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APPLIED GEOPHYSICS U.S.S.R.

APPLIED GEOPHYSICS Ei ake

Edited by NICHOLAS RAST, B.Sc., Ph.D., F.G.S.

Liverpool University

PERGAMON PRESS eee THE MACMILLAN COMPANY NEW YORK

PERGAMON PRESS INC. 122 East 55th Street, New York 22, N.Y. 1404 New York Avenue N. W., Washingion 5, D. C.

PERGAMON PRESS LTD. - Headingion Hill Hall, Oxford 4 and 5 Fitzroy Square, London W. 1

PERGAMON PRESS S.A.R.L. 24 Rue des Ecoles, Paris

PERGAMON PRESS G.m.b.H. Kaiserstrasse 75, Frankfurt am Main

Copyright

© 1962

Pergamon Press Ltd.

Library of Congress Card Number 60-53385

Printed in Poland to the order of PWN Polish Scientific Publishers by Drukarnia Narodowa, Cracow

10.

ine

CONTENTS

aTUGrISe ORC WOLGN ee tls asc a a a Sa ee ee ee

Part I SEISMOLOGY

. Intensities of Refracted and Reflected Longitudinal Waves at

mncleson Incidence below Critical, Siegal eniduce, syed V. P. GORBATOVA

. Method and Techniques of Using Stereographic Projections for

Solving Spatial Problems in Geometrical Seismics .... . E. I. Gav’perin, G. A. Krasit’sucutkova, V. I. Mrronova and A. V. FROLOVA

mibiultiple mettected, Waves... ..:) Huu. age co iste suse er

S. D. SHusHAKOV

BH ACIeG se SEISMIC NVAVES). . ¢ ck c./sunee at. eet nee IE Sliuiee ahaa

T. I. OBLocINA Part II GRAVIMETRY

. The Influence of Disturbing Accelerations when Measuring the

Force of Gravity at Sea using a Static Gravimeter .... . K. Yr. VEsELov and V. L. PANTELEYEV

. Evaluating the Accuracy of a Gravimetric Survey, Selecting the

Rational Density of the Observation Network and Cross-sections of Isoanomialies of the Force of ‘Gravity . 2M. 60h: B. V. KOTLIAREVSKII

Part II] EvectricaAL SonpE MetHops

. Theoretical Bases of Electrical Probing with an Apparatus Immersed

MMM Lenihan tee iis! i a a eS» (ae E. I. TEREKHIN

. The Use of New Methods of Electrical Exploration in Siberia

A. M. ALreKserey, M. N. BerpicHEevskit and A. M. ZAGARMISTR

. The Method of Curved' Electrical Probes ..........

M. N. BERDICHEVSKII

The Use of the Loop Method (Spir) in Exploring Buried Structures

I. I. KRoLENKO

Allowance for the Influence of Vertical and Inclined Surfaces of Separation when Interpreting Electric Probings ..... .

V. I. Fomina

R192

ll

44,

15

99

123

139

14.

15).

16.

Lee

CONTENTS

Part IV Ort GEopuysics

. Some Problems of Gas Logging Estimation of Gas Saturation

of Rocks

L. A. GALKIN

Luminescence Logging

T. V. SHCHERBAKOVA

Optical Methods of Bore-Hole Investigation

T. V. SHCHERBAKOVA

Determining the Permeability of Oil-Bearing Strata from the Specific Resistance :

S. G. Komarov and Z. I. Keven

New Types of Well Resistivity-Meters

kK. A. Potyakov

The Use of Accelerators of Charged Particles in Investigating Bore-Holes by the Methods of Radioactive Logging

V. M. ZaporozHetz and E. M. FiLippov

AuTHOR INDEX Supyect INDEX

4.23 426

PUBLISHER’S NOTICE TO READERS ON THE SUPPLY OF AN ENGLISH TRANSLATION OF ANY RussIAN ARTICLE MENTIONED BIBLIOGRAPHICALLY OR REFERRED TO IN THIS PUBLICATION.

The Pergamon Insiitute has made arrangements with the Institute of Scientific Infor- mation of the U.S.5.R. Academy of Sciences whereby they can obtain rapidly a copy of any article originally published in the open literature of the U.S.S.R.

We are therefore in a position to supply readers with a translation (into English or any other language that may be needed) of any article referred to in this publication, at a reasonable price under the cost-sharing plan.

Readers wishing to avail themselves of this service should address their request to the Administrative Secretary, The Pergamon Institute, at either 122 East 55th Street, New York 22, N.Y., or Headington Hill Hall, Oxford.

EDITOR’S FOREWORD

SPECTACULAR successes achieved by the Soviet scientists in the field of applied physics have focused attention on the vigour of scientific research in the U.S.S.R. As a result a concerted attempt is being made to make the extensive Soviet scientific literature available to Western readers. Although, at present, several institutions are concerned with translations from Russian only rarely are aspects of applied, as distinct from pure, science given their rightful place. Thus, the attempts of the Pergamon Institute to redress this situation are especially welcome. It must be remembered that in the U.S.S.R. scientific workers often follow what can be called an American tradition in not separating sharply the fundamental research from its technological applications. As a consequence many Russian papers dealing with specific industrial problems contain much of general scientific interest. This is especially true with respect to geophysics. Every new method of geophysi- cal exploration is valuable since it provides a new possibility of inspecting the unseen parts of the Earth. In any case in a new science practically every investigation is of some significance if only because it adds to the relatively meagre store of factual data.

In the U.S.S.R. the methods of geophysical research have been extensively applied not only in an effort to find useful minerals, but also in order to accumulate information on the geological structure of that vast country. Furthermore, the accuracy and reproducibility of the geophysical methods has been widely checked with the aid of numerous bore-holes systematically located at critical points. As a result very notable advances have been made in developing the so-called electrical, seismic and gravimetric methods, while the existence of the numerous bore-holes has led to an extensive, application and improvement of the geophysical methods of logging. In the present compendium a selection of papers published in the volumes 18 and 20 of the Soviet journal “Applied Geophysics” are being presented to the Western scientists. The intention is to illustrate some of the achieve- ments of the Russian applied geophysicists by translating their recent publi- cations. Although a fairly wide range of topics is being covered there is a bias towards the application of geophysical methods to the search for oil. In this respect the editor, who was responsible for selecting the papers to be translated, followed the tendency discernible in the original journals. Nonetheless the orientation of many of the included articles on seismic, electrical, gravimetric methods and logging techniques is such that the

iy

8 Ep1tTor’s FOREWORD

volume should appeal not only to the oil geophysicists and geologists but to everyone interested in modern developments in geophysical methodology. Since the translation of Soviet scientific literature is as yet in a pioneering stage it is, perhaps, not inappropriate to add a few remarks of a purely linguis- tic nature. The relatively prolonged isolation of Russian scientists from their western colleagues has led to differences in terminology. For instance, the Russian term equatorialnyi sond does not mean an equatorial sonde, but probes with a quadrilateral arrangement, while the word podniatie in various contexts implies an elevation, a culmination or an upfold. In the present volume, where necessary, the Russian usage is indicated and it is hoped that such editorial remarks will be of use to the translators of scien- tific Russian. Responsibility for these remarks rests with the editor, but in certain instances Professor R. M. Shackleton and Dr. C. D. V. Wilson of the University of Liverpool were consulted and made suggestions, for which

the editor wishes to express his gratitude. NicHOLAS Rast

PART I. SEISMOLOGY

or “Ay tel hih aay Me eee Me eine ty Pa i NN ag

CHAPTER 1]

INTENSITIES OF REFRACTED AND REFLECTED LONGITUDINAL WAVES AT ANGLES OF INCIDENCE BELOW CRITICAL

V. P. GorRBATOVA

THE dynamic properties of waves can be effectively utilized in interpreting seismic prospecting data, since these properties, in conjunction with the velocity components enable us to recognize the nature of any particular wave recorded on the seismogram.

The solution of problems connected with the dynamic propagation of waves presents difficulties which are well known. While part of the work done in this field by PETRASHEN’ and the team of mathematicians headed by him has already been published, the theory which we have in mind has been fully worked out only for ideally elastic horizontally laminated media.

Each of the layers is presumed to be sufficiently “‘dense’’, that is to say the travel time of a disturbance in the layer is substantially longer than the duration of the pulse transmitted. The velocity of propagation of longi- tudinal and transverse waves, however, as well as the densities, is constant inside the layer and assume new values on the boundaries of the layers.

Quantitative comparisons made up to date have not revealed any sharp discrepancy between theory and experiment. The theoretically discovered qualitative laws also show good agreement with seismic prospecting practice. We suggest that there would be undoubted advantage in introducing the theory, in the form in which it has been worked out to date, into the inter- pretation of field data.

A method for calculating the intensities and shapes of seismic traces for different waves propagated in media with plane-parallel boundaries has been worked out in detail in the Leningrad Section of the Institute of Mathematics (Academy of Sciences of the U.S.S.R.) @%). The Section has also compiled tables for fairly accurate calculations.

In this paper we offer a number of simplified methods for determining the intensities of purely longitudinal waves (in the media referred to) and discuss how the different parameters of the medium affect their frequency rate.

11

12 V. P. GorRBATOVA

In the main these simplifications mean that the frequency rates of the waves under consideration are determined not by accurate tables but simply by a small number of typical graphs, which we shall give later. Further, the assumption has been made that layers possessing a higher longitudinal velocity have also a higher ratio of transverse velocity to longitudinal veloc- ity and a greater density. This would appear to be true for most real media.

The elastic properties of two neighbouring media (the ith and the 7+1th) are characterized by the following parameters: «—the ratio of the lower longitudinal velocity to the higher; 7—the ratio of the transverse velocity to the longitudinal velocity in the layer where v, is the smaller; A —the ratio of the transverse velocity to the longitudinal velocity in a layer where v, is the greater; o—the ratio of the lower density to the higher.

Tf vj, p < Vi+1, p,

Vi, p Vi, s Vi+1, s i e=——§ , y=, A= Ge iy

Vi+1, p Vi, p Vit+1, p Qi+1 If vi, p > +1, p,

Uv; 1 V; il, UV; . 1 ES Doky NERDS ee eae Nig eel

Vi, p Vi+1, p Vi, p Ci That is, y <A, o <1,0, «<1, on all occasions.

The parameters a, y, A and o of adjacent layers, on whose boundaries. refraction occurs, are chosen within the following limits:

0:3 <4 09) 03 <7 < 016; 04 00, 05 <6 0

But the boundaries of adjacent layers characterized by the parameters. y = 0.3, J = 0.6 are excluded. For the ratio of transverse velocities in the boundary media therefore we shall always assume: 0.3<0<0.9,

| Vi, s : 5 Oh oe Hen, js

where 6= hy OA, w 2 | Vi, s Table 1 shows values for the parameters of adjacent layers on whose bound- ary reflection occurs if the reflection takes place from the layer in which the travel velocity of longitudinal waves is higher than in the layer through which the wave has passed. If the reflection occurs from the layer with the lower group velocity the values for the parameters will be found. in Table 2.

13

REFRACTED AND REFLECTED LONGITUDINAL WAVES

[ wavy],

01-20 01-20 0'T-2°0 OT OT OT 0'T-L'0 O'T=20 OTS 9 L99°0-V'0 Sé9'0-SLE"0 ji 9°0-9€°0 60-20 6'0-L'0 6°0-L'0 LiOTE:0 LAOS EAU) Ln0=e 0 % v0 sO 9°0 v0 s0 9°0 v0 s'0 = 9°0 oe £0 v0 s'0 v0 cb) 9°0 v0 s'0 9°0 4 @ WIavy, : 0O'T-L°0 0'T-2'0 01-20 01-20 01-20 OT OT OT 0'T-2°0 OT-Z0 O'T-L'0 o G0 >” SVO>% |L99°0-7'0 |SZ9°0-SZE'0} 9°0-9€'0 | 60-20 6°0-L'0 60-20 L'0-€'°0 L'0-€'0 L'0-€'0 2 0) 9°0 v0 $0 9°0 v0 s'0 9°0 v0 s'0 9°0 V £0 v0 €0 v0 0 v0 s0 9°0 v0 s'0 9°0 4

14. V. P. GoORBAOVA

The angles at which simple and multiple waves strike the reflecting boundaries are assumed to vary from zero to the value of a, where sin «, = 0.75 sin @,,. But in the case of reflection from media where the group velocity wave is travelling, the angles of incidence onto the reflecting bound- ary are treated as having a sine less than or equal to 0.6.

In this way then, we can investigate the intensity“ and shape of vertical displacements of points on a free surface caused by the arrival of purely longitudinal waves. The source of excitation chosen will be a shot fired im the upper layer of the medium.

In our analysis of the intensities of the head and reflected waves we shall always assume that the conditions of the excitation remain unchanged and that the velocity of propagation of the longitudinal waves and the density of the medium in the layer where the shot is fired are kept constant. The head waves are examined at a distance from their points of emergence, while the simple reflected waves will be examined at a distance from the points of emergence of the head waves formed at the same interfaces as the simple reflected wave under consideration.

By the expression “‘at a distance from the points of emergence”’ of a head wave, we mean. all points on the ground surface lying at a distance 7 from the sbot point, which satisfy the equality (1).

lr ra Ow ry eee eaten (1) cos? &; COS? K

Where ry is the abscissa of the point of origin of the head wave under consideration:

i, is the wave-length when the wave is travelling along the refracting boundary; A, = v,T. Here T is the predominant period of the vibration being propagated, and v, is the boundary velocity of the head wave under consideration;

h; is the thickness of the ith layer of the covering medium;

«, is the angle at which the wave under consideration (head or re- flected) passes through the ith layer;

h,q is the shot depth.

* By intensity we mean a quantity proportional to the amount of maximum displacement of points on the ground surface caused by the arrival of the wave under consideration. If the same conditions of excitation are maintained, the coefficient of proportionality is constant for all primary waves. It is also constant for all reflected waves, but is not the same as for primary waves.

REFRACTED AND REFLECTED LONGITUDINAL WAVES 15

Multiple reflected waves are examined at a distance from the points of emergence of like reflected-refracted waves. We evaluate the distance with h;tan a; cos? &; is each of the items repeated as many times as the multiple wave under consideration passes through the ith layer.

the help of inequalities similar to (1), where only in the term

METHODS FOR CALCULATING INTENSITIES Head Waves

The intensity of a head wave propagated in a medium (Fig. 1) from the n—Ith and nth boundaries of the layers is determined by the expression

n-2 (OF Eres) (Pi41 P3) [pp (p)

l Ast (Ay + 2ptg) Vr (779) where C, is some multiplier dependent on the properties of the zero layer* IPP (p) is the coefficient of the head wave formation at the boundary of the nlth and nth layers; r is the distance between shot point and the observation point along the

; (2)

Jhead a

free surface;

To is the abscissa of the point of origin of the head wave under consid- eration ;

Ag, and fg are the Lamé constants for the layer where the shot is fired;

(P; P;.3) (P41 P,) are the coefficients of refraction in the intensities when the wave is passing through the boundary between the ith and the 7+1th

layers from above to below and from below to above. n-2

In the product JI (P; P;,,) (P;,1P;) in formula (2) the coefficients of i=0

refraction of the wave under consideration at all the intermediate interfaces are taken into account.

The coefficients of refraction, the coefficients of head wave formation, and also the multiplier Cy depend both on the properties of the interfaces themselves and also on the angle at which the primary wave under consider- ation strikes these boundaries. Values for these coefficients are given in the detailed tables compiled by the team of mathematicians headed by Petrashen’ at the Leningrad Section of the Institute of Mathematics. Using these tables to determine the quantities mentioned we have compiled graphs

* The advisability of using this lies in the fact that if the calculations are suffciently approximate, we can assume that Cy = 2 when sin @ < 0.9. 0 0

16 V. P. GoRBATOVA

on which the sines of the angles of incidence with which we are concerned are plotted against the quantities which interest us or quantities which differ from these by having different multipliers.

Fic. 1.

The process of calculating the intensities of head waves can be sum-

marized as follows: 1. Using the law of wave refraction we determine the sines of the angles of incidence of the primary wave under consideration at all the interfaces:

Yj, Pp

sin ai = fs n,p

O O1 O2 O03 0-4 05 06 O7 0-8 0:9 LO

SIN Bp Fic, 2. Curves for the multiplier C).

2. From the ordinates of the curve (Fig. 2) which is characterized by the

V0, s

parameter yp = we take the value for the multiplier with Co at the

V0, 0 V0, ©

point x = sin & = Vn, 0

REFRACTED AND REFLECTED LONGITUDINAL WAVES Lig

3. To determine the value of the coefficient of refraction (P; P;,.,) (P:.1 P;) at the point « = sin @; (where «; is the lesser of the angles @; and @;,1) from the curve in Fig. 3 corresponding to the parameter which is the same as at the boundary of the ith and ilth layers, we take the value of the ordi-

Jigecn, Gao nate [(P; P,,,) (Pj, P)] rel and multiply Bi geel fo co which is the product of coefficients of refraction (P;P;,,) (P;,1 P;), of the plane waves when incidence is vertical.

SIN @, i O06 0 Bee? ! 0-4 7 RSs : 0:8 a=0'9 O9 | + hi iy 2 ae a- = O08 ae 7 a a- (at a=0°5 a) 0-7 a=0:3. 0-6

Fic.3. Curves for determining the coefficient of refraction [(P; P;+4) (P;+P)).

The product obtained thus gives the value for the coefficient of refraction (P; Piss) Pisa P)- 4. The coefficient I? (p) is taken at the point x = sin o,, = —2-hP : ony Pp from the curve corresponding to the parameters y, A and o which charac- terize the boundary of the nlth and nth layers (Fig. 4). Fig. 4 (@ and }) shows the JP? (p) curves for two values of the parameter o (dotted lines—

o = 0.7; thick lines—o = 1.0).

Applied geophysics 2

18 V. P. GoRBATOVA

5. The values obtained for Cy (P; P; 4) (P;41 P;), LP? (p) and 7p are inserted

in formula (2).

Reflected Simple and Multiple Waves

The intensity of a simple wave reflected from the boundary of the nlth and nth layers is determined from formula (3)

n—-2 cl UPit)) apse ey) an tare)

Jref = An (A aL 9 ; = 0 Lo) Yo, p (3) sin Gp na 3 V: 9 h;tan @; “, hq tan @ 2 cos’a; = Cos" &

Similar formulae are given in MALinovsKata’s work ©).

The intensities of multiple waves are determined from formulae similar to (3). Here again we must take into account all the refractions and reflec- tions at the intermediate interfaces. Therefore if the multiple wave under consideration intersects the interface between the ith and i+1th layers while it is travelling from above downwards m times and the same number of times when it is travelling from below upwards, then the multiplier (P; P;., ;) (P;,1P;) is repeated in the product JJ (P; P,,,) (P;,1 P;) also m times. It the wave suffers n reflections from the given interface, the coefficient of reflection corresponding to this boundary is raised to the nth power. We introduce coefficients raised to the appropriate powers for reflection from all the interfaces at which the wave under consideration suffers further reflections.

We shall henceforward adopt the following notation: if the reflected wave is travelling in the mth layer and is reflected from its lower boundary, the coefficient of reflection will be denoted by (P,, P,,); but if the reflection takes place from the upper boundary of the mth layer the coefficient of reflection will be denoted by (Pv Pm)

In the sum y h; ae %i each item is repeated as many times as the multiple : COS” &;

wave under consideration passes through the ith layer. If this wave is

propagated from the shot point upwards and then is again reflected from the

tan &o

COS” Xp

introduced into the sum 2 eee o/h) ee in formula (3). cos? a; “COS? Go

free surface and goes downwards, a further term, 2hsa must be

REFRACTED AND REFLECTED LONGITUDINAL WAVES 19

The method of calculating the intensities of reflected waves can be sum- marized as follows.

1. First we determine the sines of the angles of incidence of the reflected wave under consideration at all the interfaces.

These angles will vary from point to point along the ground surface unlike the corresponding angles for the head waves. It is best to have sin @

a a

so

Fic. 4a. Curves for the coefficient IPP (p) of primary wave formation (4 =0.6 and 4 = 0.4)

given and then the angles of incidence of the ray selected will have the same sines at all the interfaces, and these will be determined by expression (4). - _s Vi, p F} sin @ = -—~ sin &&, (4) VO, p while the point of emergence of the ray under consideration will lie at a distance r, determined by the formula

ae

20 V. P. GoRBATOVA

n-1

i=0

from the shot point along the profile. In a case of multiple waves, the term h; tan @; in the sum >) fh, tan a; in

expression (5) is repeated the same numbers of times as the wave under consideration passes through the zth layer.

6 =06

J 0:04, A

a

t 5:0 °

4:0

Bee iS

* 126

Y, +O 02 wt Un-1 Pp

Tap | SIN@cr

Fic. 4b. Curves for the coefficient [PP (p) of primary wave formations (A= 0.5).

2. The multiplier Cy is determined in the same way as for primary waves. Its value is taken at the pomt x=sin a, from the curve (see Fig. 2) cor-

: v responding to the parameter yy = —*, as for the zero layer of the medium Vo, Pp

under consideration.

REFRACTED AND REFLECTED LONGITUDINAL WAVES 21

3. The coefficients of refraction (P; P;,,) (P;,, P;) are determined in the same way as for head waves. Only the ordinates of the curves (see Fig. 3)

: Vi ies : are now taken at the points x = sin a, = —*” sin a where v, ,, is the lesser of Vo, p b) the velocities v; , and v;,, ,, that is at points equal to the sines of the lesser

of the angles at which the ray under consideration meets the boundary be- tween the ith and the 1+1th layers. The curves are chosen with the same parameter as corresponds to the boundary between the ith and the 7+1th layers. Then the ordinates are again multiplied by the quantity,

40; Vi, p Ci+1 Vi+1, p cae a : Nee (0; Vi, pt Oi41 Vi+1, p)

The product also immediately gives us the value of the coefficient of refraction (P;P;.,) (P;4,P,).

4. The coefficients of reflection (P,, P,,) and (P,,,P,,) depend both on the properties of the reflecting boundary itself and also on the angle at which the wave strikes it. To determine these we have constructed graphs (Figs. 5, 6 and 7) based on the tables drawn up by the PETRASHEN’ team; the sines of the angles at which the wave under consideration strikes the given reflecting boundary are plotted against the abscissa, and the parame- ters of the reflecting boundary are used as the parameters of the curves. We shall deal with each of the following cases separately:

(a) Reflection of a wave from a layer having higher velocity of longitudinal wave propagation than the layer in which the incident ray is travelling;

(b) Reflection of a wave from a layer having a lower velocity of propagation of longitudinal waves;

(c) Reflection of a wave from the free surface of the medium.

Case a. To determine the coefficient (P,, P,,) or (P,.P) from the curve corresponding to the parameter A (see Fig. 5) which characterizes the reflecting interface under consideration, we take the ordinate value (P,, P,,)rep if Um, p< Um+t, p> OF (Pry P)rer f Ump 18 < Um—1, p Where the abscissa is equal to the sine of the angle at which the wave under consideration strikes the given reflecting boundary. The value of the ordinate is again multiplied by the coefficient of reflection of plane waves when the incidence is vertical.

The product gives the value of the co-efficient of reflection,

°m, p< Um+i, p? that is, if

(Pm Pin) <5 ae (Pm ralecl CmUm, p— Qm+1¥m+i, p Om Um, p + Om+1Um+1, p

99 V. P. GorRBATOVA

and at 2g nee.

(Prn’ Pm’) ae AW (Pm Pm’)rel Om Ym, p— @m-1Um=1, p

Om Um, p + Om-1Um-i, p Case 6. From the curve corresponding to the same parameters y and A as on the reflecting boundary (see Fig. 6) at a point with a reading along the

4-0-4

SiN @ ref

0-2 0:4

=

ala eet

Fic. 5. Curves for calculating the coefficient of reflection of longitudinal waves from the layer boundary with a higher velocity of propagation than in the layer in which the incident ray is travelling.

abscissa equal to the sine of the angle of reflection, we take the ordinate

(ee ean ait il ees aN OF (pled eres Oh, pe Uma and multi- ply it by the coefficient of reflection of planar waves when the incidence is vertical. The product gives the coefficient of reflection, that is if

Um, p z Um +1, ra)

REFRACTED AND REFLECTED LONGITUDINAL WAVES 23

Om Um, p—@m4+1¥m+41, Pp , a Om Um, p + Om4+1Um+41, p

then (Pn Py) jaane (PrP mca

and if ,.-,<Um-—1, p> then

Om Um, p— @m-1U¥m-1, p OmUm, p + Cm-1Um-1, p

(PEP) = s(n Pm)re

sin ret 0-2 0-4 06

774-05 0) pS O6 ‘ie = at E | = | 7‘ 04,4=0'5 0-5 oa 7205, 4-06 724-706

Fic. 6. Curves for calculating the coefficient of reflection of longitudinal waves from the layer boundary with a lower velocity of propagation than in the layer in which the incident ray is travelling.

Case c. The value of the coefficient of reflection (Py Py) is taken at the point x = sin a, along the ordinate from the curve (see Fig. 7) corresponding

24, V. P. GorBATOVA

v ok. : to the parameter y) =—”-5 characterizing the zero layer of the medium %0, p

under consideration.

The values found for the coefficients of refraction and reflection and also for the multiplier Cj are substituted in the appropriate formulas. The intensity obtained for the wave relates to a point at a distance r from the shot point along the profile calculated from the formula (5).

Sin 2 1-02 0:2 0:4 06 0:8 05 0:8 j w04 0-6 im Ir -g° 2 67 0°5 0:4 = %70 6 02 | y | |

Fic. 7. Curves for the coefficient of reflection (Py Py) from a free surface.

For two-layered media the formula (3) is converted into (6) and (7) for single reflected waves, and into (8) and (9) for waves reflected n times

Rak CrP eeeo) SIN & COS Ap (6) fat ai Ae (Ay + 2p) Yo, p r Gee, cos? o, - jret = 0 ( 0 0) 0 (7)

Agci(Ay = Qag)itey Laney

REFRACTED AND REFLECTED LONGITUDINAL WAVES 2

CVs Pay" Ce Po)! Sia 08 A Agr (Ay + 249) Vo, p r

Jref =

a es Cy (Po Po)” (Por Po)" _ cos * (9) DES Ag (Ag + 2M) Vo, p 2nhy—hsa

Here fy is the thickness of the upper layer. For multi-layered media, the multiplier

sin &

9 h;tan @; ] tan. Gy cos? a; cos? &

from formula (3) can also be simplified and represented in the form

SIN &

when the angles at which the reflected wave under consideration strikes the intermediate interfaces are not too great. The permissible error with such a substitution lies within the limits shown in Table 3, where «; is the largest of the angles «, (where i = 0 to n—2) at which the wave reflected from the n—Ilth and nth layers strikes the intermediate interfaces.

TABLE 3 oj | 10° | Pe BEECSO © | “BOE 2aRS0r 25° | 30° | | Error, % 0.3 15 | 34 | 45 | ai | cee | 9 12.5 |

It can be seen from the table that if @; does not exceed 25° the permissible error will not exceed 9%. This degree of error will occur if all the angles «(i = 0 to n—2) are equal to 25°. If however some of them are smaller than 25° the error will be reduced. Here the true values of the multiplier we are dealing with are lower than the approximate values. If, by analogy

SIN Gq COS hy

with a two-layered medium, we substitute the value for this

7 multiplier, we obtain still better accuracy.

_If the angles of incidence onto the intermediate interfaces of multi-layered media are not too great, then the multiplier

sin Ko

9 h; tan @; } tan Go lf (pas sq 2 cos? a; COS” Xp

26 V. P. GoRBATOVA

in the formulas for calculating the intensity of reflected waves (simple and

SIN % COS &

multiple) can, be replaced by —— . Formula (3) is then converted

into (10)

n—-2 Co Hl (PiPi+s) (Pit1Pi) (Pn=1Pn-1)

Je ce ota (10)

SIN Gp COS By ny ha If we apply eqn. (4) to eqn. (3) and go over to a single independent variable sin %, and then proceed to the limit sin @—>0 we shall obtain formula (11) for determining the intensity of waves which reflected once above the shot point when reception is vertical:

n—2 2 eed) (Pi+1 Pi) |v (Pn—1 Pra-i)o ice = aT ae a ee, (11) Agt (Ag + 2p) Vo, p 2 h;

peat

Where [(P;P;,,) (P:4,P)], is the coefficient of refraction of planar waves when the incidence of the wave is vertical; that is [(Pi Pi+1) (Pi+1 Palo = ——_— tite __ x Qi Vi, pT Ci+1Vi4+1, p (12) 20i+1Vi+1,p GiVi, p+ C:41%i41,p

(es eas

incidence is vertical on to the reflecting boundary under consideration of

is the coefficient of reflection of planar waves when the

the n—l1th and nth layers equal to the following expression

= p=, ja Vn, (Pig: SPI igh AE ia ean RO: (13) @n-1Un-1, pT OnUn, p h; is the distance traversed by the wave in the i-th layer; v;,p is the group velocity of longitudinal waves in the i-th layer. When we determined the intensities of multiple waves above the shot point, formula (11) is transformed in the same way as formula (3). All the coefficients of reflection and refraction of the multiple wave under consider- ation are introduced into the numerator, each of these coefficients being

REFRACTED AND REFLECTED LONGITUDINAL WAVES 2H

equal to the coefficient of reflection or refraction of plane waves with vertical incidence of the wave on a given interface, and being determined from formulas (12) and (13). The distance h; travelled by the multiple wave under consideration in the ith layer is introduced into the denominator.

The formulas thus obtained add to the knowledge we already have from the theory of plane waves, about the intensitites of simple and multiple reflected waves over a shot point, the possibility of taking into account the weakening of reflected waves due to the divergence of a spherical wave.

ACCURACY OF THE PROPOSED METHOD OF CALCULATION

The method proposed is approximate. We shall now evaluate this method by comparing it with accurate solutions, and with the values obtained when the Leningrad tables and methods were used.

The basic formulas are merely another way of writing out the expressions for intensity which were given in the papers @%); no new errors are therefore introduced. The values obtained from the tables for the multipliers Cy, L??(p), (Po Po’) have been plotted in figures 2,4 and 7. We may regard their graphical values as being determined with a sufficient degree of accuracy. The coefficients of refraction and the coefficients of reflection on the other hand have been found approximately by means of the graphs shown.

Let us now evaluate this approximation. The possible error in determining each of the multipliers (P;P;,,) (P;,,P;) for interfaces characterized by the quantity lying within the limits 0.7 <a@ <0.9 does not exceed 2%; if this quantity lies between the limits 0.5 <a@ <0.7 the error is 5% and finally if it lies within the limits 0.8 <a < 0.5 the error is 10%. This estimate has been made for refracting boundaries with the parameters indicated above. Only for boundaries with y = 0.4 and A = 0.6 are the errors in determining the refraction coefficients slightly higher. The refraction coefh- cients for such boundaries can nevertheless be calculated by the method referred to, the errors being reduced as the angle of incidence becomes smaller or approximates to critical. For boundaries characterized by the parameters y = A = 0.6, o = 0.7 to 1.0, the error will never exceed 3%.

We shall estimate the error entailed in determining the coefficient of reflection separately for the following cases.

Reflection from a layer with high acoustic rigidity—When the sine of the angles of incidence on the reflecting boundary is equal to 0.75a (a being the parameter of the reflecting boundary under consideration), the error in determining the coefficient of reflection does not exceed 10%. The degree of accuracy rises rapidly as the angle of incidence becomes smaller.

28 V. P. GoRBATOVA

The approximation method indicated can be used for reflecting boundaries with the parameters shown in Table 1. :

Reflection from a layer with lower acoustic rigidity—For reflecting boundaries with parameters as shown in Table 2, the magnitude of error in determining the coefficient of reflection is given in Table 4, from which it can be seen that when the angles at which the wave under consideration strikes the reflecting boundary are not too great it is permissible to use our approximate method of calculation.

TABLE 4

Values of parameters Error in determination of coefficient of reflection, % y A sin ref < 0.6 sin Gref < 0.5 | sin Gret < 0.40 0.6 0.6 5) 3 | 1

0.5 0.5 6 2 1

0.4 0.4 9 5 4

0.5 0.6 33 13 10

0.4 0.5 8 6 By

0.3 0.4 2 2 115)

ANALYSIS OF THE INTENSITIES OF HEAD (REFRACTED) WAVES A Two-Layered Medium

The intensity of head waves in two-layered media is determined from the formula

Co LP? (p) Art (Ay + 2g) Vr (r—79)""* The value of the multiplier Cy (see Fig. 2) depends slightly on the values

Jnead a (14)

of the parameter yg and is near to 2 when sin %& <0.9. The behaviour of TP (p) the coefficient of the head wave formation (see Fig. 4) will therefore illustrate the dependence of the intensity of head waves in two-layered media on the parameters of the interface at distances 7, which are sufhciently far from the point of emergence of the head wave, when we can set Vr (r—rp)'!2 Mee

It can be seen from Fig. 4, moreover, that when y, A and o are fixed, the intensity of the head wave increases in inverse proportion to the difference in the longitudinal velocities at the refracting boundary. At distances r > the intensity of a refracted wave increases with reduced sharpness of the refracting boundary just as the coefficient J? (p) grows.

At distances r comparable with r9, when it is not possible to set Vr (r—ro)"! ie

REFRACTED AND REFLECTED LONGITUDINAL WAVES 29

the increase in intensity of the refracted waves with decrease in the drop in velocities of propagation of the longitudinal waves at the refracting boundary occurs still more rapidly than the growth of the coefficient I”?? (p). Even if the comparison is made at equal distances from the point of emergence, then when r—ry > 0.8h, refracted waves with higher amplitude will correspond to boundaries with less difference in the velocities of propagation, although for these the points of comparison are at a greater distance from the shot point. For boundaries where the difference in propagation velocities is slight, the intensity of a head wave at some distance from its point of origin will be greater than for a sharp interface at the same distance from its point of origin. The curves shown in Fig. 4 show how the intensity of head waves depends on the values of the parameters y, 4 and o at the refracting boundary. We can however choose these parameters to be such that when the drop in the propagation velocities of longitudinal waves is slight, the head waves will have a lower intensity than in a case of greater difference in the velocities of propagation at the interface (but with other parameters y and A). It can be seen that the intensity of the head waves increases in direct proportion to A and in inverse proportion to y.

The density ratio at the interface also affects the intensity of the head waves. For boundaries A = 0.4 and A =0.5 the head wave intensity increases as the difference in densities decreases, while for boundaries where A = 0.6 and a > 0.35 it decreases.

The damping of the head waves with distance is determined by the multi- plier 7~/* (r—ry)~*. The influence of the depth of the refracting boundary on the intensity of the head waves has a substantial effect only at distances r comparable with ro. If the comparison is made for several two-layered media which differ from one another only by the parameter hy, (r being fixed and the same for all the media), we arrive at what seems to be a contradictory conclusion: namely that the greater the bedding depth of the interface the greater the intensity of the primary waves. If however we compare the intensity of the head waves at uniform distances from their points of origin, everything becomes clear. We find that to get head waves of the same intensity at the same distance from their respective points of origin in the case of much deeper interfaces, a much more violent causé of excitation is required. At distances r>>7q the bedding depth of the interface does not influence the intensity of a head wave. The head waves will dampen with distance as r—?.

Multi-layered Media Of the many problems connected with the origin and propagation of head waves in multi-layered media, we shall here treat only the following:

30 V. P. GORBATOVA

(a) the influence of the velocities at which transverse waves are propagated on the intensity of longitudinal primary waves;

(b) the effect of adding an ith layer, and changing the longitudinal velocity in it, on the intensity of the head waves excited in layers of greater depth;

(c) the effect of a sharp principal refracting boundary on the intensity of a head wave excited in it;

(d) the damping of the head waves with distance and the influence of the bedding depth of the main refracting boundary on the intensity of these waves.

We shall examine all these questions in order.

(a) Formula (2) is used to determine the intensity of the head waves excited in multi-layer media. As has been shown above the multiplier Cy as well as the coefficients of refraction (P; P;,,) (P;,, P;) at all the intermediate interfaces depend only slightly on the parameters y and A, that is on the values of the transverse velocities in the covering layer. The intensity of the