SIMPLIFIED NUMERICAL MODELS FOR THE GENERATION OF SYNTHETIC REFLECTION PROFILING SEISMOGRAMS AND SYNTHETIC REFLECTION/REFRACTION SEISMOGRAMS by Charles David Lodge United States Naval Postgraduate School THESIS SIMPLIFIED NUMERICAL MODELS FOR THE GENERATION OF SYNTHETIC REFLECTION PROFILING SEISMOGRAMS AND SYNTHETIC REFLECTION/REFRACTION SEISMOGRAMS by / /3Z£4 3 Charles David Lodge April 19 70 Tku> document hcu> been approved fan. public iz- lzcu>e. and 4>alt; lti> dUt/iibutlon JU> untmLttd. Simplified Numerical Models for the Generation of Synthetic Reflection Profiling Seismograms and Synthetic Reflection/Refraction Seismograms by Charles David Lodge Lieutenant Commander, United States Navy B.A., Rice University, 1962 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL April 1970 ABSTRACT Two numerical models were developed which generate synthetic seismograms in formats similar to those of records obtained in field exploration. The models solve the hyperbolic wave equations in one and two dimensions by finite difference approximations in initially undeformed solution domains of transversely-isotropic layered media subjected to a time varying, dilatational forcing stress applied at the surface. A velocity attenuation term was included in the models to inhibit strong boundary reflections. The one-dimensional model produces synthetic reflection profiling seismograms for arbitrary horizontal or dipping layers. The two- dimensional model generates synthetic reflection/refraction seismograms for horizontal layered media with arbitrary distribution of wave velocities, Poisson's ratio and density. Several sample records were produced for some representative velocity structures. The synthetic seismograms were interpreted and gross correlation was carried out as if they were actual field records. OSTGKADUATE SCHOOL Y, CALIF. 93940 TABLE OF CONTENTS I. INTRODUCTION 11 II. THE SEISMIC REFLECTION PROFILING MODEL 14 A. THE MATHEMATICAL MODEL 14 B. THE SOLUTION DOMAIN 16 C. THE DIFFERENCE EQUATIONS - - 16 D. STABILITY AND CONVERGENCE 20 1. Stability 20 2. Convergence 21 E. DESCRIPTION OF THE PROGRAM 22 1. Distribution Parameters 22 2. Numerical Parameters 23 F. SCALING THE PROBLEM 27 III. THE REFLECTION/REFRACTION MODEL 30 A. THE MATHEMATICAL MODEL 30 B. THE SOLUTION DOMAIN 33 C. THE DIFFERENCE EQUATIONS 33 D. STABILITY AND CONVERGENCE 37 1. Stability -- 37 2. Convergence 38 E. DESCRIPTION OF THE PROGRAM 38 1. Distribution Parameters 39 2. Numerical Parameters 44 F. SCALING THE PROBLEM 45 IV. INTERPRETATION £7 A. THE REFLECTION PROFILING SEISMOGRAMS - 47 1. Variable Depth Records 51 2. The Seismograms 54 B. THE REFLECTION/REFRACTION SEISMOGRAMS - — 71 1. The Seismograms 74 2. The Contour Output --- 91 V. SUMMARY - - - 99 VI. FUTURE WORK - 101 COMPUTER PROGRAMS - 103 THE REFLECTION PROFILING MODEL 103 THE REFLECTION/REFRACTION MODEL - 109 SUBROUTINES - - 120 LIST OF REFERENCES 126 INITIAL DISTRIBUTION LIST - - 128 FORM DD 1473 - 129 LIST OF TABLES TABLE PAGE I. Sample Horizontal Station Distances 26 11. Sample Velocity Defining Array 26 III. Sample Attenuation Parameter Polygon IV. Sample Forcing Function Polygon 27 V. Scales Available for C - 0.8 = 1.5 km/sec -- 29 VI. Sample Wave Velocity and Density Polygons 40 VII. Sample Forcing Function Polygon 40 VIII. Sample Attenuation Parameter Polygons 40 IX. Scales Available for C = 0.6 = 1.5 km/sec 46 X. Scale of Contoured Quantity 95 LIST OF FIGURES FIGURE PAGE 1. The Solution Domain 17 2. Mesh Detail 18 3. Sample Velocity Structure for Seismic Profiling 24 4. Sample Distribution Parameters 25 5. The x-y Solution Space 34 6. Isometric Detail of Three-Dimensional Mesh 35 7. Sample Distribution of Wave Velocities and Density 41 8. Sample Distribution Parameters 42 9. Sample Distribution of the Attenuation Parameter 43 10. Velocity Structure C3H 48 11. Velocity Structure C4D 48 12. Velocity Structure C5B 49 13. Velocity Structure C6C 49 14. Velocity Structure C7A 50 15. Velocity Structure C8A 50 16. Forcing Function F7 ■ 52 17. Attenuation Parameter A14 52 18. Stress versus Time at Various Depths 53 19. Uncorrelated Reflection Profile C3H F7 A14 55 20. Reflection Profile C3H F7 A14 57 21. Reflection Profile C4D F7 A14 59 22. Reflection Profile C5B F7 A14 61 23. Reflection Profile C6C F7 A14 63 24. Reflection Profile C7A F7 A14 65 25. Reflection Profile C8A F7 A14 67 26. Precision Reflection Profile C4D F7 A14 69 27. Velocity Structure CA 72 28. Velocity Structure CB 72 29. Velocity Structure CC 73 30. Forcing Function F5A 75 31. Attenuation Parameter B3 75 32. Reflection Seismogram CA F5A B3 L2 76 33. Refraction Seismogram CA F5A B3 L2 81 34. Reflection Seismogram CB F5A B3 L2 83 35. Bottom Detector Seismogram CB F5A B3 L2 85 36. Reflection Seismogram CC F5A B3 L2 87 37. Theoretical Time-distance Curves for Velocity Structures CA and CB 89 38. Portion of the Printer Contour Plot Showing Generation of the Dilatational Wave 93 39. Portion of the Printer Contour Plot Showing Transmission of the Dilatational Wave into the Lower Media 94 40. Portion of the Printer Contour Plot Showing the Development of a Dilatational Head Wave 96 41. Portion of the Printer Contour Plot Showing Generation of a Shear Wave at the Transition Zone by the Dilatational Wave 97 42. Portion of the Printer Contour Plot Showing Development of an Interface Shear Wave 98 ACKNOWLEDGEMENTS The author wishes to extend his sincere appreciation to Professor Robert S. Andrews who provided invaluable assistance in delimiting the scope of the problem and in interpretation of the results. Apprecia- tion is also extended to the staff of the Naval Postgraduate School computer Facility, especially Mr. David F. Norman, for assistance in running and debugging the programs. Particular thanks are due to the author's wife, Diana, whose moral support and secretarial skills aided t lie project immeasurably. I. INTRODUCTION A major problem in exploration geology is the determination of structural relationships for areas in which the rocks cannot be directly examined. This is usually a subjective process, involving the extra- polation of lithologic facies from a region where control is available, such as surface outcrops, mines or wells, across an unknown or inaccessible area to another point with known structure. Indications of subsurface structure, including magnetic, seismic and gravity measurements as well as interpretation of surface morphological features, are frequently relied upon to aid this process. The problem becomes particularly acute in oceanic regions where the only available control may be continental outcrops several hundreds of miles away, a few bottom grab samples and shallow sediment cores and a rough bottom topography. Remote measurements may then become the only method of initial structure determination. Frequent use is made of the manner in which the ocean bottom and sub-bottom rocks reflect and transmit acoustic or seismic energy, not only for the correlation of near-shore with on shore structures, but also for initial determination of sub-bottom structures. When con- ducting seismic exploratory operations, the information obtained is usually presented as a two-dimensional graph (seismogram) on which three variables are shown. These variables are: horizontal distance, time and intensity of energy received. Depth information may be inferred from the time scale with a known or assumed vertical distri- bution of sonic velocity. A high level of intensity is usually assumed to represent a point or surface at which the physical 11 parameters of the media change more or less abruptly, reflecting energy or providing a surface along which energy might be propagated. Interpretation of the sei sinograms and the development of a sub- surface structure can be aided by the use of a mathematical model of the supposed structure which would produce a synthetic seismogram. The synthetic seismogram could be compared with the field record for correlation. This paper presents the development of two numerical models which produce synthetic seismograms for layered media subjected to an initial time varying, compressional (dilatational) stress input applied at the surface. The models were designed to have fairly wide variability while remaining simple enough to preclude the use of inordinate computer time and storage requirements. The one-dimensional model produces a reflection profiling seismogram. It is applicable to the case of vertically incident dilatational waves reflected and transmitted by arbitrary bottom profile and sub-bottom layered structure of hori- zontal or dipping interfaces. The two-dimensional model yields a set of seismograms applicable to a reflection/refraction problem where the receivers are distributed in a horizontal array. Both dilatational and shear stresses are treated. The region modeled is one of horizontal layered media with arbitrary distribution of shear and dilatational wave velocities and of density. No attempt was made here to include the wide variability of possible media which is an integral part of the large SLAM (Stress waves in LAyered Media) code model developed at the Illinois Institute of Technology (IIT) [Constantino 1969] primarily to treat the neighbor- hood of high energy (nuclear) explosively generated ground shocks. 12 The LIT model has the capability of simulating a wide variety of conditions but requires a large amount of computing time and space in order to produce a single trace. It is felt that the two models pre- sented here have a greater applicability to marine and terrestrial seismic exploration problems. These models were developed on the basis of numerical approxima- tions to the solutions of the wave equations in one and two dimensions by finite difference methods. The computer programs were written in the FORTRAN IV language and were run on a dual IBM 360/67 system. The plotting subroutines used to drive a CALCOMP 765 plotter were prepared by the staff of the United States Naval Postgraduate School Computer Facility [l969j 13 II. THE SEISMIC REFLECTION PROFILING MODEL The problem of synthetic seismic profile generation was approached as one of repeated transmission of normally incident longitudinal (dilatational) disturbances through transversely-isotropic media. The seismogram was developed by changing the velocity structure in the solution domain between pulse transmissions, corresponding to a change in position of the shot point as the profiling vessel moves across an area of changing water depth and sub-bottom structure. The solutions are presented as successive traces as though from a pressure hydro- phone located at any depth below the shot point. A. THE MATHEMATICAL MODEL It was assumed that the wave velocity (C) is continuous throughout the domain and either varies linearly with depth (x) or is constant. Additionally, it was assumed that in transition zones, where the wave velocity does change, density remains constant. Under these assump- tions, the equation of motion in one dimension for dilatational waves can be written (modified from Wolf [1937J ) : ht* by {?%)+ A fc . (i) where u is longitudinal particle displacement and A is a velocity attenuation parameter which is a function of x. Carrying out the indicated differentiation of the right side of equation (1) yields: E: *h> - cz s£u +. 1C 33l£ $£ 4- a ^V- . (2) H1 c a** + 1C ^* ^y + A ^E U) 14 Four collateral conditions are required in order to complete a well-posed problem. The following conditions were selected: initial conditions: II: u^,0) = 0'! (3) 12: ot(y; oN- = C , (4) boundary conditions: Bl: C(L,0 - O t (5) B2: UK(C,-t ) - fit-/} , t^ "ti • (6a) V (O, M =: , t>ti , (6b) where L is the distance to the lower boundary. The initial conditions allow the solution domain to be initially undeformed and at rest. The first boundary condition requires that displacements are zero for all time at the lower boundary, which then becomes perfectly reflective. The second boundary condition allows a variable stress to be applied to the upper boundary for t( t,, For t > t , normal stresses are not transmitted across this boundary. It then becomes a free surface. The second term on the right side of equation (1),A ^rr: , represe- sents an attenuation factor, dependent on depth and particle velocity, which was used to reduce the magnitude of the disturbance in the lower regions of the domain so that reflections from the lower boundary would not mask late arriving multiple reflections from points nearer to the surface. The function A(x) is assumed to be zero in the region from which reflections are to be studied. 15 B. THE SOLUTION DOMAIN The solution domain (Fig. 1) is divided by N equally spaced mesh points into N-l subintervals each of length £x. A sample wave velocity distribution is shown in the figure. The upper boundary (x = 01 is shown on the left. In the region O^x^x., the wave velocity is constant and equal to Cl. For x„4 x ^ L, the velocity is equal to C2. In the transition zone, x. < x < x„ , the velocity varies linearly from Cl to C2. The time-depth mesh is an array of N by M mesh points. Each rectangular mesh area is then of size S x by £t. This mesh is the full solution domain and the objective of the model is to find a solu- tion of equation (2) at each mesh point under the collateral conditions, equations (3) through (6). C. THE DIFFERENCE EQUATIONS A central difference form for each of the partial derivatives in equation (2) can be written in terms of the (i,j) mesh point [ Smith 1965] . Figure 2 shows the (i,j) and neighboring mesh points. The required derivatives, in central difference form, become: ^S ± Cut- Ci- 16 12 3 4 5 t VELOCITY = CI VELOCITY = C2 Xl X2 Figure 1. The solution domain N t u-l 17 \( j+1 A\ i.i+i j 7T\ -Pcnr—fti fen j-1 c "n Vij-i i-1 L i+1 Characteristic curves Figure 2. Mesh detail. 18 Substituting these into equation (2) gives the difference equation of motion for the (i,j) mesh point: ■"■- ) + .-2cX,j-n>t|j.i _ cf / ^i^-l^^u^jN + Lq ( Cwi - O-A /O^J^U-y, A + fr. (U^-O^-iN (7) The collateral conditions become: II: Util = o (8) 12: UL.a.- OC.o - 0 (9) Bl: UMjj = O . (10) B2: "\i "°°j = Fj ' (11) where F. = F(t) , 0(ti t , fj ■ ° . •= > ti . Solving equation (7) for u. .,-, gives: E. o-tlj4i = [2UL,j (l- CiV2-) Hr UL+x.j (C?fc* + ic^(c^,-c,.-i)c.a) ■ -cJ.r,(l+i.A;St)]/[l-iALS^ , (i2) where: £2 = ( ^-M * Writing equation (12) for j=l and substituting u- n from equation (9) yields the initial conditions: II: L>-til = 6 (8) 12: 0„z - O . (13) 19 Writing equation (12) for i=l and substituting u . from equation (11) o > J yields the boundary conditions: Bl: °^,j + i " ° » (10a) B2: 0L,ri = [20Lij(i-C4V) + 2 0a,j (CfR1) -:,r:(l^Ai-t)^^(^)]/[l-iAt- (14) The equations (12), (8), (13), (10a) and (14) then form a well- posed problem for numerical solution. Initially, the j=l and j=2 rows are set to zero following equations (8) and (13). The j+1 row can then be solved utilizing equation (10a) for the lower boundary (i=N) , (14) for the upper boundary (i=l) and (12) for the interior points. Note that the value of u anywhere along the interior of the j+1 row is dependent only on the values of u at three points along the j row and one on the j-1 row (shown as circles in Fig. 2). D. STABILITY AND CONVERGENCE 1. Stability Because of the large number of arithmetic operations involved 9 (approximately 10 for the solution spaces used), it is necessary to insure that there is a stable decay of errors introduced in the various computations. Studies of the stability of the centered difference formulation of the basic wave equation: TZ- ' c -5** ' (15) have been carried out by several investigators [ Smith 1965j . Smith investigated the stability by the Fourier series method and showed 20 that the centered difference formulation is stable for the mesh ratio: ^ 4 c * <16> The addition of the velocity gradient and velocity attenuation terms in equation (2) alters the stability conditions somewhat. A large velocity gradient, approaching discontinuous velocity change, generates instability in the gradient region. Investigation of the stability conditions showed that if the condition: &C, - Ci+i - Ci-i 4 C'l , (17) were imposed on the gradient regions, the requirement on the mesh ratio [equation (16)1 becomes: £t y 0-89 . bx * C (18) The addition of a negative attenuation term tends to decrease the right side of equation (18). This would lead to a more stringent requirement on the mesh ratio. This effect was found to be quite small for the values of A(x) necessary to damp lower boundary reflections. Empirically, it was found that stability would be assured by requiring that: St y 0-S5 - 4 -£- • (19) 2 . Convergence A finite difference scheme is said to be convergent if the exact solution of the difference equations approaches the exact solution of the partial differential equations as Sx and St 21 approach zero together. Forsythe and Wasow [1967] have shown that convergence of a finite difference scheme such as equation (12) with boundary conditions such as equations (8), (13), (10a) and (14) is convergent as long as the mesh ratio is such that the (i, j+1) mesh point lies within the intersection of the characteristics through the (i-l,j) and(i+l,j) mesh points (see Fig. 2). The equation defining the characteristics for equation (2) is: (£f ' 1 4. -r * (16) It is apparent then that as long as equation (19) holds, the finite difference formulation will be both stable and convergent. E. DESCRIPTION OF THE PROGRAM The program requires 140K bytes of main core storage. An average run with a 251x1800 mesh grid with 41 shot points requires approxi- mately 13 min of main run time exclusive of output time required for the plotting subroutines. 1 . Distribution parameters The advantage of this model lies in the wide variability of the conditions which may be easily simulated. The primary parameters used to model various conditions are the distribution parameters. They are: a. wave velocity distribution, C(x,t), 22 b. forcing function, F(t) , c. attenuation parameter, A(x). The forcing function and the attenuation parameter arc put into Liio model as single-valued, continuous, polygonal line functions of the indicated dependent variable. The wave velocity is put into the model as a two-dimensional array of points which describe the depths to isovelocity layers at successive horizontal stations. Figure 3 shows a sample distribution of isovelocity horizons with horizontal stations and shot points indicated. Horizontal station distances and the velocity defining array for this sample are shown in Tables I and II. At each shot point, the wave velocity is a single-valued, continuous polygonal line function of depth. An example of wave velocity distri- bution at a single shot point is shown in Fig. 4 along with a sample attenuation parameter and forcing function. The polygons for A(x) and F(t) are defined in Tables III and IV. 2 . Numerical parameters The numerical parameters, which remain constant throughout the problem, are used to control the computation of the distribution para- meters and to set the arrangement and size of the solution domain. They are: a. mesh size in the x direction, (DX) , b. mesh size in the t direction, (DT) , c. horizontal distance between successive solution stations (shot point spacing), (DK) , d. number of total x mesh points, (N), e. number of total t mesh points, (M) , f. number of solution stations, (KK), 23 o as PLH < in o 2 2 CO O M W M Z 2 H i-i O i— i tn N CO U CO O Q H S3 P-c 53 O 2& i— i O i— l H O M P-, i— i M CJ> CO H O H 2 O g CO > CO H o— A H O > > > > o >■ t> > > ► 0 > t> o l> t> > O — 1> t> t> > t> o — > r-l III CJ /// II // / ^ /// M / / / W // / o // / •J // / w // / > /// / / CN CO o II II II £ E >* H M (— 1 1— 1 C_> c_> O O o O hJ .J hJ w w W > > > I H 60 4-1 O 5-1 a e CO •H CU w J-i O u-i aj u ■u o •u ex I CO CU (-1 X 24 (a) f 0 C3 — CI 20.0 (b) 10.5 20.0 ■2.0 (O -0.15 t- t p- Figure 4. Sample distribution parameters (a) Wave velocity at one shot point, (b) Attenuation parameter. (c) Forcing function. 25 g. number of points defining the distribution parameter polygons : (1) wave velocity polygon, (NC) , (2) attenuation parameter polygon, (NA) , (3) forcing function polygon, (NF) , h. number of horizontal stations in the velocity defining array, (NG) , i. mesh point at which the output stress is to be measured, (IP), j. horizontal distance to the first shot point, (EEE) . Additional numerical parameters are put into the program which control the size, scale and arrangement of the output traces. For details of their use see the explanatory notes in the program. Table I. Sample horizontal station distances Station Distance 0.0 2.0 10.5 20.0 Table II. Sample velocity defining array Station Numbers 12 3 4 Velocity Depth 0.0 0.0 0.0 0.0 CI to 2.0 2.0 4.5 9.8 CI isovelocity 2.2 2.2 4.7 10.0 C2 horizons 4.7 4.7 4.7 10.0 C2 4.9 4.9 4.9 10.2 C3 20.0 20.0 20.0 20.0 C3 26 Table III. Sample attenuation parameter polygon Depth A(x) 0.0 0.0 10.5 0.0 20.0 -2.0 Table IV Sample forcing function polygon Time F(t) 0.0 0.0 0.10 0.0 0.40 -0.15 0.70 0.0 2.0 0.0 F. SCALING THE PROBLEM All wave velocities and depths were put into the model in non- dimensional form. The wave velocity in the upper layer and the depth to the first transition zone were chosen as the definitive values for non-dimensionalizing the problem. Non-dimensional time was then determined by these values through the mesh ratio. Practically, the scale of the problem is determined by the wave velocity values and one of the mesh sizes. In general, the wave velocity is entered as C distance units per time unit. For a mesh ratio of: St _ 0-025 _ i - - -±- «t C??1) the stability condition [equation (19)1 allows the wave velocity to vary from 0.0 to 3.4 distance units per time unit. If a value of C of 0.8 is to represent dilatational wave velocity in water (1.5 km/sec), 27 then the scale of the problem would depend on a choice of either time or depth mesh size, chosen in view of the definition desired. Table V shows some scales where the time mesh size was specified. 28 II 00 « o o o o CM CM cO o t3 x: o u « /-v a- 6 0) J-i -^ Q CD o •J e CO o oo aj B a to tO CD w •H hJ CU WD O o 00 00 00 cu i— i X> co H CJ> o M-l cd to -u cj g -H CD •H C W H 3 ^ CD O C cO 4-1 to p <1- O m r» m S *"> m r-~ m 00 r^ x e oo m oo 29 III. THE REFLECT ION/ REFRACTION MODEL Synthetic reflection/refraction seismogram generation requires solution of the wave equation in at least two dimensions. A rectangu- lar solution space was chosen with three rigid, reflective boundaries and one free boundary corresponding to the air-water or air-ground interface. In order to produce a model with wide variability as well as one that may be easily manipulated without inordinate computer time and storage requirements, it was assumed that the media in the solution space is horizontally isotropic. The y-derivatives of density and Lame's constants (and hence of shear and dilatational wave velocities) will then be zero. A. THE MATHEMATICAL MODEL It was assumed that density, o , and Lame's constants, A and ll , are continuous throughout the solution domain. Further, it was assumed that density and the dilatational and shear wave velocities, C and S, are either constant or vary linearly with depth. Under these assumptions, the equations of motion in two dimensions for the propaga- tion of shear and dilatational disturbances can be written (modified from Officer [ 1958] and Takeuchi [l966J ): 6c El + ¥" H^ + 4 5 E2: ? + i Y $1 5H (23) at (24) 30 where u and v are the particle displacements in the x and y directions and B is a velocity attenuation factor. The stresses are: Dilatational stress: VcT* ?>u" ) Shear stress: / \"d* bu~ These stresses are computed in the program and are used, in addition to displacements, to form the seismograms. Carrying out the indicated differentiation on the right side of equations (23) and (24) and collecting terms yields: El i ^^ /SO . "&>A 2- ^A ^ . B^o E2: Vv _ / V^A^v where it has been assumed that: (25) £rt (26) 31 The eight collateral conditions chosen to complete a well-posed problem were: initial conditions: II: oCx^.o^) = O , (27) 12: ^0,^,0^ - O , (28) 13: ^^H,o) - O , (29) 14: \f^,L\,o) = O , (30) boundary conditions: Bl: L)L*,0,-\.) - O 9 «J (*,fA,0 = O , B2: v(K,0,-t) = O , V(L(H,-t) = O , (32) B3: Ox (0lH,4r) + ^(O^.-fc) = F lH .t) * (33) where: F(y,t) = 0 for t > t B4: °H (°iHi"t^ +vxl°1H.t) " O • (34) The initial conditions allow the solution space to be initially unde- formed and at rest. The first two boundary conditions (Bl and B2) require that displacements are zero for all time at the subsurface boundaries, which then become perfectly reflective. The other boundary conditions (B3 and B4) allow the x=0 boundary to become a free surface. Shear and dilatational stresses are not transmitted across the surface for all time except for an externally applied, time-varying dilatational stress, applied at a point or combination of points for t v. t, . As in the one-dimensional model, an attenuation term was included in the equations of motion (El and E2) in order to keep boundary 32 reflections from masking multiple reflections in the central regions. The function B(x,y) is non-zero only in the vicinity of the three rigid boundaries and is zero in areas where disturbances are to be studied. B. THE SOLUTION DOMAIN The solution domain is a three-dimensional rectangular array of L by M by N mesh points dividing the domain into (L-l) by (M-l) by (N-l) equal subdivisions, each of size ox by &y by £t. Figure 5 shows the x-y solution space. The total solution domain is formed from N succes- sive solution spaces or planes. A detail of the distance-depth-time mesh in the vicinity of mesh point (i,j,k) is shown in Fig. 6. C. THE DIFFERENCE EQUATIONS As in the one-dimensional model, a centered difference form for each partial derivative in equations (25") and (26) can be written. Making use of the definitions for wave velocities: c ST — ^ / P (35) (36) finite difference forms of equations (25) and (26) may be written in terms of wave velocities and their derivatives. Carrying out this substitution and solving for the time leading (i,j,k+l) term for u and v yields: El: O-^.iw, =[iU-L.j,k(l-eRA-sfRc) + Ud+1ijiL(ct+^ +^)KA + UKj^(c':-^-^)KH^(;uLir,,L-^U;|-l|OS"Xc +VA(c\-s!)R£ (37) 33 - — i cn ro " I I o 1— 1 H EC u CO psi II II II H >- H H B U U 1— 1 o o CO hJ J 2 fa w w > > Q hJ Pd 1 1 o ■x 1— 1 CO H < H CN CM CM O CO § II II ii SK H M M M y u CO q o 55 g £ w > > ga 5 w •. I X u (30 •i-l fa X 34 / I i,j,k+l i-lf j»k i» j-l»k - + ■ i + - i.J+l.k h J,k ■* /- i,j,k-l t i+l,j,k + - Figure 6. Isometric detail of three- dimensional mesh. 35 + v/i-'.j.te. U*-^:)^ + Wij+lik.-».M-lij-,1jc.)c"riCC ■+ UA(c"-s")kS ■v Co;lj+.1k-o:1)-i,k)a&OM. ->/;,),k.^i (i + |r£%t)j / f i- §^&" where: RA - (^ , *6 = ^fg_ , Re = /it " OA - Uf+i.JH,,!.. + pt,,j.lik - O^.j-.k - U,,ifl|k The collateral conditions become: II: Ul(J1 - O , 12: Vj,j(1 - , 13: Ui/,,1 - 14: Bl V; P B2: Ufcj.k. - ^-W - ^uk - ^L,jk = V o o o o o o o o B3: Od B4: Vkj.t-M - i2N/i.j.k.(i-ci«C-SiRA) - v/,,j, k (* Si |2A) (38) (39) (40) (41) (42) (43) (44) (45) (46) 36 A well-posed problem for numerical solution is then formed by equations (37) and (38) with collateral conditions (39) through (46). Initially, the k=l and k=2 planes are set to zero following equations (39) through (42). The k+1 plane can then be solved using equations (43 ^ through (46) for the boundaries and equations (37) and (38) for the interior mesh points. D. STABILITY AND CONVERGENCE 1. Stability Under the conditions that the attenuation and velocity gradient terms are zero and that Sx = Sy, the difference equations (37) and (38) reduce to those for which a stability analysis was carried out by Alterman and Karal [ 1968 J , and Alterman and Rotenberg [1969 J The former paper showed that the stability criterion is given by: 8* C (47) In the latter paper, the authors indicated that stability was guaranteed under the condition that: g < °-86 ' (48) with the assumption that S/C = 0.55, corresponding to a Poisson's ratio of 0.28. Poisson's ratio is defined in terms of wave velocities as: ox) reduced the left side. An approximate stability criterion was established, given by: b^b^-f < ^ wir ' (») under the condition that the restrictions on velocity gradients (for C and S) given by equation (17) apply. 2. Convergence Following the methods of Forsythe and Wasow [1967] and of Fox et al. [ 1962 J , it was assumed that convergence would be assured as long as the mesh ratios were sufficiently small to insure that the (i,j,k+l) mesh point lies within the characteristic surfaces through adjacent mesh points in the k plane. Fox equated the characteristic surfaces with successive positions of a wave front. The most restrictive condition on convergence is then given by the maximum dilatational wave velocity in the solution space. This would give a less stringent mesh ratio condition than the stability condition. Therefore, as long as equation (49) holds, the method will be convergent as well as stable. E. DESCRIPTION OF THE PROGRAM The program produces a set of four seismograms: shear and dilata- tional stress and vertical and horizontal displacement. The data is written on a temporary storage device (tape, disk or drum). A second program then reads the data and plots the seismograms. An average run with a 71x81 grid, 800 time mesh points and 25 detectors requires 278K bytes of main core storage and a run time of 18 min. The output requires 325K bytes of temporary storage space. The primary limiting 38 factor on a given problem is the time required for input/output to and from the temporary storage device. 1. Distribution Parameters As in the one-dimensional model, the distribution parameters are used to model the physical conditions in the solution space. They are: a. dilatational wave velocity, C(x), b. shear wave velocity, S(x), c. density, RH(x), d. forcing function, F(y,t), e. attenuation parameter, B(x,y). Wave velocities and density are put in the program as single- valued, continuous, polygonal line functions of depth. The forcing function is put in as was done in the one-dimensional model along with horizontal stations at which F is to be applied. The attenuation parameter is put into the program as two single-valued, continuous, polygonal parameters. They are: el. vertical attenuation parameter, BA(x) , el. horizontal attenuation parameter, BB(y). The full attenuation parameter, B(x,y), is then computed as follows: B(*.cO = MAX[BA(X), BB(H^] • (51) Tables VI through VIII give sample numerical input used to describe the various distribution parameters. Their distributions in the solution domain are shown graphically in Fig. 7 through 9. 39 Table VI. Sample wave velocity and density polygons Depth C(x) S(x) RH(x) 0.0 0.6 0.0 1.0 3.0 0.6 0.0 1.0 3.3 1.2 0.82 2.1 7.0 1.2 0.82 2.1 Table VII. Sample forcing function polygon Time F(t) 0.0 0.0 0.1 0.0 0.7 -0.15 1.30 0.0 2.50 0.0 Table VIII. Sample attenuation parameter polygons lepth BA(x) Distance BB(y 0.0 0.0 0.0 -2.5 4.0 0.0 1.0 -1.0 7.0 -2.5 2.0 0.0 10.0 0.0 11.0 -1.0 12.0 -2.5 40 t >^ o o o • • • O o 1—1 II II II u CO Pi fa' o u 00 c •I-l o (X C o •H J-l CO o * •H >! i—l u a •H a. CO fa H > CO & <+-l o c o •H 4J •H M ■U CO ex E CO CO cu u 3 60 •r-l fa 41 (a) 0.15 (b) °l 7.0 1 x — > -2.5 BA (c) 0 12.0 0 BB •2.5 — Figure 8. Sample distribution parameters. (a) Forcing function (b) Vertical attenuation parameter (c) Horizontal attenuation parameter 42 43 2. Numerical Parameters These parameters are used to control the computation of the distribution parameters and to set the arrangement and size of the solution domain. They are: a. mesh size in the x direction, (DX) , b. mesh size in the y direction, (DY), c. mesh size in the t direction, (DT) , d. number of total x mesh points, (L), e. number of total y mesh points, (M) , f. number of total t mesh points, (N) , g. number of points defining the distribution parameter polygons : (1) wave velocity and density polygons, (NC), (2) forcing function polygon, (NF) , (3) attenuation parameter polygons: (a) vertical, (NBA), (b) horizontal, (NBB) , h. x mesh point at which the output is measured: (1) stress, (IA), (2) displacement, (ID), i. number of horizontal output stations, (KK) , j. y index of the first output station, (JB) , k. mesh increment between output stations, (JJ), 1. shot point or area location: (1) first shot mesh point, (MA), (2) last shot mesh point, (MB), m. last time mesh point at which F(y,t) is to be applied, (LL) 44 F. SCALING THE PROBLEM The scale of the problem is determined by choices of wave velocity values and either &t or ox and £y. For mesh ratios of: St Q-Q£ _ i_ 8* o.io 2. (52a) fit - o^o| = i , (52b) and under the condition on shear velocity that S = C/ V3" (0~ = 0.25), the dilatational wave velocity may vary from 0.0 to 1.86 distance units per time unit, and shear wave velocity may not exceed 1.07 distance units per time unit. If a value of C = 0.60 is to represent dilata- tional wave velocity in water of 1.5 km/sec, then the scale of the problem would depend upon a choice of either time or distance mesh size. Table IX shows scales available for various distance mesh sizes specified. 45 o o o o OJ ^~, o o CN CJ >-, £ o CN i—l CTJ CM r-l a .—1 on cu ti N o •r-l •H CO 4-1 o o O 3 •-S o o r- i—l X £ o r^ o r~- CO Ln o o 00 CO u jz O MO g c VD X o r-( 5-i ,0 O rt m • • H co cu J-l i—i CD ,a J-J CO a) .—l e ■H cO cO J-i > cO cO CX CO .— ( a> CO i—i o CO •H o u CO cu ^ e a x e o- o o o o o ■ 22 o o 5 10 o ^^ C = 1.4 ^. C - 0.8 ° C = 2.2 =^\^ o C = 2.8 u a- ■ 22 C = 0.8 C = 1.4 5 10 r - 1 0 ^"^=:::::::5=: x: 4-1 a 0) I 15 C = 2.0 C = 3.0 20 25 Figure 13. Velocity structure C6C 49 15 J 20 25 distance C = Q.8 ?9 Figure 14. Velocity structure C7A distance 22 20 C = 0.8 Figure 15. Velocity structure CSA 50 (a) High velocity bed (C7A) . (b) Low velocity bed (C8A) . All traces utilized forcing function F7 (Fig. 16) and attenua- tion parameter A14 (Fig. 17). 1 . Variable Depth Records As an aid to interpretation, several single traces were developed with a single velocity structure, where the depth of the detection point was varied between traces. The traces are shown in Fig. 18 along with the mesh point at which each was computed. These various detection points are shown as circles in Fig. 10. The traces were generated using a source located at a horizontal distance of 0.5 units for the problem designated C3H F7 A14. The first disturbance in each trace, designated d, is the direct wave. Pi and P_ designate the reflected pulses from the first and second transition zones, while the first primary multiple (P-, reflected from the free surface) is shown by P,P . Other multiples are marked in a similar manner. These traces show'clearly the development of a train of alter- nating negative and positive pulses behind the main pulse. The change in pulse shape is also quite evident. As the pulse progresses, the amplitude decreases and the pulse width increases. Additionally, the base and tip of the pulse form tends to become more rounded. These are primarily mesh effects and are dependent on mesh size (DX) . When one particle is unloaded as the pulse passes by, the adjacent particle is still under stress, causing a reverse stress between the particles. In gradient zones, where the derivative of C is non-zero, there is also an elastic rebound effect which reinforces the pulse train. 51 0.1 0.4 0.7 i • i ■0.03 Figure L6, Forcing function F7 52 p p P?P<=+P4P<;P1 \ *115 ^Detection mesh point time 6 10 Figure 18. Stress versus time at various depths 53 2 . The reflection profiling seismograms Synthetic reflection profiling seismograms are shown in Fig. 19 through 26. Figure 19 is an uncorrelated version of Fig. 20, included so that lineations may be observed without the influence of correlation lines shown on the remaining figures. Solid lines on the seismograms mark lineations which were correlated with first reflections from a particular horizon. Dashed lines indicate multiple reflections which, if used as primary reflections, might lead to erroneous interpretations of structure. For each correlation, a particular phase was chosen and followed laterally through successive traces. Figures 20 and 21 show clearly the false dips introduced by the sloping bottom, increasing the pulse travel time in the lower velocity upper layer. The bottom slope causes a change in thickness of the top two layers (water and the first subbottom layer). The false dip introduced is a function of the bottom slope and the ratio of velocities involved. Note that there is a change in dip at each point where a layer pinches out. Figures 22 and 23 show similar false dips and horizons intro- duced by variations in the thickness of a single layer. The high velocity wedge causes a doming up to be introduced into a horizontal transition zone below, while the low velocity wedge has the reverse effect. The most obvious feature of the records for velocity structures C7A and C8A (Fig. 24 and 25) is the relatively high amplitudes in the records in the areas where the dipping beds pinch out against the flat bottom. This is caused by a combination of factors. In velocity structure C7A (Fig. 24), there is a sharp change in velocity contrast across the bottom as the dipping bed outcrops. The steeper gradient then reflects more energy, causing an increase in amplitude in both 54 Figure 19 Uncorrelated reflection profile C3H F7 A14 DX: 0.10 DT: 0025 DK: 0.50 N: 251 M: 1800 IP: 2 Horizontal distance (d) versus time (t) 55 ■H U-A O u cu c 0 •H o 0) 56 Figure 20 Reflection profile C3H F7 A14 DX: 0.10 DT: 0.025 DK: 0.50 N: 251 M: 1800 IP: 2 Horizontal distance (d) versus time (t) First reflections Multiple reflections 57 A O.'r A f o 1 * fi '*: i : p Aft MW alii 'XT fen. sr H 0) iH •H >4H O M a c o •H J-l O QJ QJ o CM OJ M 3 6C 58 Figure 21 Reflection profile C4D F7 A14 DX: 0.10 DT: 0.025 DK: 0.50 N: 251 M: 1800 IP: 2 Horizontal distance (d) versus time (t) First reflections Multiple reflections • Location of precision trace (Fig. 26) 59 o 1-1 a o •H 4J O CD •H 62 Figure 23 Reflection profile C6C F7 A14 DX: 0.10 DT: 0.025 DK: 0.50 N: 251 M: 1800 IP: 2 Horizontal distance (d) versus time (t) — First reflections — Multiple reflections 63 64 Figure 24 Reflection profile C7A F7 A14 DX: 0.10 DT: 0.025 DK: 0.50 N: 251 M: 1800 IP: 2 Horizontal distance (d) versus time (t) — First reflections — Multiple reflections 65 -1- < Pn < CJ + o C = 0.60 S = 0.35 RHO = 1.00 C = 1.40 S = 0.82 RHO = 2.10 -+ Reflection spread o Refraction SDread CA Figure 27. Velocity structure o 4 + C = 0.60 C = 1.40 S = 0.0 RHO - 1.00 S = 0.82 RHO =2.10 + + Reflection spread • * Bottom spread Figure 28. Velocity structure CB 12 72 1 C = 0.60 S = 0.35 RHO = 1.00 12 C = 1.10 S = 0.65 RHO = 2.10 C = 1.80 S = 1.05 RHO = 3.00 + + Reflection spread Figure 29. Velocity structure CC 73 (2) Water layer over elastic layer, (CB). (3) Three elastic layers, (CC). With the exception of the water layer in CB, all velocity structures were designed with Poisson's ratio of 0.25. All traces were produced utilizing forcing function F5A (Fig. 30), attenuation parameter B3 (Fig. 31> and shot point location L2. At this shot point location, the surface dilatational forcing stress is applied at mesh points j=15 and j=16. This is marked by the letter "F" in Fig. 27 through 29. 1. The Sei sinograms Selected correlated synthetic seismograms are shown in Fig. 32 through 36. Figure 32 is a complete set of four synthetic seismic records for a close (reflection) spread of 25 surface detectors for velocity structure CA. The detector spread is shown in Fig. 27. Com- parison of these records shows the interdependence of the displacements and stresses. Actual land field records would include only vertical and/or horizontal displacements, or particle velocity, while marine records would only consist of the dilatational (pressure) record. The additional synthetic seismograms enable the proper phase arrivals to be quickly identified. A set of theoretical time-distance curves was developed for velocity structures CA and CB (Fig. 37), where it was assumed that the width of the transition zone was zero. There is good agreement between these theoretical arrival times and correlated linea- tions on the synthetic records. The first trace on the dilatation record corresponds to a normally incident wave. It bears a very close resemblance to traces obtained with the one-dimensional model for similar velocity structures. 74 0.1 0.7 1.3 f -0.15 Figure 30. Forcing function F5A y ^ 0 0 12 B = 0 v£> X vO CN 1 CN II \ II pq PQ 7 B = 2.6 Figure 3. L. Attenuation parameter B3 75 Figure 32 Reflection seismogram CA F5A B3 L2 Dilatational stress Shear stress Vertical displacement Horizontal displacement DX: 0.10 DY: 0.15 DT: 0.05 L: 71 M: 81 N: 800 IA: 2 ID: 1 JB: 16 J J : 1 Horizontal distance (d) versus time (t) Direct wave arrivals First reflections Multiple reflections Shear-dilatation conversion wave 76 o o o A* d_v. -Yui--\ ^.ux/\^r^A. — ^^a ■J A!\iy wv/sj -^ — J-44- [A- p^ A_^p=r,._.A^ r m Ttm 1 J±Al 4- /■■ -...v.-.^^.^^- -r ^.p-v^^^^---^^^-:,^-^ /-..'-!_ — k, vr_ t_ ^-^d^-'j 4=— /■^Jv^/v .'^v/v'-'^v p'^' '- -••""H'v-"rr-"-^ t ^V'/'-'f^^ -Mr-A i^ f \ / X J M-V^f -X/-- -K-^;/v- r^^iV^N^t^t^ ,^\/-> ^v-^ivf -v r ■ r -l ■ J /- \ / X£ -r x ■ - 'v u I vl f ' Af ■A7+: - > ' W -V^- '-i-';i~t=p-"\-/ - ^v-^vy-K tl -u— = u •u CO o ■H CO •u c8 CNI hJ m P3 k:. r ■;...■ ^.cd£ ^// : k — T^ • ly" ■4 / V 4- / 1' " - X r v. • ^1 -' .'"v. V^ ■ . A I - 1 ■■■Kk-fi. 1 c 0) a 0) u 03 .-I P- W •H Cfl O •H •U S-i CD > Csl ro PQ -l b0 O S CD •H QJ W C o •H j-J o 0) Q CO 0) M •H 80 Figure 33 Refraction sei sinogram CA F5A B3 L2 Shear stress DX: 0.10 DY: 0.15 DT: 0.05 L: 71 M: 81 N: 800 IA: 2 ID: 1 JB: 22 JJ: 2 Horizontal distance (d) versus time (t) 0 Direct wave arrivals First reflections Refracted arrivals 81 o 00 o CO t o CO b AT" A /■ / M v ^v — AA^ l-l CD 43 CO CM hJ CO pp < m fa I too O 6 W •r-l cu C/3 C o •r-4 4J O 0) i—l m cu erf CO cu 3 too •H fa 84 Figure 35 Bottom detector seismogram CB F5A B3 L2 Dilatational stress DX: 0.10 DY: 0.15 DT: 0.05 L: 71 M: 81 N: 800 IA: 20 ID: 20 JB: 16 JJ: 2 Horizontal distance (d) versus time (t) 0 Direct wave arrivals Surface reflected multiples Interface wave conversion 85 CNJ o to ■D o ro CO w 0) u •u a o 4J tO i-H •H Q CM hJ CO pa < m fa CQ C_) i u 60 O B w •r-i 0) CO S-i o •u u 4J LCl-CCCCCCCCC t- KGCCOCCCGGGGGCCGGOGO'JuC 6 CiuQGuUGGGGGGCCK.KCCGGGGGGGCCCGCCCCGGCGQGG 7 GGGuOGGGGuGCGLDGCGGGGJuCGOOGaGCCGCGGGGGG 5 gggccccccccgcccgOcucccgccccccgccgcgoogcc i GCQGCGCCGCCCGtGGfcGOCCOGCOCCGCOCCCCCGGGGC : j GCGGOGGCOCGGGuGGGGGGGOGCQCGGOGOCCOOGOOCG LI CGCCCCCCCCDCCrCOCCJt GCCCCGCCOCeCCCCCCCGCG 12 GGuGuGGGGCGCOUGGGUGGOGGGQCOGOOGCCGGCCQCG i 3 CGCClClGGCGCGCGCCCLGGCGCGCCGGGCCCCOGGGGG I -t GGGGGtCCCC GCGCGCCGCCCCCCCCCCGCCCCCGCCCGG 1 15 GGGCGGGGGCCGGCGGCGGGGGGCOCGGOOCCCGOQGGGG 1 iQ GG'uCGi I CGCCGGCGGt CGCCGCCOCCOCCOCCQSQOUGG 17 GGGGCGCGCCGGGCGl GUOGGGGCOCGGOOCCCCGGGGGC 1 ]^ GGOGGICGGCCCCCCGOCOGCOGCQOGOOOOGOGOGGGOG j * 19 CCCCCCCCCCCCGCCi*CCGCCCCCGCCGCCCCCCCCCGGO 21 GGUGGGUGGGGGuLG{3GGUdGGGGGOQGOGGCOGG-GGGGO\ 21 CCGGGCGCCGCCnCCCCCCCGGCCGCCOGOCCGCGCCGOG \ ' 22 GGGGGCGGnGGCGCGdGGGt CCCCLCCrCGCGCCCCCuCG \ ;2 GGOCGGGGGaCCGC GCCdOGC.GCCGGCGOGGCGCUGGOQG^^JX 2h GuJL CGGGLCGCCGGCCOGCGGCCOCGOOOGCGGGGGGOG T ? 5 LGuGGGCGuGGCGCCGCGGGOOCCGQGOGQQCCCCGCGGC 2o CCGGGGGC.GLCCCCGQCCGGOOCCOOOQCaOGQGOQG'JGa 2 7 GGGGCC.CCCGGGCGCGCGOCCCCCCCGCCGCCCCCGCGGG •< GGGGGGGQGGGGGCGC CGC GC GGCGCGQGGCCCCCCGGGO Z 9 CCGGGGCGCCGCGCCCCCGCCOCCGCCCGGCCCGGGCGGC _ - GGGGGGGGCCOGGGGGGCCCCOCCGOGGGGCCCCOGCOCC H lull CI GGGG'CCGCGGGGl.'CCGCCOGGGGGGCC0GGGGGC • - 2 GGGCCGGCCCCCCCCl CGGCl GCCGCCGGGGGCOOGGGGG _ - GCCcCLLLL"CCCLLGGCL.rCGGCCCCCGCCCCCCCCCGGC 3<* GGGCGGCCCCGCGCCC CCLGGGCGOGCOOOOGCQQQGGGO 33- LGCCCCCCGCGGGCG.C CCl llCC CCCCGCCCCCCCGbGGL 36 GGGCGGCGGCCCGCGnCOGGGOGCGGOOGCGCCGGGGG'CC 37 GGGCGCCCCCCCGXC LCCGCCGCCCCCGOCCCCGGGGOQG it1 GUuGGCGCGGCCCf CCCGGCCGCCCCCCCQCCCCCGCGCC 3 T CCQGGCGuGGCCGGGi GGCCGCCCGOCGOGCGCOGGOGGC 4c GCGCGGGCC CC CCGGC CGGCCGC C GGGGGGGCCOOGGaGG 4 1 OCUGCCCCCGGCGGGOCGGQC GCCGGCGGGCCGCOG'CGCC h2 GLGCCGCGGGCCGGGOCGGCl GCCGGCCGGCCGGGGGGOO -» 3 CEOCCCCCGCCCOGCGCCQCC CCCCGCCGCCCCCGGCGCC *4 UGJCGGGuGCGGGCGGCOGGGUCCCOCOGGCGCCGCCGGC + b GGGCCCCCCCGCCCGCCGGCCCCCOCGCCGCCGGGGGGGG 4 c GGGGGGGGGCaCGCGCCCGCCGCCCCCCCCCCCGGOGOCC 47 CCGCCCCUOGCCGCCGCGGCCGCGGCGOG.CCCCCGCCGGC 4 8 GGOGGCGOGCGCGCGGCGGGCCCCfJCGCGGGCGCQOGGO.G 4 ; c.rGCCLGi:;'GGG!JLJGGCnGCCGLGCGCCOCCCCCCLQGG o G L GGGCCCGGCGCGCGC CGGCCOGCOOGGGOGOGOGGOOOG 0 12 3 4 : c c c c i — **■ T Transition zone F Forcing function application point Figure 38. Portion of the printer contour plot showing generation of the dilatational wave. . ... ... (see Table X for scaled values) ■ ■- - . . 93 DILATATION: CB F5A B4 L2 K=91 FF i oppononr,Prnpnofi ononnnonrnpnnnrinQnooGO inn > j r, o\> \" "H bbonp ww zzfv bnhn •j •-• < r n COO i j on w p mh h oono "■G.TJPJ m n n \ rn r' n k ijW P7C KOOOQOC '"jj k m\ kkk mm n n N ph oonobo in pj km vj p < M 00 PS30 000000 11 HJ K" P J.\P KPNN"1 C P 0 J F JK COOOOO 1? 3 K f 0 N J 00 Hmm mht- H K 000000 i i ! : P n IM KK SOKGFG 0000000 I {* ^K KO M >>QM n «< MP h JK 0000000 lp h h mj n KG G ooonooo 16 P K KO 0 F M r)J FFNPQ H JK OPOOOPOn 17 0 KJ KT GH HA 0 QP N FG 00000000 l?. •: < JJ OP gkjzgp o mjt hjk oooonoooo 1"! ~n k hh v v 7 u mr khf G nnpnppooo ?0 n0 G 0 wCMWG H C F JK noOOOOOOOO ?l pqp Kjh ;-,r, mw wF OAC FG 000000000 ->-> Q00 X G FKM W UWNi 00 D£ H K 00000000000 ?'} 0000 K KKOV ° OMn jj KK 00000000000 ->u hnnr k n QM np KKKK nppnpOOOOOOO ■>*; oooo kkk co )Gc.o kkkkk onooooooooor ~>i Onpo kkk kkkkkk oooooooooooq >-r nonno kkkkk kkkkkkk ooonooooonooo ?8 oruion kkkkkkkkkkkkkkk nnonoooooooon 29 oobooo kkkkkkkkkkkkkk onnnoooooonooo "*C riOOOOO KKKKKK KKK KKK OOOOOOnOOOOOOO II 0000000 KKK KKKKKK GOOOOOOOOOOOOQO 3? 0000000 KKKKKKK 0000000000000000 3i oooooooo rcmooooonooooooo ?4 oooronooo '">Gnooooooopnoooono 35 oooooccoooo corocoooobooooooobo *f> nnonopooooooo nonononononnooooonnnoo 37 ooPDoroonrocooopooooooocoaoooooboooooobo is uoo noon cooooooocjconooooooooooooooooooooo 3 ooooooooooooooot ooroooocoroooooooooooooo 4 6 oooooooooooooooocooooooooooooooooooooooo 47 oonooooroooooooroooorcooocoooooooooooooo 4P 000 000000 0OGOO0C00OOOO00O0O00OOOOOOO0O0O 49 100 oono co rn nonno crirornorpcnopononooopooo 50 000000000000000c oobooooboobooooooooooooo 0.1 ? 3 4 0 G C 0 C T Transition zone F Forcing function application point Figure 39. Portion of the printer contour plot showing transmission of the dilatational wave into the lower media (see Table X for scaled values) 8^ 94 Table X. SCALE DF CCNTCURED QUANTITY I SYM(I) VALUE 1 A -0.0085417 2 -0.0081250 3 B -0.0077083 4 -0.0072917 5 C -0.0068750 6 -0.00 64583 7 0 -0.0060417 8 -0,0056250 Q E -0,0052083 10 -C, 0047917 11 F -0.0043750 12 -0.0039583 13 G -0.0035417 14 -0.0031250 15 H -0.0027083 16 -0,0022917 17 J -0,0018750 18 -0,0014583 19 K -0.0010417 20 -0.00 06250 21 0 -0.0002083 22 0.0002083 23 M 0.0006250 24 0,0010417 25 N 0.0014583 26 0,0018750 27 P 0,0022917 28 0.0027083 29 Q 0,0031250 30 0.0035417 31 R 0.0039583 32 0.0043750 33 S 0.00479 17 34 0,0052083 35 T 0,0056250 36 0.0060417 37 U 0,0064583 38 0,0068750 39 V 0.0072917 40 0.007708? 41 w OVEPSCALE 42 z UNDERSCALE 95 DILATATION: CB F5A B4 L2 K=200 i gogoooogooooopoooooooopopooooooooooooooooo 2 WH7GWWS77ZF qu OGOGnoooGooooooooooGG ^ j p p ophg nnGGnnnonncoonccnnnnnn 4 ZCWWW 7ZC GGGGGGGGOGGGOGGGGOOGOn 5 GNH K VyjF K GGGGGGGGGGGGGGGGOGGGGOGG ft J NM nM o nGGGGGGGOGGGGGGGGCOGGOOGG 7 0 R wWPO JK ^GGnOGGGGGGorooOGGGOGGGGG => N G G K GGGnnGOGGOnnGGGPOGGGOGOGO o M NJ PS F JK GOOGOGOPnOGOGGOGGGOnGOOGG if pqrg e 9mh gh k oggoonopogoooogoooogooppp 11 pppj p g ggh k gcggogogngonogogogpqgqogg 12 mkj qh gh pogggoooogggoooooogoooopoo H M R JH p K G CGGGPPGGGGGGGOPOPOQOGOOOCO 14 M K P GG K OGGGGPGGnOOOGGOGGGOGOGGGG 1 ^ QM G N G K GGnPGnGGnnGPnGOGPGGOQHC©«- 16 0 NMM M0J HJ K GOGGOGGGOGGPrGGJ^efn^riQPGn 17 K P NNNM HG JKK 00^O0G0Oh PGOPP KK OOOn0P PGP POO OOOPOPOOnPOO 2 7 POOOO KK OPOOGG GOO GOG GOPOCOGOOOGO ?3 00000 nnngO0PPO0O npp OOPPGPGOOOPn ?o nnnOG GnnccCGOOOG OHO ObOGOOOGOCOn 3,: QPGOGO OGGGGCGGPnP POO OGGGGOGOOOGn n GGGGPP POOGPOnOGOGG GOG GGOGOOGOGPnp 2? OGOOOO CPGOCCGGOHGr1 PGG OOGGGOOnGGGn *"* GOOOOO GGGGOGOnGGGG riGO OOnoCOGOOGGG ^4 nOOPOP OGPGGppPOO^Gn GPG GGGOOOOGOGOGG 3^ OOHGOG OOGOOCCGGPnGG POG OOOCOCOOOOOGG 16 CnoOOO GGnOGGPOGOGOGGG GGOO POOOOOGGGGPGP ^7 POOOOOOOOOOOOCOOOOOOOO 0000 GGOGOPOGGGGCG 38 n- T Transition zone H Head wave Figure 40. Portion of the printer contour plot showing the development of a dilatational head wave . (see Table X for scaled values) 96 SHEAR: CB F5A B4 L2 K=84 FF 1 OOOCOGOCCCaCOCOOOOOCCOCCOCOOOOOCOOOOOOOO 2 0000000000000000000000000000000000000000 3 OOOCOCOCCGOCOOOOCOOOOOOOOOGOOOOOOOOOOOOO 4 OOOOOOOOOQOCOOOOCOOOOOOCOOOOOOOOOOOOOOOO 5 ooooooooooo noooooonncooGoooooooooooooQOo 6 noocooooococoooooocoooocoooooooooooooooo 7 0OOOO00O0C00OGOOCOOC000GO00O0000GOG00OO0 8 OCOOOOOOOOOOOOOOGOOOCOOCOOOOOOOOOOOOOOOO 9 OOOGOOOOOOOCOOOOGCGOOOOCOOOOOOOOOOOOOOOO 10 0000000000000000000000000000000000000000 1 1 ooooooooocooocoocoooooocoooooooooooocooc 12 OOOO00OO000GO0OGCOO0000GO00O0O0C0O00OOOO 1 3 0000000000000000000000000000000000000000 14 OOOOOOOOOOOCOOOOCOOOGOOCOOOOOOOOOOOOOOOO 1 5 OOOOOOOCOOOCOOOOCOOOGOCCOOOOOOOCOOOOOOOO 1 6 GOOOOOOOOOOCOOOOCOOOOOOGOOOOOOOOOOOOOOOO 1 7 OOOCOOOOOCOCOOOOOOOCOOOCOCOOOOOOOOOOOOOO 1 a OGOOOOOOOOOOOOOOL'OOOOOOOOOOOOOOOOOOOOOOO 1 9 COOCOOOOOQOOOOOOOOOOCOOOOOOOOOOOOOOOOOOO 20 OOOOOOOOOOO CGOOOOOOOCOOCOOOOOOCGOOOOOOOG 21 000000 M NNM K JJ K 000000000000000 22 OOOOC MM NM KO JKJJ K 00000000000000 23 GQOOOOO 00 KO 000000000000000 24 000000000 G 0 000 0 0000000000000000 25 00000000 COGCO OGOOOOOOOOOOOOOO 26 00000000 00000 00000000000000000 27 OOOGOGOOO QOOOO 00000000000000000 28 OOOOOOOOOOO OOOGOOOG OCCOOOOOOOOOOOOOOOO 29 OOOOOOOOOCOCOOOOOOOOGOOGOOOOOOOOOOOOOOOO 30 OOOOOOOOOCOOOOOCOOOCCOOCOOOOOOOOOOOOOOOO 31 OOOOGOOOOOOOOOOGCOOOOOOCOOOOOOOOGOOOOOOO 3 2 OOOOOOOOOOOCOCOOOOOOOOOCOOOOOOOQOOOOOOOO 33 OOOOOOOOOCOCOOOGOOOOOOOGOOOOOOOOGOOOOOOO 34 OOOOOOOOOGOOOQOOOOOOOOOOOOOOOOGOOOOOOOOO 3 5 OOOOOCOOGOOCOOOOOOOOOOOCOOOOOOOOOOOOOOOO 36 OOOOOOOOOOOOOGOGGOGOOOOGOGOOOOCCCOOOOOOO 37 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOGOOOOOOO 38 OOOOOCOOCCOCOCOOCOOOOOOCOOOOOOOOOOOOOOOO 39 OOOCOOOOOOOCOOOGGOOOCOOCOGOOOOOCCOOOOOOO 40 OOOGOOOOOOOOOOOOOOGOOOOGOOOOOOOOOOOOOOOO 41 GOOOOOOOOOOOOOOOCOGOOOOOOOOOOOOOOOOOOOOO 42 OOOOOOOOOOO COOOOOOOOOOOOOOOOOOOOOOOOOOOO 43 OOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOO 44 OOOOOOOOOOOOOGGGOOOOOGCOOCOOOOCCOOGOOOOO 45 QOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 46 OOOOOOOOOOOCOOOOOOGCOOOCOOOOOOOOOOOOOOOO 47 OOOOOOOOOOOCOOOOCOOGOOOCOOOOOOGCOOOOOOOO 48 OOOOOGOGOOOCOGOOOOOOOOOCOOOOOOOOOOOOOOOO 49 OOOGOOOCOCOOOGOOOOOCOOCCOOOOOOOOOOOOOOOO 50 OOOOOOOOOGOOOOOOOOOOOOQOOOOOOOOOOOOOOOOO 0 12 3 4 ooooo T Transition zone F Forcing function application point Figure 41. Portion of the printer contour plot showing generation of a shear wave at the transition zone by the dilatational wave. (see Table X for scaled values) 97 SHEAR: CB F5A B4 L2 K=178 1 ? 3 4 5 6 7 R 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 29 29 30 31 32 33 34 35 36 37 38 TO 40 41 42 43 44 45 ^6 47 48 4Q 50 cpgococooocooooooooooooocoooooooooooooooooo oooooooooooooocooGoooooooooooooooocoooaoooo oooocooooooonooonooooooooooooooaaoooonooooo GOOOCOCOOCOOOGCOOOOOOOOOCOOOCOOOOOCOOCCOOOO oooooooooooooooooooooaoooooooooooocoooooooo CGOCCOCGGOOOCOCOOOOOOOOOOOOOOGGOGOCOOOOOOOG OGOOCGOOOOOGOOOOGOOOOCOOCGOOOOOOOOGOOOOOOOO GOOOCOCGGGGOGOOGOOOOOOOOOOOOOGGQGOOOOOOOOOO GGOGCOCOOOGGGOCOOOOOOOOOGOOQOOOOOOOGGGOOOOG GOOGCOCOOGOGOGCOOOOOOOGQOOOOOGGOOOCOGOGOOOG GGOGCGGOOOGOOOOGOOOOOOOOGOOOOOOOOGOOOOOQOOG CGCGCOCOOOOOGOOOOOOOGOOOGOOOCGOGGCCOOGOOGOO GGOOGOOOOOOOGOOOOOOOOOOGGOOGOOOOOOGOOOOOOOO OrOOCOCOOOOOQOCCOOOOOGOOOOOOOOOOOOOOOOOOOOO OQOOCOOOOOGOaOGGGOQaOOOGGOOOOOOCOOOGGGOCOOG QGOOGGOGCOOOCOGOOOQOaOOOOGOOOGOOOGOOOOOOOOO OOGGCOCCGGOCGOOOOOOOOGOOOOOGOOOOOOGOOOOOOGO OGOCCGOOOOOOOOGOGOOOOOGOOOOOOGOOOOCOOOGG COOOCOOOOOOOOOOOOOOOOOOGOOOOOOOOOOOOOOqSflOO onaococGoooQcooooooooooooooocooooccoogifoooc 0 00 JKKJJK JGD ZACE JK OOOOOSTJOOO G FFF CNPORR NMMQ WWW R MMMMM C 0 OGGGOOGOG OGOOCOOOO QOOOCOOC OOOGGOOO cooorooo GOGGCGC GG00G 0000 G 0000 OOHQO COOGOOOOOO OCOOCOOGOC 000000000 CQOOCOOOO GOOGGOOOO OOOOGOOOO OKKQ KJ 0 N 0 0 000 0 MMO K OGG GGOOOO M 0 KK 000 00 GO u H M M OGO 0 GGCQOO 000000 0000000 00 00 GO 00 G OGG GGCC OOOG aooocoooooooocc 000000000000 GC 00000000 OOOOGOOOOOO OG OOOOOCO OOOOCOOGOOO OG 0000000 OOOOCGOOOn 00 COOOCOOGO 00 OGCOCOOC OOG OGOOOOOOOOO oroocoooo 0 KK 0 KK 0 KK KK KK KK K 00000000 OG OOOOOOOOOOG 000000 00 0000000000 0000 oc 00000000000 OOG OOG OOG 000000 2 3 4 coo OOOCOOOOGOGO 0000000000000 0000000000000 oooocoooooooo 0000000000000 OOOOCGOQGOOOO OOOOCOOOOCOOO 0000000000000 0000000000000 0000000000000 K OOOOGOOOOOOOO OOOOGOOOOOOOG 00000000000000 ooooocooocoooo OOOOG00O0OOOOG 00000000000000 OOOOOGOOOOGOOOO OOOOOOOOOGOOOOO 0000000000000000 ooocoooocooccoooo 00000000000000000G OOOOOQOOOGOOOOOOOOGGOOOOOOO OOOOOOOOOOOGOOOOOOCCOOOCOOOO OOOOOOOOOOOCOOUOOOOOOOOOOOOOO > 0 0 0000 00000 OGQOO 00000 000000 0 0000 00 0000 00 0000 0000000 0000000 0000000 OGOGGOOGCGCOO 0000000000000 OOOOCOOGGOOOOO COOOOCGGOCOOOG 5 6 0 0 T Transition zone I Interface wave Figure 42. Portion of the printer contour plot showing development of an interface shear wave. (see Table X for scaled values) 98 V . SUMMARY The two seismic models presented here were developed primarily as tools to assist in marine geophysical field exploration. The models produce synthetic seismograms similar in format to that of actual field records in order to facilitate direct comparison. Although both models present two-dimensional records (horizontal distance versus time) , the reflection profiling model is made up of a series of one-dimensional model solutions. The complete solution simulates a moving source of vertically incident pulses where the velocity structure is slightly altered between transmissions. In contrast, the reflection/refraction model treats the propagation of energy in two dimensions. Each complete solution simulates a single pulse travelling through the solution space. The synthetic seismograms are produced as records from a series of detectors in a horizontal array. The models are based on numerical solutions to the wave equations in one and two dimensions in bounded domains. The equations are expressed in unsimplified forms which are not amenable to direct analytical solution. The stability of these numerical solutions and their convergence to the analytical solutions was investigated. Sta- bility and convergence criteria were established in terms of mesh ratios and wave velocities. In order to provide a wide variability of physical simulation, the solution domains are modeled by a number of easily varied non-dimen- sional parameters. These parameters are used to set the size and sub- division of the solution space as well as the distribution of elastic parameters and the shape and extent of the forcing function. 99 The structures modeled in this paper were, in general, based upon realistic distributions of the non-dimensional elastic parameters in marine environments. Some of these models have inherent resonance with the type and width of the forcing function used and the mesh sizes. Records are produced with some features which reflect this resonance. In particular, there is some tuning of the pulses in the narrow transi- tion zone in some of the models. Care should be taken when trying to extrapolate features found in these records to cover a more general case. 100 VI. FUTURE WORK The applicability of these models to a wide variety of studies was the primary consideration in their design. The interpretation of various velocity structures in this study was limited to a gross overview of simpler problems that might be encountered in field exploration. Al- though a full presentation of the use of all of the variable parameters was not attempted here, there are areas in both models where improvements could be made which would increase their variability. A first step in improvement of the one-dimensional model might be the inclusion of a variable depth source. It was felt that this was not warranted in the model due to the complexity of a forcing function at depth that could not be included as part of one boundary condition. The same improvement might be applied to the two-dimensional model. However, this would involve an additional order of magnitude of com- plexity and the model could no longer be called "simplified." A more reasonable improvement could be made which would remove the requirement that all transition zones be horizontal. The wave velocities and density could be put into the model as two-dimensional arrays instead of vectors. Terms involving the y-derivative of Lame's constants and of density would then be retained. This improvement would approximately double the main core storage required (to 550K bytes) and increase the main program run time by about one-third (to 24 min) . Although the attenuation parameters used in both models are con- sidered adequate for the purpose for which they were included (elimi- nation of boundary reflections), they are by no means definitive. 101 These models could be used to study the effects of various velocity attenuation functions by entering A(x) or B(x,y) as functions of other physical parameters (e.g. overburden pressure, Lame's constants or particle displacement). Only a single forcing function is presented for each model. Several functions were examined in the development of the models including a flat-topped pulse, spike pulse and "sine" function. The models could be used to study the reflection and/or refraction of a wide variety of forcing functions, limited only by the size of the time mesh increment (DT). In Section III, it was noted that an interface wave was observed in shear stress on the contour plot output. This model could certainly be used to study the various classes of such waves under a variety of conditions. Some preliminary work was done in generating particle motion plots from the vertical and horizontal displacement outputs available. Although elliptical particle motions were observed as pre- dicted I Alterman and Rotenberg 1969 J , limitations of distance from the generating point did not permit identification of the interface phases. Neither model is by any means limited to the homogeneous layer- transition zone structure presented in this paper. The velocity structure could consist entirely of transition regions of various gradients. This could be applicable to investigation of the effects of the gradients on reverberation. A collection of records is on file at the Naval Postgraduate School, Department of Oceanography (Code 58Ad) . 102 LUX • LU O ic LL) > t/) LU h-^Uj X •— i LU Z o> • h- K o »— 1 *-» o d. a. <. 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IP LP IP LP t LIST OF REFERENCES Alterman, Z.S. and Karal, F.C., 1968, Propagation of Elastic Waves in Layered Media by Finite Difference Methods, Bulletin of the Seismo- logical Society of America, v. 58, n. 1, pp. 367-398. Alterman, Z.S. and Rotenberg, A., 1969, Seismic Waves in a Quarter Plane, Bulletin of the Seismological Society of America, v. 59, n. 1, pp. 347-368. Bullen, K.E., 1959, An Introduction to the Theory of Seismology, Cambridge University Press. Constantino, C.J., 1969, Two Dimensional Wave Propagation through Non- linear Media, Journal of Computational Physics, v. 4, n. 3, pp. 147- 170. Dobrin, M.B., 1960, Introduction to Geophysical Prospecting, McGraw-Hill, Forsythe, G.E. and Wasow, W.R., 1967, Finite Difference Methods for Partial Differential Equations, John Wiley. Fox, L., and others, 1962, Numerical Solution of Ordinary and Partial Differential Equations, Pergamon Press. Grant, F.S. and West, G.F. , 1965, Interpretation Theory in Applied Geophysics, McGraw-Hill. Gupta, R.N., 1966a, Reflection of Elastic Waves from a Linear Transi- tion Layer, Bulletion of the Seismological Society of America, v. 56, n. 2, pp. 511-526. , 1966b, Reflection of Plane Elastic Waves from Transition Layers with Arbitrary Variation of Velocity and Density, Bulletin of the Seismological Society of America, v. 56, n. 3, pp. 633-642. Officer, C.B., 1958, Introduction to the Theory of Sound Transmission, McGraw-Hill. Sengbush, R.L., Lawrence, P.L. and McDonal, F.J., 1961, Interpretation of Synthetic Seismograms, Geophysics, v. 26, n. 2, pp. 138-157. Smith, G.D., 1965, Numerical Solution of Partial Differential Equations, Oxford University Press. Takeuchi, H. , 1966, Theory of the Earth's Interior, Blaisdell. Tooley, R.D., Spencer, T.W. and Sagoci, H.F., 1965, Reflection and Transmission of Plane Compressional Waves, Geophysics, v. 30, n. 4, pp. 552-570. 126 United States Naval Postgraduate School Computer Facility, 1969, Techni- cal Note 0211-03, Plotting Package for NPGS IBM 360/67, by P. C. Johnson. Wolf, A., 1937, The Reflection of Elastic Waves from Transition Layers of Variable Velocity, Geophysics, v. 2, n. 4, pp. 357-363. 127 INITIAL DISTRIBUTION LIST No. Copies 1. Defense Documentation Center 20 Cameron Station Alexandria, Virginia 22314 2. Library, Code 0212 2 Naval Postgraduate School Monterey, California 93940 3. Professor R. S. Andrews 5 Department of Oceanography Naval Postgraduate School Monterey, California 93940 4. Department of Oceanography 3 Naval Postgraduate School Monterey, California 93940 5. Oceanographer of the Navy 1 The Madison Building 732 N. Washington Street Alexandria, Virginia 22314 6. LCDR C. D. Lodge 2 P.O. Box 274 Pebble Beach, California 93953 7. Professor J. J. von Schwind 1 Department of Oceanography Naval Postgraduate School Monterey, California 93940 128 Security Classification DOCUMENT CONTROL DATA -R&D Security « lassification of title, body of abstract and indexing annotation must be entered when the overall report is classified) on oin A ^ ing AC Ti vi ty (Corporate author) Naval Postgraduate School Monterey, California 93940 it. REPORT SECURITY CLASSIFICATION Unclassified 2b. GROUP REPORT TITLE Simplified Numerical Models for the Generation of Synthetic Reflection Profiling Seismograms and Synthetic Reflection/Refraction Seismograms DESCRIPTIVE NOTES (Type ol report and, inctusi ve dates) tester's Thesis; (April 1970) *u tmoRiSi (First name, middle initial, last name) Charles David Lodge REPOR T D A TE Ipril 1970 7a. TOTAL NO. OF PAGES 128 7b. NO. OF REFS 17 • . CONTRACT OR GRANT NO h. PROJEC T NO 9a. ORIGINATOR'S REPORT NUMBER(S) 9b. OTHER REPORT NOISI (Any other numbers that may be assigned this report) 3 DISTRIBUTION STATEMENT :his document has been approved for public release and sale; its distribution .s unlimited. I SUPPLEMENTARY NOTES 12. SPONSORING MILI TAR Y ACTIVITY Naval Postgraduate School Monterey, California 93940 ABSTRACT Two numerical models were developed which generate synthetic seismograms in :ormats similar to those of records obtained in field exploration. The models iolve the hyperbolic wave equations in one and two dimensions by finite difference ipproximations in initially undeformed solution domains of transversely-isotropic .ayered media subjected to a time varying, dilatational forcing stress applied at he surface. A velocity attenuation term was included in the models to inhibit trong boundary reflections. The one-dimensional model produces synthetic reflection profiling seismograms or arbitrary horizontal or dipping layers. The two-dimensional model generates ynthetic reflection/refraction seismograms for horizontal layered media with rbitrary distribution of wave velocities, Poisson's ratio and density. Several ample records were produced for some representative velocity structures. The ynthetic seismograms were interpreted and gross correlation was carried out as f they were actual field records. D ,r»"..1473 N 0101 -807-681 1 (PAGE 1) Security Classification 129 A-31408 Security Classification KEY WORDS Numerical Seismic Model DD .'£" .1473 S/N 0101-807-6821 ROLE W T 130 Security Classification A- 31 409 Thes i s L7912 I Thesis L7912 c.l 117688 ^"Simplified numerical Jels for the 9«ne« • „ rtf synthetic «c f,ection P"*1'1"* seismograms and syn thetic reflection/ ^fraction seismograms 117688 L°d9S^plHied numerical models for the genera- tion of synthetic re flection P"-°f,Vn?n- seismograms and syn- thetic reflect. on/ refraction seismograms ,hesL7912 S^num^msaSJJiS 3 2768 002 12610 4 DUDLEY KNOX LIBRARY