Bulletin of the Seismological Society of America, Vol. 85, No. 6, pp. 1879-1887, December 1995

Simple Elastic Finite-Difference Scheme

by J i~ Zah radn fk

Abstract This article investigates which features of the elastic finite-difference schemes are essential for their accuracy and which ones allow simplifications. It is shown that the schemes employing the geometrically averaged parameters are more accurate than those using local material parameters, mainly when a discontinuity passes between the grid lines. It is also shown that the accuracy of the mixed spatial derivatives at the internal grid points does not degrade when the number of the implicitly employed stress values and the geometrically averaged material parameters decreases from four to two (the so-called full and short forms, respectively). The short and full forms give the same numerical results, while 50% of the arithmetic operations are saved with the short one. However, at the free-surface points, such a simplification is not permitted, and the full form should be used. Based on these results, a new simple elastic scheme (called PS2) is suggested.

Introduction

Several second-order finite-difference (FD) schemes for propagation of elastic waves have been suggested recently, e.g., by Kummer et al. (1987) and Sochacki et al. (1991) and compared by Zahradnik et al. (1993). The recent schemes are characterized by an increased complexity with respect to the older ones, such as that of Kelly et al. (1976). Our interest is to see which features are essential for the accuracy of the schemes and which can be simplified. The schemes differ mainly in the way how the material param- eters are treated (local or averaged values) and at which lo- cations the material parameters are taken. The latter problem is tightly connected to the approximation of the stress com- ponents, although the stresses do not explicitly appear in the calculation. The aim of this note is to address two questions: (1) Are the geometrically averaged material parameters pref- erable over their local approximations? (2) Is the accuracy of the second-order schemes dependent on the number of the material parameters employed? Answering these questions will allow us to construct a new simple elastic formulation.

FD Schemes

Consider the elastodynamic equations for displacement components. For simplicity, deal only with representative terms: the nonmixed derivative (3/3z)[a(Of/Oz)] and the mixed derivative (O/3z)[a(OflOx)]. Here, f stands for the dis- placement components and a for the material parameters. As the approximation of density is much less problematic than that of a, we keep the density constant throughout this note. A square grid with the spatial step size h is used. The time derivative is approximated by a standard second-order cen- tral difference with the time step size k.

Nonmixed derivatives. The nonmixed derivatives are ap- proximated here as in Boore (1972), Kummer et al. (1987), and Zahradnik and Hron (1992) (see Fig. 1):

~z (a OJ) - Oz ._. g o A / 2 - - , (1)

where the stress component g is introduced and differenti- ated. The integration of g/a = Of/Oz along the "southern" leg S introduces the geometric average as and provides

f s ~ dz as go, v2 -- - - -- (.fo,, - fo,o),

s a

h

as - f,_dz.a (2)

Similarly for go,-u2, introducing the geometric average aN along the "northern" leg N. Note that in equation (2), not only the mean value theorem was applied but also g(t/) with an unknown t /E S was further approximated by g0,u2, which is an important trick of this approach (not mathematically justified, as stressed by Kummer and Behle, 1982). Finally,

- - a --" Oz -~z -~ [as(fo,1 - fo.o) -- au(fo,o - fo,-0]. (3)

Using the geometrically averaged material parameters

1879

1880 Sho~ Noms

O, -1

&o,o

Figure l . The grid stencil used for approximating the nonmixed spatial derivative (O/Oz) [a(Of/Oz)] by the scheme of equation (3). The material parameters are geometrically averaged along the grid legs marked by ellipses S and N. The stress values (not explicitly used in the final schemes) are shown by diamonds, while the displacement values are indicated by circles.

a s, a N, instead of their local values ao.m, a o - m , has a long history (see the references above and therein) but a poor justification. We prefer that approach for two reasons. The first one is theoretical, saying that the local parameters vio- late the traction continuity at a discontinuity passing between grid lines, while the geometrically averaged parameters per- form well in that case. This can be shown by the Taylor expansion method similarly to equations (7) and (8) of Zah- radnN et all (1993). For this purpose, let us denote F z = Of/

Oz and ~ the fraction of the grid step expressing the distance from the boundary. Then, solving a 1D wave equation by the FD scheme with equation (3), and using symbols with and without prime for the two contacting media, the leading terms of the expansion yield

as(~F£ + (1 - ~)Fz) - aNF~ = 0

+ O(h) + O(k2h), (4)

where, from equation (2), a s = aa ' / [a~ + (1 - ~)a'] and a N = a ' ; hence,

a F z - a'F~ = 0 + O(h) + O(k2h). (5)

This equation is the first-order accurate approximation to the stress continuity condition at the interface, as desired. On the other hand, using ao, m = a, a0,-1/2 = a ' (for 0 -< ~ < 1/2), instead of a s , aN, the leading terms read

( a), a(1 -- ~ ) F z - a ' 1 - ~ - 7 F" = 0

+ O(h) + O(k ih) . (6)

Evidently, the violation of the stress continuity occurs for 0, increasing with ~ and a /a ' . For ~ = 1/2 and ao, u2 =

(a + a ' ) /2 , similar results can be obtained showing that the spurious stress discontinuity is maximum.

The second reason to prefer the geometrically averaged parameters is numerical, saying that the scheme employing the local parameters yields a numerical inaccuracy (in ad- dition to the usual numerical dispersion), not present in the schemes with the geometrically averaged parameters. To prove the latter, we perform the following numerical exper- iment.

E x p e r i m e n t 1. We consider a 1D model of a single layer over a half-space with a relatively large velocity contrast (200 to 865 m/sec, i.e., 1:4.3). The layer is excited by a plane wave vertically incident from below. Its time variation is fno(t) = sin(2~t/T) - 0.5 sin(4nt/T), t E (0, T), T = 0.4138 sec, with a predominant frequency close to I / T = 2.4 Hz. The free-surface motion is studied. The FD calculations are with the grid and time steps h = 4 m and k = 0.0027 sec. Therefore, at the predominant frequency and in the low-ve- locity material, the FD solution is sampled at 20 grid points per wavelength. Three cases are considered: the layer thick- ness of H = 12 m, i.e., the discontinuity coinciding with a grid line (~ = 0), and two positions of the discontinuity not coinciding with a grid line, H = 11 m, (~ = 1/4), and H = 10 m (~ = 1/2). Two FD solutions are compared in Figure 2. Solution 1 employs the derbzative (3) with the geometri- cally averaged parameters as, aN. Solution 2 uses also equa- tion (3), but, instead of as, aN, the local values of the param- eters are used, i.e., ao,1/2, a0_ 1/2. A reference solution 3, also shown in Figure 2, is the exact solution computed by the matrix method (Miiller, i985). In both FD solutions, the free surface is treated by the vacuum formalism (Zahradnik et

al., 1993, 1994). It means that the same FD formula as for the internal grid points is employed also at the surface points, while the material parameters and the displacement values above the surface are zero valued (a vacuum), i.e., with f0 _1 = a N = a o _ l / 2 = 0. Because a s = a0,1/2 in the surface grid point, the two solutions 1 and 2 operate with exactly the same formulas there. Also, the noureflecting bottom of the model is treated by the same formulas of Emerman and Ste- phen (1983) in both FD solutions.

Therefore, for ~ = 0, there is no difference between the two codes, and, correspondingly, they provide identical re- sults. As seen from Figure 2, they well agree with the exact solution, meaning that the grid is fine enough to keep the numerical dispersion negligible in this small model. Using the same grid step size for ~ = 1/4 and ~ = 1/2, the codes are unequal due to their interface treatment and the results differ. While solution 1 is again in a good agreement with the exact solution, solution 2 is not. This remains true also when halving the grid step sizes while keeping the same ~'s (not presented). These results show that the theoretically pre- dicted violation of the stress continuity when using the local parameters yields a significant numerical inaccuracy being

Short No~s 1881

10.0

0.0

-10.0

10.0

I I

0.0

' ' [ ' ' ' ' I ' ' ' ' I

0.5 1.0 1.5

0.0

-10.0

10.0

" . H = 1 1 m

~] ! / , .

' ' ' ' [ ' ' ' ' r ' ' '

0.0 0.5 1.0 1.5

0.0

-10.0

0.0

H = l O m

• , / /

' ' ' ' I ' ' ' ' I ' ' ' '

0.5 1.0 1.5 T i m e ( s e c o n d s )

I Solution 1 1 - - - Solution2

-]- Solution 3

Figure 2. Synthetic seismograms for the free-surface receiver of the layered model (experiment 1). The layer and the half-space velocities are of 200 and 865 m/sec, respectively. Three interface depths H are studied on the grid with the 4-m step: (a) H = 12 m, (b) H = 11 m, (c) H = 10 m. Three solutions are compared: solution 1 (solid line), the FD solution with the geometrically averaged parameters; solution 2 (dashed line), the FD solution with the local parameters; and solution 3 (crosses), the exact solution. The position of the interface with respect to the grid is shown as the heavy horizontal line,

added to the numerical dispersion. On the other hand, also in agreement with the theory, solution 1 is free of this in- accuracy, and it well represents the depth variation of the discontinuity between the grid lines.

What is behind the success of the geometrically aver- aged parameters? It is both the use of the elastic parameters

and the stress terms g ' s between the displacement grid points, as in the staggered velocity-stress methods (e.g., Vi- rieux, 1986), and in the so-called new formulation of Ste- phen (1988), but also it is how the stress terms are approx- imated, in particular, avoiding a simple factorization go, v2 =

ao,1/z( Of] Oz )o,1/2.

1882 Short Notes

For these reasons, we suggest use of the geometrically averaged material parameters in all the following 2D elastic schemes, although such an approach does not automatically guarantee the traction continuity for any position of a dis- continuity with respect to the grid lines. Full treatment of this problem is beyond the scope of this short note. (For discussion, see also Muir et al., 1992.)

Mixed derivatives. To model a "simple" and a "complex" second-order scheme, we approximate the mixed spatial de- rivatives in two forms, called the short and full forms. The short form starts from (Fig. 3a)

O {aCgJ~ Og . g o , 1 / 2 - - g o , 1/2 (7) ~x - Oz - h '

where, from the integration of g/a = Of/Ox, we get

f s Of dz Ox a s [ Of

h s~xdZ" (8) gO, l/2 "-- f s adZ

The geometrical average as, defined in equation (2), is kept here. The following approximation of f OflOx dz employs the same above-mentioned trick as used when approximat- ing f g/a dz:

as(Of) h - a s f l . l / z - f - l . u 2 g0,1/2 ...r_. h - ~xx o,1/2 h 2

• as = 4--h (f~o + fl,1 - - f - - l , O - - f--l,t)" (9)

Another altemative in equation (8) would be to expand Of/ Ox as a function of z before integrating over S (Kummer et al., 1987). Therefore, the approximation (9) is analogous to the zeroth term of the mentioned expansion, with the only (important) exception that Of/Ox is taken at (0, 1/2), not (0, 0). As the approach of Kummer et aL (1987) does not pro- vide better accuracy at a discontinuity than O(h), we suggest equation (9), which also gives O(h) (shown below) but is simpler.

Treating go,- 1/2 in a similar way, we finally get the short form as

- - a - [as(fl,o + fl,~ - f-l,o - f-~,,) Oz - aN(f~,-, + f~,o - f - l , - , - f~t,O)]" (10)

To investigate theoretically its behavior at an internal discontinuity, consider one of the 2D equations of motion for P-SV waves, e.g., the equation updating the horizontal component u,

O, -1

N

0 ,0

S

0,1

1,0

(a)

NW

O, -1

i" 1' NE

hO, O d I F •

1,0

(b) lhO, 1 •

Figure 3. The grid stencils used for approximating the mixed spatial derivative (O/Oz)[a(Of/Ox)]: (a) short form, equation (10), and (b) full form, equation (16). The material parameters are geometrically averaged along the grid legs marked by ellipses S, N, SE, SW, etc. The stress values (not explicitly used in the final schemes) are shown by diamonds, while the displace- ment values are indicated by circles.

0-7 (2+2#)7 . +Tzz 7zz oz/

- - = - -

+ OZ P Ot 2"

For simplicity, study an internal discontinuity passing through a grid point (i.e., the case of ~ = 0). Let the dis- continuity be horizontal at least within a plus/minus one grid step around the grid point (a locally horizontal discontinu- ity). In that case, for the scheme given by equations (3) and (10), the highest-order Taylor expansion terms correspond-

ShonNo~s 1883

ing to the derivative (O/Ox) [2 (Ow/Oz)] vanish because the contributions weighted by 2e = 2w = (2 + 2')/2, for the "eastern" and "western" legs, cancel each other. The deriv- ative (O/Ox)[(2 + 211)(Ou/Ox)] vanishes for the same reason. Then, the interface behavior is determined solely by the re- maining derivatives, (O/Oz)[fl(Ou/Oz)] and (O/Oz)~(OwlOx)]. Denoting Wx = Ow/Ox, U z = Ou/Oz, etc., and assuming fur- ther W~ = W" (Zahradm~ et al., 1993), the scheme approx- imates the continuity of the tangential stress component with first-order accuracy:

/~uz +/~w~ - / ~ ' u ; - /~'w" = 0 + O(h) + O(kZh). (12)

The analysis of the equation of motion updating the vertical component proves the continuity condition for the normal stress component.

However, at a horizontal free surface, treated by the short form (10) in the vacuum formalism, the leading terms of the Taylor expansion are

1 tzU: + /zWx + ~ 2Wx = 0 + O(h) + O(k2h), (13)

1 - - '~ [ase (fl.o + f1,1 - fo,o - fo,1)

+ asw (fo.0 + f0.1 - f~l,O - - f~l,1)

- - a N ~ ( f ~ , - 1 + f l , 0 - - f o , - i - - f0 ,o)

-- amy (fo,-1 + f0,0 -- f - 1, 1 -- f-l,o)].

(16)

This form is quite analogous to the schemes of Sochacki et aL (1991) and Zahradm~k et al. (1994) with the only excep- tion that the material parameters ase, etc., that are geomet- rically averaged, as in equation (2), appear here as sche- matically shown in Figure 3b. The close similarity with the referenced schemes is appealing, as they were based on the integral formulation of the equations of motion and, there- fore, can be considered as physically justified. On the other hand, the scheme (16) is more complex than (10), and we would like to know if there is enough reason for employing such a complexity.

The Taylor expansion of the scheme defined by equa- tions (3) and (16) at the internal grid point located at a (lo- cally) horizontal discontinuity shows again that the traction continuity condition is satisfied with order O(h), as in equa- tion (12). Moreover, using now the vacuum formalism for a free-surface point, the present scheme gives the vanishing tangential stress component, as desired:

meaning that the stress-free condition is violated by the term 1/4 2Wx. It comes from (O/Ox)[2(Ow/Oz)], which, despite of the equality 2 e = 2 w = 2/2, does not vanish with the ap- proximation (10).

The full form of the mixed spatial derivative starts sim- ilarly to equation (11), but, instead of the two g's (i.e., two stress values), four g 's are employed (Fig. 3b):

- - a

oz =Tz

1

½(g-1/2,,/2 + g~/z, v2) - ~(g ~,2.-1,2 + gl/2.-l/2) (14)

gzu~ +/~wx = 0 + O(h) + O(k2h). (17)

The analysis of the equation of motion updating the vertical component proves that the normal stress component van- ishes, too.

As the next step, we want to numerically verify the the- oretically predicted similar properties of the full and short forms at the internal grid points and their dissimilar prop- erties at the surface. As the preceding analysis was for a 2D wave field at a (locally) horizontal discontinuity, we start with a sedimentary basin model having an extended hori- zontal part of its bottom (experiment 2). Moreover, that model was already studied by several other methods. Next we will proceed to a more complicated basin model (exper- iment 3).

Similarly to the short form, we take

. asE(OJ) h - ase g,/2,,/2 - h ~xx ~/2,,/2 -h- (f~,,,2 - fo,~/2)

• __ ase t¢ - 2 h w ~,0 + f1,1 - f0,o - f 0 , 0 ,

etc., for the other g's, which finally yields

(15)

Experiment 2. For this model, we use a symmetrical sedi- mentary basin (Fig. 4) previously studied by the Discrete Wavenumber Boundary Element (DWBE) method of Ka- wase and Aki (1989). The basin is excited by a plane P wave vertically incident from below: f ,c = (2zero t2 - 1) e x p ( - zc~ot2), with the predominant frequency f0 = 0.5 Hz. The space and time grid steps are h = 62.5 m and k = 0.0075 sec, respectively. So, inside the basin and at the predominant frequency, the FD solution is sampled at 32 grid points per shear wavelength. The overall size of the grid (representing one-haft of the symmetrical model) is 131 × 65 space points

1884 Short Notes

E

.__.

0 o o

0 N

0 "1-

Horizontal coordinate (km) -6 -3 0 3

Vp = 2 km/s, Vs = 1 km/s

o -2 Vp = 5 km/s, Vs = 2.5 km/s

5 10 15 20 Time (seconds)

I - - S°lut]°nl I ( a ) '4= Solution 2

25

• 4

o o

c o N t- O -r-

N 10 15 \ 20 Time (seconds) \

I - - Solution 1 Solution 3

(b)

2 5

Figure 4. Synthetic seismograms for the free-surface receivers of the basin model shown in the inset (experiment 2). Three FD solutions are compared: (a) solution 1 (solid line) employing the full form of the mixed derivative at the surface and the short form elsewhere and solution 2 (crosses) employing the full form everywhere; (b) so- lution 1 (solid line) as in (a) and solution 3 (crosses) employing the short form every- where. Note the coincidence of solutions 1 and 2 in (a).

and 3467 time levels. For reducing spurious reflections from the grid edges, the conditions of Emerman and Stephen (1983) are used.

The synthetics are studied for the vertical component only, the horizontal one providing analogous conclusions. We already computed this model in Figures 5 and 6 of Zah- radnN e t al. (1993) by the method essentially equivalent to the full form presented here (with the only difference in the treatment of the material parameters), and a good fit with the DWBE method was found. Then we switched to the geometrically averaged parameters (full form of the present note), and a good fit has been found, too. This was also the result when perturbing the model by shifting the horizontal bottom half a grid step down (~ = 1/2). The observed small effect of the material parameters treatment was interpreted as due to the smaller velocity contrast compared to Figure 2. To save space, these results are not shown. Therefore, we consider our full form FD solution checked by the DWBE

method and concentrate here on the comparison between the full form and the short form.

The present experiment compares three FD solutions: solution 1 employs the full form at the free surface, while the short form is used at the internal grid points; solution 2 employs the full form everywhere, including the free sur- face; and solution 3 uses the short form everywhere, includ- ing the free surface. At the free surface, the vacuum for- malism is used for all. The three solutions are performed on the same grids specified above. As seen from Figure 4a, solutions 1 and 2 are identical, while solutions 1 and 3 (Fig. 4b) show significant discrepancies. Note that the agreement between solutions 1 and 3 occurs despite the fact that the regions in which the full and short forms are used, respec- tively, covers the entire basin and that the wave field is lat- erally varying. On the other hand, the discrepancy between solutions 1 and 3 occurs although the codes differ only at a single top row of the grid only (i.e., at the free-surface

Short Notes 1885

points). Before making conclusions and recommendations, we proceed to a more complicated, larger model.

Experiment 3. It is a basin model with a mostly nonplanar bottom, filled by horizontally layered sediments (Figs. 5a and 5b). It is excited by a plane S wave vertically incident from below, f ,c(t ) = (~2 _ 7 ) e x p ( - 7 ) , 7 = re2( t - to) 2/t2, with the predominant frequency at 0 Hz. The space and time grid step sizes are h = 40 m and k = 0.00425 sec. The FD solution can be considered accurate enough up to 1 Hz, where, in the material of the lowermost velocity, the solution is sampled at more than 25 grid points per shear wavelength. The overall size of the grid is 516 × 91 space points and 9000 time levels. For reducing spurious reflections from the

artificial grid edges, the conditions of Emerman and Stephen (1983) are used. Both components provide analogous re- suits; the vertical one is presented because of the remarkable basin-induced surface waves. The same three solutions 1 through 3 as in experiment 2 were used. In spite of the in- creased lateral heterogeneities and the larger model size, compared with experiment 2, the results confirm the preced- ing ones: the identical solutions 1 and 2 (Fig. 5a) and the unequal solutions 1 and 3 (Fig. 5b).

In te rpre ta t ion and a N e w FD S c h e m e

The identity of solutions 1 and 2 agrees with the theo- retical prediction that the full and short forms of the mixed

o 8

'g I

20.00

16.00

12.00

8.00

4.00

0.00

0.00

0

~ 2 121

3

Horizontal coordinate (km)

0 5 10 15 20

' ; , :~:Vs =. .~; '7 ; :57~7i : : :

Vs = 2.800 km/s

(VplVs = 2 everywhere)

10.00

- - Solution I |

-~- Solution 2

§

!

20.00

16.00

12.00

8.00

4.00

0.00

20.00 30.00 Time (seconds)

(a) I - - S O l u t i o n ,

4- Solution 3

(b) Figure 5. The same as in Figure 4 but for a more complicated, larger model shown in the inset (experiment 3). Note again the coincidence of solutions 1 and 2 in (a), kept throughout the entire 9000 computed time levels.

1886 Short Notes

spatial derivatives are of the same behavior at the internal discontinuities. As the theoretical considerations were for a (locally) flat discontinuity only, the numerical results for the laterally varying models serve as a generalization. The prac- tical implication is that using twice as many material param- eters and stress values (g) and twice as many arithmetic op- erations in the full form compared with the short form has no effect. Obviously, this result encourages the use of the short form for its simplicity.

There is, however, also another reason for preferring the short form, though not apparent from the present examples. When the full form is used at a horizontal discontinuity pass- ing in the middle between two grid lines (~ = 1/2), the scheme becomes unstable, mainly in models with large ve- locity contrasts (e.g., 1:5). In general, the full form can be driven into instabilities (independently of the choice of h/k) much more easily than the short form. As an illustration of typical numerical problems with a complex scheme, see the discussion of the so-called SGES scheme, analogous to the present full form, in Zahradnik et al. (1993, 1994). Many other examples could be listed where slight changes in the material treatment, the nonreflecting boundaries, and the other features of the FD methods may serve as the instability triggers, when using complex second-order schemes.

The discrepancies between solutions 1 and 3 are also in good agreement with the Taylor expansion in which we showed the short form of the mixed derivative (when used in the vacuum formalism) to violate the stress-free condition, while the full form was shown to work well. Therefore, the practical recommendation for the free surface is to use the full form in the vacuum formalism. The only numerical problems we detected in some experiments were instabilities starting in a surface point at which an internal discontinuity reached the free surface with a slope smaller than 1:4 be- tween the first and second grid row. Such problems were avoided by a modification of the model between the two uppermost grid rows. Another alternative would be to treat the gentle "rock outcrops" locally by the short form, while using the full form at the remainder of the free surface.

Finally, we suggest a new elastic FD scheme (called PS2) as follows. The nonmixed spatial derivatives in form (3) are to be used everywhere. The full form of the mixed derivatives (16) is to be used at the free surface, together with the vacuum formalism. The short form of the mixed derivative (10) is to be used everywhere except the free sur- face. The PS2 scheme is equivalent to what was called "so- lution 1" above. For the complete PS2 formulas, see Zah- radnik and Priolo (1995).

Efficiency. An important feature of any FD scheme is its efficiency, given by the number of grid points, time steps, and arithmetic operations per a point, needed to propagate a wave a given distance with a prescribed accuracy. Using, for example, a kinematic criterion of a quarter period permis- sible error in the arrival time, the efficiency of several sec- ond- and higher-order FD schemes has been investigated by

Sei (1993). As regards the PS2 scheme, the number of nec- essary grid points and time steps is controlled by the standard dispersion and stability relations for second-order explicit schemes (Alford et aL, 1974; Sei, 1993). In this respect, of course, the PS2 scheme cannot compete with the advantages of the schemes fourth-order accurate in space, where the main saving is due to the less pronounced dispersion, i.e., due to the decreased number of the grid points.

Concentrating on the comparison between the short and full forms, it is of interest to evaluate their respective effi- ciency. As a basic fact, we employ the numerical result that the dispersion properties of solutions 1 and 2 are the same, even in complex 2D models (cf. Figs. 4a and 5a). Therefore, the only difference between the two solutions is in the num- ber of operations per grid point. As this number for the short and full forms is 1:2, the short form is 50% more efficient than the full form. Of course, such a saving can be docu- mented only by measuring the computing time directly in the subroutine realizing the FD scheme, because most of the other code operations (the subroutine calls, the nonreflecting boundaries, the output operations, etc.) remain the same when switching from the full to the short form. Also bear in mind that the discussed algorithms differ in heterogeneous media only. If homogeneous parts of the model (e.g., below the basins) are treated by the standard second-order formulas for mixed derivatives with constant coefficients, the overall time saving of the short form may be negligible (only 4% saving for the model of Fig. 5a, measured on PC486/66, where the total computing time was 17 hr).

Anyway, we believe that our result of the numerical equivalence between the full and short form is important. It implies a caution that new algorithms (e.g., fourth-order or 3D) should not be developed from relatively complex sec- ond-order schemes, as the complexities involved do not yield the accuracy improvement. Even more, they often yield unnecessary numerical problems.

The recommendation to use the geometrically averaged parameters instead of their local values does not seriously degrade efficiency. The parameters are computed in advance and stored in six arrays (three arrays for 2, #, and 2 + 2¢t for all horizontal legs of the grid and three arrays for the vertical legs). For the algorithm applicable for computation of the averaged parameters in polygonal block structures, see Zahradnfk and Hron (1992). The time needed to compute the parameters is usually negligibly small. For example, in experiment 3, the parameters required only about 0.3% of the total computing time.

Even simpler scheme ? The short form of the mixed deriv- ative we suggest is still more complex than that proposed by Kelly et al. (1976). However, with the latter scheme, the Taylor expansion method would give the same negative re- sult as in equation (13) of Zahradnfk et al. (1993), showing that the traction continuity would be violated even when a horizontal interface coincides with a grid line. A numerical inaccuracy, most likely associated just with this problem,

ShonNo~s 1887

was illustrated in Figure 3 of Sochacki et al. (1991). So far, an elastic scheme of reasonable properties simpler than PS2 is not known to the author.

Conclusion

As recent second-order elastic schemes are character- ized by a large complexity, we have studied which features of the schemes are essential for their accuracy and which ones allow simplifications. The results are as follows:

1. It has been shown that the schemes employing the local approximations to the material parameters have a nu- merical inaccuracy (Fig. 2), appearing in addition to the usual numerical dispersion, when a discontinuity passes between the grid lines. The geometrically averaged pa- rameters were shown preferable.

2. It has also been shown that the accuracy of the mixed spatial derivatives at the internal grid points does not de- grade when the number of the implicitly employed stress values (i.e., grid legs with the averaged material param- eters) decreases from four to two. The short form of the mixed derivative (10) gives the same results as the full form (16), and 50% of the arithmetic operations are saved. Moreover, the short form avoids instabilities en- countered in some models solved with the full form. However, at the free-surface points (when treated in the vacuum formalism), such a simplification is not permit- ted, and the full form (equation 16) should be used. These results apply even for strongly varying 2D wave fields, as in Figures 4 and 5.

3. Based on results 1 and 2, a new simple elastic scheme (called PS2) is suggested. The nonmixed spatial deriva- tives (equation 3) are used everywhere, and the mixed derivatives in short form (equation 10) are used at the internal grid points, while the full form (equation 16) and the vacuum formalism are employed at the free-surface points.

A more general implication is that relatively simple sec- ond-order schemes are competitive compared with the com- plex second-order ones. New schemes that are fourth-order accurate in space and/or 3D schemes should not be devel- oped from the complex second-order ones. The complexities do not necessarily improve accuracy and often yield addi- tional numerical problems.

Acknowledgments

The author thanks Dr. P. Moczo and the reviewers for their critical read- ing of the manuscript. Financial support was provided from the Charles University grant GAUK-321, the Czech Republic grant GACR-0507, the NATO ENV1R.LG 940714 grant, and the NATO Science for Stability GR- COAL grant.

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Department of Geophysics Faculty of Mathematics and Physics, Charles University V Hole~ovi6k~ich 2, 180 00 Praha 8, Czech Republic

Manuscript received 4 October 1993.