series capable of originating a complete train of multiple reflections. Figures 60b and 61b

Abstract The use of mechanical sources in seismic prospecting is not new, but vibrating sources only recently have become important in seismology, since its utilization is governed by the feasibility of processing long signals in a commercially profitable manner. Seismic prospecting with vibrators is attractive from various aspects: flexibility of implementation, simple stacking, elimination of shotholes; no fracturing, hence the possibility of prospecting in a built up area; choice of suitable signals which allow only frequencies useful in seismic work to be transmitted in the ground; good quality of reflection character in favorable cases. On the other hand, it displays certain drawbacks with respect to conventional seismic work with explosives which should not be underestimated: the surface noise generated by the vibrators is often stronger than that given by buried shots. Hence the source and geophone array are more important than in conventional shooting; the lack of seismic quiet periods in industrial areas: the inability to employ a gain control important in land recording, because of the long signal utilization, the dynamic range is therefore limited in practice to that of the instrumentation without AGC or programmed gain. Hence the need to sacrifice shallow horizons if we wish to obtain deep horizons; rather weak output of vibrating plates, hence the need to sum a large number of vibrations to compensate for the weak seismic energy of each vibration; necessity to conduct rapid correlations in the field in order to evaluate the chosen signal and array; small dynamic range of the correlators. These last four points demand a field instrumentation

Abstract In this chapter, spectra of functions with a single variable will be studied, for example those that are calculated when seeking the frequency content of an individual seismic trace. These few pages should be regarded only as an introduction to spectra. The special properties of spectra encountered in seismic work will be described in the chapter on signal and noise. Furthermore, the theorems which are given in the present introduction will be supplemented by others which may be stated only after spectral distributions and convolution are described. At this time, a more systematic account of spectral properties will be given. For the purposeo of introductiona, an example will be chosen which will serve us later in the description of noise and in the explanation of seismifc filtering. The concepts of frequency and the representation of periodic phenomena by systems of rotating vectors will be assumed to be known. Let us supposeth at a large number of very closely spaced seismometer S are located on a straighlt line ( L ), in order to study the waves generate dy a sufficiently distant source with constant characteristic. The source may be an eccentric rotating device M , which excites the ground in sinusoidal fashion at each turn (Figure 1). Let us first assume that M is located at a very large distance from the profile ( L ) and that propagation takes place without distortion, the ground surface being flat and the subsurface homogeneous and isotropic. This case is similar to that in radio-astronom, M being a source in space and S the receiver of hertzian waves; it is also analogous to conventional astronomy with M as a bright star and S as one of many p oto-sensitive cells (Jennison, 1961). We can even imagine that M generates surface waves in a large expanse of water, the ideal case being that of a well-defined swell ( M is then at infinity) observed from the line ( L ) far from the shore in a deep sea without wind or currents. In a similar situation, the waves received at ( L ) are practically plane and their wavefronts form a constant angle α with ( L ). A given wavefront that arrives at ( L ) with the velocity V , which is a function of the medium, will traverse the line of seismometer ( L ), with the apparent velocity. Since the excitation produced by the source is sinusoidal, we write it as being the angular frequency. x being the distance on ( L ) measured from an origin, S 1 for example.