IC/97/178

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ANALYTICAL DERIVATIVES OF EIGENFUNCTIONS, ENERGYINTEGRAL AND GROUP VELOCITY FOR P-SV WAVES

Zhijun DuDipartimento di Scienze della Terra. Universita degli Studi, Trieste, Italy,

Department of Geological Sciences, University of Durham,Durham DH1 3LE, United Kingdom1

andInternational Centre for Theoretical Physics, SAND Group, Trieste, Italy,

Giuliano F, PanzaDipartimento di Scienze della Terra, Universita degli Studi, Trieste, Italy

andInternational Centre for Theoretical Physics, SAND Group, Trieste, Italy

and

Ludvik UrbanDepartment of Geophysics, Charles University, Prague, Czech Republic.

ABSTRACT

We present a fast and accurate analytical procedure to compute partial derivatives ofenergy integral and eigenfunctions with respect to the structural parameters. The methodadopts the P-SV waves modal formalism for laterally homogeneous layered structure.In addition, we propose an asymptotic fast method to compute analytically the groupvelocity derivative with respect to structural parameters. These developments allow todevelop an efficient algorithm for computing the differential seismograms, which can beused in waveform inversion.

MIRAMARE - TRIESTE

October 1997

1Present address.

1. Introduction

Waveform inversion has become increasingly useful in the investigation of the

earth seismic velocity structure, aided by the large quantity of accumulated broadband

waveform data. Prior studies {e.g. Nolet, 1990; Gomberg and Masters, 1988; Zielhuis

and Nolet, 1994) have demonstrated that it is possible to obtain reliable layered earth

velocity structure by matching the seismogram waveforms. Such studies have used

either a trial and error technique or a linearized inversion technique that calculates

the differential seismogram, dehned as the partial derivative of a seismogram with

respect to the structural parameters, using a numerical finite difference scheme, which

is rather time-consuming, and suffers from numerical noise and instability. An efficient,

analytical method to compute the differential seismogram for the layered structure is

highly desired.

A key step in the development of an accurate and computationally stable analytical

method to compute modal summation differential seismogram requires the efficient

computation of the partial derivatives of the phase and group velocities, of the

cigenfunctions and of the energy integral with respect to the structural parameters. To

this end, Urban et al. (1993) have developed a fully analytical algorithm for obtaining

the derivative of the phase velocity, fn the first part of the paper we present an efficient

technique to compute analytical partial derivatives of the eigenfunctions and of the

energy integral with respect to the structural parameters for P-SV wave modes (Knopoff,

1964; Schwab, 1970; Schwab et al. 1984). In the second part of the paper, improving

Rodi et al.'s hybrid approach (Rodi et al., 1975), we develop a fast asympotic method to

compute analytically group velocity derivatives with respect to structural parameters.

2. Derivatives of eigenfunctions and energy integral

We consider the computation of the derivatives of the eigenfunctions, i.e.

displacement-depth and stress-depth functions, for P-SV (Rayleigh-wave) modes with

respect to the structural parameters, /? (S-wave velocity), a (P-wave velocity) and p

(density).

With the notation of Schwab (1970) and Schwab et al. (1984), the evaluation of

the partial derivatives of the eigenfunctions u(z) (radial displacement), w(z) (vertical

displacement), o{z) and r{z) (normal and tangential stress, respectively), is reduced to

the determination of the partial derivatives of the layer constants Am, Bm, Cm and Dm

for the layers above the homogeneous half-space, and the constants An and Bn for the

deepest structural unit.

We compute the derivatives of the elements of the (1x6 ) matrix, appearing in the

basic interface-matrix multiplication of Knopoff :s method (Schwab and Knopoff, 1972):

[Um+\- iVm+\ Wm+1, Rm+\ iSm+1, -Um+1] (1)

where U\ V\ W\ R*, Sl are the i-th interface elements in the fast form of Knopoff's

method for Rayleigh-wave computation.

For the first interface and a continental structure the elements are:

" l ) (2a)

V° - 0 (2b)

W° = ( 7 l - l )2 (2c)

R» = 7 l (2d)

SQ = 0 (2e)

where 71 = 20ljc2, >3i is the S-wave velocity in the upper layer, and c is the phase

velocity. The derivatives of (2) are:

( 3 a )

)dii (3c)^ ^ (3d)dpj dpi

^-S° = 0 (3e)dp

where the derivative | ^ {pj = ,8j, aj or pj) corresponds to:

\.0i dc dji 48i 4,3? dc . , ,~T~T,— [when j •£• 1} = —r — [when j = 1) (4a)

2 ^ (4b)

^7i 4/3? 5c ,, .

and 4^- (pj = fij7 otj or pj) can be computed with the method described in Urban et al.

(1993).

The next step consists in the evaluation of the derivatives for the quantities ê .

(k = 0 , 1 , . . . 16) and C^m) (k = 1,2,... 15), given by Table 2 of Schwab (1970) or Table 1

of Urban et al. (1993), which are the elements that make up the rn—th (m > 1) interface

(1 x 6) matrix (equation (1)).

For the m—th layer the derivative with respeet to j3j, aj and pj of

for (j 7̂ m,j ^ -

for (j = m)

for (j = 771+1)

cda 7

dc for

for

for

m.j ^ m + 1)

T , , - , . c (ml (ml , (in)

The derivatives ot t2 , t\ and e4 are:

and

where pj = 5j. aj and pr

The quantities Q" are:

1)

for (j = m)

Am)(=2 = cos

d m ) =,im) _

Vramy

(5a)

(5b)

(5c)

(6)

(7)

(8)

(9)

sin

where Pm = dm • coc/c • rftm, Qm = dm • ucjc • ram . dm is the thickness of the m-th layer,

and

if m < n

if (c>W

if (ĉ am 1 cĉ

i : ^ # ^ for (j- =

(10)

(11)

(12)

(13)

(14a)

(14b)

z-c(nZs~C) for__ I -lam «m

1

where in equations (14a) and (14b) d = ^ , while in (14c) and (14d) d = ^ , and in

(14e) and (14f)

d (m) _ J fjr^SinQm+rpm-^QmCOftQm JOT (j^TTlAm] I ii*- Tar,, t " t >-""- OOi ^ " " " " * •' \J I ••• I t i G \Q = \ i m _ (16e)

) L .P̂ -. .. -̂ CITI {^} \ .. fn't* (i —A m l

;cosQm — c as' f ~— sinQm)^— /or (j = m)

where c' denotes the derivative of the phase velocity with respect to pj.

Similarly, the derivatives with respect to OLJ and p7 are

(17a)

d C(-) A Q i Q m ( l 7 b )

for (j ̂ 7n)•^Tji - v-'"* —j (~\ 70^

cjc'am-c) 1 • p i _3_p f»nC p fm. fi — m l̂

(17d)^ - for (j = m)

(17c)

where c' denotes the derivative of the phase velocity with respect to a,-.

(17f)

(18a)

•4~QmSmQm (18b)dpj

tmj-_PmcosPm (18c)

-^J-*,P.)J- (IM)

C"0 ^ Q + Q Q m (18e)

cosQra - ^ - L s i n g m ) - i - (18f)

where d denotes the derivative of the phase velocity with respect to pj.

The other derivatives of e™ (k > 4) and Q (^ > 6) can be computed easily from

4 m ) (k = 1,2,3 and 4) and Cfcm) (k = 1,2,... 6) (see Table 2 of Schwab (1970) or Table

1 of Urban et al. (1993)).

This allows us to express the derivatives of the interface-matrix elements

Op3 dPj

(where Pj = j3j,

phase velocity derivatives with respect to frequency is performed numerically, which is

computationally expensive because it requires the determination of additional nearby

roots for each mode and at each frequency. In the following, we propose an asymptotic

fast method, which allows the analytical calculation of the group velocity derivatives.

It is well known that group velocity, u, is equal to the partial derivative of frequency,

u, with respect to the wavenumber k:

u = -1 tjj dc I (20)

Using the implicit function theory, the partial derivative of group velocity with

respect to the model parameter p.j (pj = !3j. ctj, pj) can be written as:

du = u/2_u\dc_UJ c \ c) dpj

u1 d f dc(21)

where the partial derivatives of the phase velocity with respect to pj can be computed

analytically with the method described in Urban et aZ.(1993). Instead of computing

JL foe J numerically (Rodi et al, 1975), we compute it analytically as follows.For a given Rayleigh-wave mode, the expressions of -§^-\w (Levshin, 1989) are

dch k dz

dc

da.j uh

2ft

k dz

dc c 2 - b / ? + v22]

(22a)

(22b)

(22c)

where yi = ~ ^ , V'i

where u* =

3^f and 7j is the energy integal given by

r \ l 2

w(0)\u*(z)}2}[w(0)\ j dz (23)

10

The eigenfunctions can be represented through a linear combination of exponential

functions (Haskell 1953; Levshin, 1989) which, at a given angular frequency, LJ, can be

written as

exp ±i-d.mr0cr to

m\ and exp \±i—dmrc\ V c

(24)

Therefore. ^ (equation (22)) is a linear combination of exponential functions.

Exponential function (24) can be approximated through a series expansion, that

when the argument is small, may correspond to a low order polynomium.

In the actual calculations, for each mode we are given the analytical values of

at a discrete number of frequency points, w; (I = 1, 2 , . . . , A"), and we choose to

approximate locally the derivatives y^ using a cubic spline polynomium:

dc= ajuJ3 + + c\bj + (25)

where the coefficients a;, &;, Q and di are determined from dodp, at each frequency,

using the two adjacent frequency values (i.e. w;_! and UJI+I) by imposing the smoothness

of the first derivative and the continuity of the second derivative both within the interval

and at its end points (Press et at, 1988). To properly constrain the interpolation, at

the two end frequencies of each mode Rodi et al.'s method is applied.

Therefore, if A.LJ = U>I — W;-i = i is the frequency interval used in the

computation of the spectral quantities (e.g. Panza, 1985), recalling that the maximum

value of jr.9m| or \ram\ is equal to 1, the error in the calculation of -j^ I J^-

upon the quantity

dmc

depends

(26)

around each frequency point w/.

For Aw = 0.005 hz, 3 digits precision of j can be achieved satisfyingJ \Pj

the condition Aw

can be achieved when

< 0.01 or dm < 2c, while for Aw = 0.0005 hz, 4 digits precision

< 0.001 (or dm < 2c).

11

4. Tests and results

The upper 250 km of the structure used to generate the P-SV waves spectrum using

multi-modal algorithm (e.g. Panza, 1985) is shown in Fig. 1. The notable difference of

our model with respect to PREM is that the Moho depth is at 38 km. Here, the S-wave

velocity increases by about 0.6 fan/s to model the sharp change in the elastic properties

of a continental structure between crust and upper mantle. The low velocity channel

occupies the depth range from 96 to 245 km.

In Figs. 2 and 3 the analytical calculations are compared with the numerical

evaluations, for the fundamental mode at 0.025 hz. The small dots correspond to the

analytical calculations, whereas the larger open squares indicate the results of the

numerical calculations. The numerical derivatives are determined by adopting a first

order centered numerical differentiation; therefore, each numerical derivative involves the

calculation of two spectra and guarantees some stability in the numerical differentiation,

when the numerical increment does not exceed 0.02% of the perturbed parameters.

In general the agreement between analytical and numerical calculations is of at least

3 digits, therefore, in the following figures we only show the results of the analytical

calculations.

In Figs. 4 and 5 we show examples for the fundamental mode at 0.1 hz. Fig. 4

represents the partial derivatives of the eigenfunctions, Fig. 5 those of phase, group

velocities and energy integral. The results for the higher modes, the first and the ninth

higher modes respectively, at twTo different frequencies of 0.2 and 1.0 hz, are shown in

Figs. 6 to 9.

Since the amplitude of the derivative depends on the layer thickness, the derivatives

shown in Figs. 2 to 9 are normalized with respect to this parameter.

Depending upon the wave penetration, non-zero derivatives reach different depths,

which are mode and frequency dependent. Only at shallow depth, the derivatives with

respect to a (dotted lines in each figure) are comparable and, in some cases, larger

12

than the derivatives with respect to (i (solid lines). The derivatives with respect to p

(dashed lines) penetrate to larger depths than the derivatives with respect to a, but

with much smaller amplitudes compared with those of the derivatives with respect to j3.

The major structural discontinuties are responsible of well marked amplitude changes

in the derivatives.

In comparison with the numerical approach, with the analytical computation

we obtain approximately a two order of magnitude reduction of CPU time in the

determination of the derivatives of the eigenfunctions. In the computation of the

derivative of group velocity, we obtain more than two order of magnitude of CPU time

saving with respect to Rodi et a/.!s hybrid method {Rodi et aL, 1975).

5. Conclusion

We developed and successfully tested analytical procedures for the evaluation of

the partial derivatives of eigenfunctions, energy integral and group velocity with respect

to the structural parameters using modal representation. Our analytical method is

accurate and about two order of magnitude faster in the calculation of the derivatives

of the eigenfunctions and of the energy integral, in comparison with numerical methods,

and more than two order of magnitude faster in the calculation of the derivative of

the group velocity with respect to Rodi et al.'s hybrid approach (Rodi et at, 1975).

The developed technique permits the construction of an efficient algorithm for the

computation of differential seisinogram, to be implemented in the linearized, iterative

waveform inversions for the determination of the internal elastic properties of the earth.

Acknowledgments

The research has been carried out in the framework of the activities planned by the

Central European Initiative Committee for Earth Sciences, partially supported by the

NATO Linkage grants ENVIRXG 931206, SA.12-5-02(CN.SUPPL 940S80)453: the EEC

contract COPERNICUS CIPA-CT94-0238, Italian CNR grant 96.00318.05, CZECH

Ministry of Education contracts ES-003 and OK-171, and MURST (40% and 60%).

References

Gomberg, J. S. and Masters. T. G., 1988. Waveform modelling using loeked-mode

synthetic and differential seismogram: application to determination of the

structure of Mexico, Geophys. J. R. Astron. Soc.94, 193-218.

Harkrider, D. G., 1968. The perturbation of Love wave spectra, Bull. Seism. Soc. Am,.,

58, 861-880.

Haskell, N. A., 1953. The dispersion of surface waves on multilayered media, Bull.

Seism,. Soc. Am., 43, 17-34.

Knopoff L., 1964. A matrix method for elastic wave problem. Bull. Seism,. Soc. Am.,

54, 431-438.

Kosloff, D., 1969. A perturbation scheme for obtaining partial derivatives of Love wave

group velocity dispersion, Bull. Seism. Soc. Am., 59, 731-740.

Levshin, A. L., 1989. Surface waves in vertically inhomogencous media, in Seismic

Surface Waves in a Laterally Inhomogeneous Earth, edited by Keilis-Borok, pp

1-34, Kluwer Academic Publishers.

Nolet, G., 1990. Partitioned waveform inversion and Two-Dimensional structure

under the Network of Automously Recording Seismographs, J. Geophys. Res. 95,

8499-8512.

Novotny, 0., 1970. Partial derivatives of dispersion curves of Love waves in a layered

medium, Studia. Geophys. GeodaeL, Ceskoslov. Akad. Ved., 14, 36-50.

Panza G.F.. 1985. Synthetic seismograms: the Rayleigh waves modal summation. J.

Geophys., 58, 125-145.

Press, W. H., B. P. Flannery, S. A. Teukolsky and W. T. Vetterling (Eds): 1988.

Numerical Recipes, Chapter 4, Cambridge University Press.

Rodi, W. L,, P. Glover, T. M. C. Li and S. S. Alexander, 1975. A fast accurate method

for computing group-velocity partial derivatives for Rayleigh and Love waves,

Bull. Seism. Soc. Am., 65, 1105- 1114.

14

Schwab F.A.. 1970. Surface-wave dispersion computations: Knopoff's method. Bull

Seism. Soc. Am., 60, 1491-1520.

Schwab F.A., Knopoff L.; 1972. Fast surface wave and free mode computations. In.:B.

A. Haskell, N. A., The dispersion of surface waves on multilayered media, Bull.

Seism. Soc. Am., 43, 17-34.

Schwab F.A., Nakanishi K.; Cuscito M., Panza G.F., Liang G., Frez J., 1984. Surface

wave computations and the synthesis of theoretical seismograms at high

frequencies. Bull. Seism. Soc. Am.. 74, 1555-1578.

Urban L., Cichowicz A., Vaccari F., 1993. Computation of analytical partial derivatives

of phase and group velocities for Rayleigh waves with respect to structural

parameters. Studia geoph. et geod., 37, 14-36.

Zielhuis, A. and Nolet, G., 1994. Shear-wave velocity variation in the upper mantle

beneath central Europe, Geophys. J. Int., 117, 695-715.

This manuscript was prepared with the AGU I£T£X macros v3.1.

15

Density(g/cm**3),2.0 2.5 3.0 3.S 4.D

i i ' ' i ' ' i—i

Velocity(km/s)2 4 e e

50 -

100

O-

150

200

250

Qs and Qp100 400 700 1000

Figure 1. Upper 250 km of the velocity model used in our computation.

16

C

«£

a

I

1 = s- 1 =

c

a °P .?

(0CO

. £Q .

CO

15t(0Q.

Figure 3. Partial derivatives of phase, group velocities and energy integral with respect

to structural parameters j5 (solid lines), a (dotted lines) and p (dashed lines) for the

fundamental mode at 0.025 hz. Small dots correspond to the analytical calculations,

whereas larger squares correspond to the numerical calculations.

L8

55a>Q•OcCO»g>

+•»o_o

CD

CO

CO

OL

co"5

O)

o

.1(1)

Q

CO

o.

Figure 4. Analytical partial derivatives of eigenfunctions with respect to structural

parameters ,3 (solid lines), a(dotted lines) and p (dashed lines) for the fundamental

mode at 0.1 hz.

19

ca

w

o

I0)

r•gq>

ocLJJ•acCOQ .

O

oCO

0.

CD

a

Q.

Figure 5. Analytical partial derivatives of phase, group velocities and energy integral

with respect to structural parameters 0 (solid lines), afdottcd lines) and p (dashed lines)

for the fundamental mode at 0.1 hz.

20

enCOfa-

BPartial Derivatives of Eigenfunction wrt S-, P-wave Velocities and Density

Displacement (u ) * d/dp Displacement ( w ) * d'dp

-0.10 -0.05 0.00 0.0S 0.10 -0.30 -0.15 0.00 0.15 0.30

Stress ( o ) * d/dp

-10 -5 0 5 10

Stress ( t ) • d/dp

-14 -7 0 7 14

N>

a

3c

Oto

70

era'

to

COpa

(15

8

en

Partial Der ivat ives of Phase, Group and Energy Integral wr t S-, P-wave Veloc i ty and Densi ty

crq(T

3o

Cto

Phase Velocity ( c ) * d/dp

-0.06 -0.03 0.00 0.03 0.06

Group Velocity { u } * d/dp

•0.10 -0.05 0.00 0.05 0.10

70

Energy Integral ( I I ) * d/dp

-80 -40 0 40 80

03

00

9

CD

Partial Derivatives of Eigenfunctions wrt S-, P-wave Velocities and Density

Displacement (u ) * d/dp

-0.2 -0.1 0.0 0.1 0.2

Displacement (vi) * d/dp

-2.4 -1.2 0.0 1.2 2.4

Stress ( o ) * d/dp

-24 -12 0 12 24

Stress ( T ) * d/dp

-120 -60 0 60 120

tr

!i-j

BcoI L

o

S -

45 -

50

era

Partial Derivatives of Phase, Group and Energy Integral wrt S-, P-wave Velocities and Density

y

r

Phase Velocity (c ) *d /dp

-0.08 -0.04 0.00 0.04 0.08

Group Velocity { u } * d/dp

-0.4 -0.2 0.0 0.2 0.4

Energy Integral ( I I ) * d/dp

-300 -150 0 150 300

BoD

50