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### NARROW

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Abstract “An elementary text and problem book containing 44 lessons in seismology arranged for selection or combination to cover the normal 36-week course, or for condensation into an 18-week course. The lessons begin without assuming more than secondary school mathematics. An elementary knowledge of calculus is desirable, though not required, for the last half of the book.”

Abstract Of fundamental importance in the discussion of seismic interpretation theory and technique is the concept of the emergence angle of a seismic wave. As a seismic wave emerges to the surface of the earth, its direction of approach is not necessarily vertical; that is to say, the wave front need not be horizontal. A moving wave front, at any particular instant is the totality of points, at that instant, which are in the same cycle and phase of cycle in the propagation. This implies that the travel-times from the source, wherever it is, to each of the points of the wave front are the same. For our purposes we may reduce the concept to a two-dimensional situation and look at a section of the wave fronts as they emerge to the surface, somewhat as indicated in Figure 1. The various positions of sections of a wave front, at times t 1 , t 2 , t 3 , t 4 , etc., as the wave travels toward the surface, are indicated by WW and the trace of the surface of the earth by SS. Closely allied to these wave fronts is the concept of wave paths. These wave paths are curves, or straight lines, which represent the direction of travel of the waves. Thus, these paths are curves which intersect the wave fronts at right angles. A series of wave fronts with the related wave paths is shown in Figure 2. In the discussion which follows, it is sometimes more convenient to speak of wave fronts and, at others, of wave paths.

Abstract The plot of travel-time of a seismic event against horizontal distance is called the time-distance curve of that event. It will be assumed that the concept of this type of plotting is understood; but if this is not the case, the reader is strongly advised to master it before going on. In fact, we strongly recommend that all matters in these lessons be the subjects of group discussion until the ideas are firmly in hand. Ex. 5: Assuming that the time-distance curves for the situations in Ex. 3 and Ex. 4 are straight lines and the earlier travel-times in both cases are each 1.035 sec, draw the associated time-distance curves. Before continuing, we must now introduce some definitions and ideas which can be found in any elementary book on the Calculus and which we must employ in what follows. (See Griffin, Introduction to Mathematical Analysis.)

Abstract In this lesson we shall discuss some of the basic ideas which form the structure of the theory of interpretation of reflection seismology. If the eye of an observer, E in Figure 9, looks into a mirror, MM , in order to see the reflection of a point object 0, he will see the reflection at the imaginary point 0' , which is called the image point of O. This image point 0' is obtained in this manner: From 0 drop a perpendicular OP to the plane of the mirror MM . Then, extend this perpendicular to 0' so that OP=PO' . Actually, although the observer at E looks towards this image point 0' , the light waves travel from 0 to the mirror MM and from there they are reflected to E . To obtain the path of the light wave from 0 to E , we join 0' to E by a straight line which will intersect MM in a point R . The point R is then the reflecting point and the light wave path is the path ORE .

Abstract We are now in a position to direct our attention specifically to seismology. Let SS again represent the trace of the plane surface of the earth. Until we are in a position to deal with low-velocity, uphole, elevation and other corrections, we shall assume ideal conditions, among which will be the supposition that the medium immediately below the surface is one for which the velocity is υ , a constant , in all directions. The shot-point, too, is assumed to be at the surface. Immediately succeeding the shot-point explosion, seismic waves are excited which, by virtue of our assumptions, spread out equally in all directions in the earth. Consider the successive positions of the foremost wave front. It has the form of an expanding hemisphere concentric with the shot-point, whose radius is increasing at a rate equal to the velocity v .

Abstract In this lesson we shall deal step by step with a specific numerical case, in order to understand better the more general results in later work. We shall consider that between the surface of the earth, SS , and a plane horizontal reflecting bed, RR , whose depth is h=6,OOO ft, the medium is homogeneous and isotropic , insofar as seismic propagation is concerned, and that the speed-or velocity-is v = 7,500 It/sec. The reader is requested to draw all the pertinent relations to scale and go through with the detailed developments which follow. The exercises will pertain to the figures and results obtained. In Figure 18, the situation described is drawn to scale and at-axis (timeaxis) is drawn through the shot-point O. The trace of the surface SS is used as an x-axis oriented positively to the right.

Abstract In the preceding lesson we paved the way for solving the general problem by a lengthy numerical example. We are now in a position to look into this general problem. Let the depth of the horizontal reflecting plane RR be h below the surface SS . If 0 designates, as usual, the position of the shot-point on SS , the image of 0 in RR is the point 0' , where 00' is perpendicular to RR and, in length, equal to 2h . Let P be the position of any receiving point, i.e., of a pickup, at a distance x from 0, so that OP=x.

Abstract The most universally accepted method of converting travel-times of reflected events to subsurface depths is that embodying the use of the so-called time-depth charts. After a set of reflection records have been studied for traveltimes and what we might term “correlatability,” and after corrections for weathering and low-velocity layer have been applied, the resulting travel-times of the reflections are interpreted in terms of depth by means of such charts. As a result of what we have learned in our preceding lessons, we are now in a position to discuss the simplest form of these time-depth charts, which is also the most useful and important. If we assume that the reflection has come from a horizontal bed at a depth h with a uniform velocity v, the travel-time t is given, as we have seen, by the relation in which x is the distance from the shot-point to the receiving point.

Abstract We continue with the assumptions that we have used heretofore concerning reflection data, namely, that the reflecting bed is a horizontal plane, that the medium intervening between that plane and the horizontal plane of the earth is homogeneous and isotropic to seismic wave propagation, and that all necessary corrections to the data have been applied. We thus have resultant travel-times for the reflected waves which depend only on the distance of the pickups from the shot-point (regardless of azimuth of the pickup points, since the reflecting plane is assumed parallel to the plane of the earth).

Abstract We propose to consider now the problem of reflections obtained from a single plane dipping reflector under ideal conditions, similar to those implied in all previous work.

Abstract Before proceeding to an analytic discussion of our problem, it might be well to discuss a numerical case. In Figure 31, we have represented a reflecting bed, R'R , dipping at an angle of 5° to the right and such that its depth below the shot-point 0 is 5,000 ft; that is, inthe figure, OT=5,000 ft.

Abstract At this point, it is probably advisable to study the problem before us from an analytic point of view. Accordingly, referring to Figure 32, we shall indicate the dipping reflecting interface by R'R whose dip is cP down to the right. The velocity of transmission of the medium between the surface S'S and R'R is indicated by 'II, and the depth to R'R measured vertically from a shot-point is h.

#### Lesson No. 12: Determination of Velocity and Interface from Reflection Data—General Case

Abstract It might already have occurred to some of the readers that the methods described in Lesson No.8 (pp. 31 et seq .) can readily be extended to cover the more general case of reflection data from a dipping interface. It is the purpose of this lesson so to extend those methods; and, as before, we shall do so by a numerical example. We might mention here, too, as we have mentioned before, that the method has its analytic analogue. The numerical method here described has been found to be the most practical for solving the problem with the type of data usually available. To be specific, then, suppose that we have a spread on one side of a shot-point o running from 2,000 ft to 3;000 ft and that the corrected travel-times to the first and last pickups are, respectively, 1.332 and 1.353 sec. Suppose, further, that on the other side of 0 and in line with this spread we have another spread from 2,000 ft to 3,00'0 ft and that the travel-times to the first and last pick-up are, respectively, 1.377 and 1.420 sec. The situation is indicated in Figure 33.

Abstract It has probably occurred to some of the readers that the basis of interpretation by means of the so-called “dip-shooting” method can be found in the last two lessons. It is therefore opportune at this stage to include some pages in which we shall attempt to outline the general ideas and, by a few exercises, to direct the reader's attention to the possibilities of the method. Suppose, then, that centering about a point P , at a distance x from a shotpoint 0, we have a spread of pickups, Δ x in overall length, in line with 0 and over which a reflected wave-front shows a difference in travel-time of Δ t (of Figure 37). If, as usual, we indicate the emergence angle of this wave-front at P by α , then we have, for practical purposes, the relation

Abstract The contents of this lesson will, for the most part, be immediate consequences of the last few lessons, insofar as computing techniques are involved. However, the lesson will furnish an opportunity for stating some broad and important considerations which, it is to be hoped, will be studied by the reader to the extent that a full understanding of the implications for interpretation is obtained.

Abstract At the end of the preceding lesson it was mentioned that we shall continue the subject under discussion by using Ex. 57 as a numerical case. The following facts can be ascertained for the solution of that problem. The travel-time of the reflection to PI is 1.778 sec, and to P 2 it is 1.592 sec. At PI we have Δ t /Δ x =0.062/1,000, and at P2 we have Δ t /Δ x =0.029/1,000. The corresponding angles of emergence are:

Abstract The preceding lessons have shown the very fundamental and important role played by the relationship existing between the slope of the time-distance curve (dt/dx) of a seismic “event” and the emergence angle of the corresponding wave. There is yet another concept of equal importance in the study of “geometrical acoustics,” to coin a phrase, which we shall now introduce. Consider a moving plane wave front which assumes the successive positions of the members of a family of parallel planes. Let its velocity of propagation in the medium be VI. Now suppose that this medium is bounded by a plane surface, called the interface, on the other side of which is a medium in which the velocity of propagation is V 2 , different from VI.

#### Lesson No. 17: Snell’s Law—Principle of Least Time

Abstract A fundamental result in the mathematical discussion of wave propagation is the concept known as Fermat*#x0027;s Principle . Stated for our purposes, it runs somewhat like this: The study of the propagation of waves may be reduced to the study of wave paths which are defined as the paths along which the travel-times are minimal. We proceed to expand on this subject. Consider a moving wave front. Corresponding to it is a family of wave paths (see Lesson No.1). Choose any two points lying on anyone of these wave paths. Of all possible paths joining those two points, the wave path is that for which the travel-time is least. In other words, the travel-time along any other path (within suitable limits) joining those two points would be greater than along the wave path .

Abstract Consider a shot-point 0 on the surface, SS' , of the ground under which we have a medium in which the velocity of propagation is VI. We shall assume this medium, or bed, to have a thickness of hI and to be separated from the next deeper medium, or bed, by a plane interface, II' . In this lower bed the velocity of propagation is V2 , which we assume to be greater than VI, (V2>VI)' (See Figure 52.) The shot at 0 generates a wave front which expands spherically. Consider the corresponding rays, or wave paths, associated with this wave front. One wave path moves along the surface SS' at a velocity VI. If detectors were placed along SS' , this wave, sometimes called “first-kick” wave, would arrive in a travel-time t proportional to the distance x from the shot-point; i.e., for the travel-time of this wave, we have