geophys. j. int. 1989 randall 469 81

7/24/2019 Geophys. J. Int. 1989 Randall 469 81

1/13

Geophys.

J .

Int. (1989) 99 469-481

Efficient calculation of differential seismograms for lithospheric

receiver functions

G. E. Randall*

Departmen1 of Geological Sciences, SUNY at Binghamton, Binghamton, New York 13901, US A, and Earth Sciences Department, Lawrence

Livermore National Laboratory, Livermore,

CA 94550,

U S A

Accepted 1989 April 1 7, Received 1989 April

17;

in original form 1987 November 2

SUMMARY

A new technique for computing differential seismograms for crustal and upper

mantle response to a teleseismic wave (receiver functions) is developed using the

matrix

formalism

of Kennett. The work was motivated by the difficulty

of

modelling

teleseismic SV-waves, and has also proven useful

for

modelling teleseismic P-waves.

This efficient me thod for calculating diff erential seismograms is based on three

separate methods

for

computing synthetic

seismograms.

Two of

the synthetic

seismogram methods save intermediate results; then the remaining synthetic

seismogram algorithm uses the stored results in an efficient calculation

of

a

new

synthetic seismogram for

a

perturbed velocity model. These developments have led

to a faster (for a 30-layer model, a 90 per cent reduction in computation time) and

more accurate linearized inversion scheme for the dete rmination of velocity models

using teleseismic waves.

Key

words: lithosphere, receiver functions, synthetic seismograms.

1

INTRODUCTION

Synthetic seismograms for the crustal and upper mantle

response to a teleseismic body wave (receiver functions) are

important tools for the interpretation teleseismic wave-

forms. Previously, receiver functions have been computed

either with ray theory for complex velocity models

(Langston 1977) or with the Haskell-Thomson formulation

(Haskell 1962) for laterally homogeneous layered velocity

models. Ray-theory synthetics can compute only specified

ray-paths, potentially missing arrivals and reverberations

whose importance may not have been anticipated. The

Haskell-Thomson technique can only compute a complete

seismogram of all arrivals; these synthetics can be as difficult

to interpret as the original observed seismogram, frustrating

attempts to explain the effects of changes in velocity models.

Furthermore, the Haskell-Thomson technique suffers from

numerical instability for frequencies and phase velocities

appropriate for teleseismic SV-waves propagating through

high-velocity layers typical of the lower crust and upper

mantle. Practical interpretation problems for teleseismic P -

waves (Zandt, Taylor Ammon

1987)

have shown the

difficulty and computational expense of specifying an

appropriate set of multiples in highly reverberant velocity

models, and

in

studies of teleseismic SV-waves (Zandt

Randall 1985) the Haskell-Thomson technique was

frequently unusable because of the numerical instability.

* Current address: Seismological Laboratory, MacKay School of

Mines, University of Nevada, Reno, Reno, Nevada 89557, USA.

Although techniques exist for dealing with the numerical

instability of the Haskell-Thomson technique in surface

wave dispersion studies (Dunkin 1965) and reflectivity

studies (Kind 1978), none

of

these techniques are applicable

to the synthesis of receiver functions. A careful analysis of

Haskell s paper on receiver functions (Haskell

1962)

shows

that the terms for the free surface displacement are not

composed of terms that are 2 x 2 minors of the original

matrix problem, as required by the methods that stabilize

the Haskell-Thomson technique.

Kennett 1983) has developed a technique for computing

the elastic wavefield in vertically stratified media. His

technique

is

numerically stable, and can synthesize a

pre-specified order of multiple internal reflections (including

a complete seismogram with all internal multiples) within a

laterally homogeneous layered structure. This paper

presents three formulations for synthetic seismograms of

layered receiver structures based on Kennett s technique.

The three formulations for synthetic seismograms are

used to develop an extremely efficient technique for

computing differential seismograms. Differential seismo-

grams are used in linearized inversion, the process of

adjusting the parameters of a velocity model to create a

minimum mean square error fit between an observed

seismogram and a synthetic seismogram for the velocity

model. A first-order Taylor series approximation

is the basis for the linearized inversion technique (Menke

469


2/13

470 G.

E.

Randall

1984) that iteratively solves for perturbations to the

parameters of the velocity model. In the Taylor series,

Soh(t) is the observed seismogram, and

S,,,[t,

c u t ) ] is the

synthetic seismogram for a reference velocity model, ~ z ) .

The seismograms are treated as vectors of time samples, and

the Tay lor expansion is taken about th e unp erturbed model

at each iteration of the inversion. The differential

seismogram,

represents the differential change in the seismogram for a

differential change of a single model parameter, or the

sensitivity of the seismogram to a single parameter in the

velocity model. In a typical inverse modelling study, the

computation of differential seismograms is a major burden,

equivalent to th e computation of a perturbed synthetic for

each layer of the velocity model to be estimated, and one

synthetic for the original unperturbed velocity model. The

technique presented

in

this paper will compute differential

seismograms for a variation of a single parameter in each

layer

of

the model for slightly more than the cost

of

three

synthetics. For a velocity model of N layers, this results in a

total time

of

slightly over 3T, where

T

is the time for a

single synthetic, as opposed to N

+

l )T for a conventional

technique. T he tim e for a single synthetic, T, is proportional

to the number

of

layers, N ,

so

the efficient technique grows

linearly with N, but the conventional technique grows

quadratically with

N.

For a 30-layer model, a 90 per cent

reduction

of

computation time compared with a conven-

tional approa ch represents a significant saving.

A major additional benefit

of

using the technique

presented here

is

an improvement

of

the accuracy

of

the

inverse modelling technique over using ray-theory synthet-

ics. The WKBJ technique has been linearized (Shaw

&

Orcutt

1985)

for efficient waveform inversion

of

refraction

profiles, but for some receiver function modelling studies

ray methods are inadequate. Ray-theory synthetics in highly

reverberant velocity models are

of

necessity incomplete and

therefore inaccurate. Previously, the results of any inverse

modelling of teleseismic waveforms with ray-theory syn-

thetics had to be appraised by computing a

complete

Haskell-Thomson synthetic. In reve rbera nt velocity models,

the Haskell-Thom son results freq uen tly revealed the

problem of truncating the infinite reverberation series in a

finite ray-theory synthetic. Previous studies

(Dr

Steven R.

Taylor, personal communication 1986) had shown that a

brute force computation of differential seismograms with the

Haskell-Thomson technique was impractical when com-

pared with ray theory. Although a previous study

(Fernandez 1965) had found a similar development for

computing differential seismograms based

o n

the Haskell-

Thomson technique, the report preceded the popular use of

inverse modelling techniques; th e result was never published

in the general literature and the potential importance was

not widely recognized. The improved efficiency of the

technique described in this paper has made it faster and

more accurate to compute complete synthetic and

differential seismograms, and now ray theory is required

only when laterally inhomogeneous media are modelled.

An Appendix presents a summary of the relevant theory

of Kennett s technique, a nd examp les of the application of

Kennett s method . This Appendix is included f or those

read ers who are unfamiliar with Kennett s wo rk, but for a

complete treatment a careful reading

of

Kennett s

monograph (Kennett

1983)

is recommended.

2 SYNTHETIC RECEIVER FUNCTIONS

This section presents two direct fo rmulations for the

computation

of

synthetic receiver functions with Kennett s

techniques, and an indirect formulation based on the result

of the two direct algorithms. The two direct formulations

compute the synthetic receiver functions by application of

Kennett s techniques, an d req uire n o results from prior

computations. The indirect technique requires that both of

the direct algorithms have been executed and saved

intermediate results, performing very modest computation

and indirectly doing the bulk of the computations by

referencing the previously computed results of the direct

techniques. For a forward modelling problem, the indirect

approach would be clearly impractical; however, th e indirect

approach ,radically simplifies the com puta tion of differential

seismograms. The indirect approach uses the intermediate

results f ro m ,th e two direct approaches t o compu te a new

seismogram for a model with a single perturbed layer, and

need not recompute the intermediate results from the

unperturbed layers. The two direct approaches will be

discussed first, setting the stage for the indirect approach. In

the following discussions, the solutions will be in the

frequency-slowness domain, and can be inverse transformed

from the frequency domain to the time domain, at a fixed

slowness,

to model a seismogram for a teleseismic arrival.

2.1 Direct bottom-up approach

The first technique presented is a straightforward bottom-up

approach . Th e description bottom-up deno tes the flow

of

computations through the velocity structure. The idea is

simple, and the algorithm follows the energy through the

model. A transmission operator, TU,propagates the energy

up from the bottom

of

the structure to the base of the

surface layer. In parallel with the computation of T,, a

reflection operator

R,,

calculates the reflectivity

of

the

layered medium from the base of the surface layer to the

bottom of the model. T he recursive com putation of T, and

R, using Kennett s technique is outlined in th e appendix.

These two operators are then combined with the free

surface reflection and diplacement o perato rs to com pute the

free surface displacement for a complete synthetic. This may

be represented by a simple matrix equation that can be

understood from right to left

d

= W(I plR~Np R)- p T~Ni

(3)

where

d

is the vector of free surface displacements, and

i

is

the wave vector incident at the base of the model. The

model consists of N laterally hom ogeneous layers, with layer

1 bounded above by the free surface, and layer

N

is a

half-space. The matrix P1 is the diagonal matrix of phase

delays for propagation through the surface layer, as

described in the Appendix. The matrix W converts the

upward directed energy to free surface displacements. At

the free surface the reflection matrix is R. Th e construction

and interpretation of the matrices for the region bounded


3/13

Differential seismogramsfor lithospheric receiver functions

47

above by layer

1

and below by the half-space,

RkN

and

ThN,

are discussed in the A ppend ix.

The usual reverberation operator interpretation is

attached to the

(I-P'RkNP'R)-'

term, with a single

reverberation in the surface layer represented by

PIRkNP'R

(see Appendix). Reading from right to left , this represents

reflection from the free surface, propagation delay down

through layer

1

reflection from the entire medium below

layer

1,

and finally propagation delay back up through layer

1 o the free surface. A schematic diagram of this bottom up

technique is shown in Fig.

1,

where a surface layer is

bounded above by a free surface and below by a layered

region with a net transmissivity an d reflectivity calculated by

the methods described in the Appendix.

As

stated in the A ppend ix, the reflection and transmission

matrices for the interfaces a re inde pende nt of frequency for

elastic media, and only weakly dependent on frequency for

attenuating media. T he frequency indep enden ce can result

in a substantial savings

of

computation because the interface

reflection and transmission matrices can be computed once

for a fixed slowness, and then used for all frequencies in a

synthesis. This saving can be applied to all three

of

the

synthetic receiver function algorithms.

This formulation was compared with the Haskell-

Thomson formulation for accuracy and completeness. The

comparisons show that this bottom-up technique is

computationally efficient an d num erically st able as predicted

by theory. Two comparisons

of

vertical component SV-wave

seismograms com puted with Haskell s and Kennett s

techniques are shown in Figs 2(a) and (b) for different phase

velocities. All computations were for the

P

and

S

velocity

structure shown in Fig. 2(c).

w

W

I

N-1

/

N

F i e 1. Schematic diagram for the reverberation operator in the

bottom-up method described by equation

(3).

This shows the direct

arrival and the first reverberation in the surface layer, using the net

transmissivity, T,, and net reflectivity, R,, for all layers below the

surface layer, computed by the technique discussed in the

Appendix. The free surface displacement operator s W, nd

R

is

the reflection operator for the free surface. The propagation delay

through the surface layer is represented by PI.

2.2

Direct

top-down approach

The top-down approach alludes to the way the algorithm

proceeds from the top of the model to bottom, but is not

related to energy flow within the model. This technique is

less physically motivated, but still can be understood as a

sequence of reverberations.

The initial step consists

of

modelling the response of a

half-space by transforming from the incident wave vector to

the free surface displacements using the

W

operator, as

described in the Appendix. For convenience, the receiver

function for

N

layers will be denoted

D',N

and the net

reflectivity operator for a region

of N

layers bounded above

by the free surface will be denoted by

RLN. For

both

D ' .N

and

RhN

layer

N

is taken to be an infinite half-space. The

matrix D I y N s the operator that transforms the incident

vector of wave amplitudes

i

into the vector

of

free surface

displacements, d . Th e excitation,

i ,

is applied at the top of

the half-space, just below the interface between layer

N 1

and the half-space layer

N.

The computations are initialized

for a half-space model with N =

1

by

D ' . ' =

W , an d

RL1

=

R ,

the free surface reflection o pera tor.

The next step consists of calculating the response to a

single layer over a half-space. Th is is simply described by

(4)

1,2 = Dl.l(I- plRb2p1Rbl)-1p1TL2

initializing the recursion for the receiver function, and

initializing the recursion for the reflectivity. The equation

for the receiver function represents the propagation of

energy through the interface at the bottom

of

the surface

layer, propagation of the direct arrival through the layer to

the free surface, and a sequence of reverberations. Each

reverberation consists

of

reflection from the free surface,

propagation down through the layer, reflection from the

base of the layer, and finally propagation back up to the free

surface to begin the sequence again. At the free surface,

each upward bound wave, e ither the initial direct wave or a

reverberati,on, is then transfo rmed into free surface

displacements by the operator W. The matrix

D',*

represents the net free surface displacement from upward

travelling energy incident at the bottom of the first layer. A

similar interpretation shows the reflectivity operator

Rh2,

t o

be the net reflection from the first layer, accounting for all

reverberations within the layer.

Now both recursions have been initialized, and the

following two equations will recursively add layers to the

model. A schematic representation for the addition

of

a

layer to the structure is shown in Fig. 3. First the receiver

function is updated by

where the reverberation now serves to compute the total

effect of the upward travelling waves at the top of layer

L - 1

caused by reverberation within layer

L - 1.

This

upward travelling energy is then transformed to free surface

displacements by

D 1 3 L - ' ,

he receiver function for the L - 1

layer model. The net result of these operations is to treat

layer

L

as the new bottom half-space, and layer

L 1

is now


4/13

472 G. E.

Randall

Comparison

of

Kennett .vs. Haskell Synthetics

I

I

I

-

1

.o

0.5

-

0.0

-

-0.51.o

+

P

S

-1.0

L

I

I

1

I I I I I I

20 40 60

80

Time (sec)

Figure 2 a). Comparison

of

vertical component of synthetic receiver functions comp uted with Haskell s an d Ken nett s metho ds using the

velocity model sho wn in Fig. 2(c).

For

comparison, both synthetics have been normalized t o a m aximum amplitude

of

unity. Both traces have

been low-pass filtered with a zero pha se (non -causal) filter with the corner fre quenc y at

1

Hz. Time

is

the time since the arrival

of

the incident

wave at the base

of

the model. These traces are computed for

a

phase velocity of 7 95 km s - ' , which means P-waves a re evanescent in the

fourth layer (from 40 to 60 m) and SPr,,P is post-critically reflected from the Moho. The Sp phase shown is from the M oho.

a finite thickness layer connected to the bottom of the

previous results by the recursion formula. This process then

proceeds adding layers until all layers have been

incorporated in the result. In parallel with the recursion

for

the receiver function, a similar recursion

RhL = T L - l s L p L - 1 1.L-1

D Ru

(7)

I

- L-IRn ,-1.LpL-'R1.# -1

1 L - 1

L - 1 . L

D

u )- p T

updates the reflectivity operator to include the reverbera-

tions within successive layers. The interface matrices

between layers

L 1

and

L

are TL-'3L,

TL-lrL,

R&-l ,L,

and

Rk-'9L

as described in the App endix. I t is imp ortant to

note that RLL

and

DlSL

epend on layer

L

only through

these interface matrices.

The top-down technique may sound more complicated

than the direct bottom-up technique, but it is computation-

ally

equivalent. T he description is mo re difficult because the

physical m otivation for the algorithm is less obvious than the

motivation for the bottom-up approach. The computational

expense of D1,N and

Rh N

is nearly identical to the

computational expense for

R hN

and TLN n the bottom-up

approach.

2.3 Indirect

approach

The indirect approach requires that th e interm ediate results

from both direct approaches be saved before the indirect

algorithm can execute. This is clearly not a practical way to

compute a synthetic seismogram; however, it leads to a

practical technique

for

computing a set

of

differential

seismograms

for

a model with many layers.

The key to this technique is the description of the receiver

function using the results from the two direct techniques.

The top-down approach provides information about the

response above a given layer, and the bottom-up approach

provides information about the response below a given

layer. The remaining task is to combine these results from

the direct approaches with a reverberation sequence within

the layer to calculate a synthetic receiver function.

Th e receiver function formulation

for

the indirect approach

is based on a reverberation sequence for a chosen layer,

and the use

of R h L

and D I S L or the region above layer

L and the use of

R k N

and

ThN

for the region below

layer L. The equation for the synthetic receiver function is


5/13

Dijferentiat seismogram s for lithospheric receiver functions 473

0

N

t

Cornporkon of Kennet t .vs. Haskell Synthetics

1

.o

0.5

0.0

0.5

-1.0

1

.o

0.5

0.0

0.5

-1.0

20

40

60 80

Time (sec)

Figure 2 b). The sam e comparison as in Fig. 2(a) , with a phase velocity of 8.15 km K which m eans that P-waves can propagate through the

fourth layer. Both this f igure and Fig. 2(a) show minor amo unts

of

time dom ain wraparound caused by frequency dom ain aliasing.

This

can be

cured by taking a longer time window o r using complex frequency to attenu ate unwanted later arr ivals .

and may be understood by expanding the reverberation

operator

as

discussed in the Appendix and examining the

individual terms. The expanded form

of

the equation for the

synthetic receiver function, with the direct arrival plus a

single reverberation is

(9)

~ N DI.LPLT N D ~ . L ~ L R N ~ L R ~ L ~ L T G . N

where

D1*N

s the synthetic receiver function for the total

N

layer model. A schematic diagram of the direct arrival and a

single reverberation is shown in Fig. 4.

The first term from the expansion representing the direct

arrival, D'sLPLT$N, s simply understood from right to left

as the net upward transmission from the base of the

N

layer

model up to the base of layer L , followed by propagation

through layer L , and finally transformation into free surface

displacements by the receiver function for the region from

the free surface down to the top of layer

L.

The first

reverberation within layer

L

is represented by

D ' ,LP LR $N P ' ,R hLP LT~ ,N ,

h e second term from the

expansion. Reading from right to left again, term by term,

this is: the net upward transmission from the base of the N

layer model up to the base of layer L followed by

propagation delay through layer L , reflection from the

region above layer L, propagation back down through layer

L reflection from the region below layer L , propagation

back up through layer L , and finally transformation to free

surface displacements by the receiver function for the region

above layer

L.

The reverberation operator is the

representation for the sum of all the multiple reverberation

terms.

3 E F FI C IE N T C O M P U T A T I O N O F

D I F F E R E N T I A L S E I S M O G R A M S

Differential seismograms can be easily computed by

carefully considering what parts of the indirect computation

are changed by the perturbation of parameters of the layer

of interest. Then, using the results previously computed by

the direct techniques for those parts of the computation left

unchanged by the perturbation of a single layer, the

seismogram for the perturbed model can be computed

without duplicating previous computations. The indirect

formulation is valid for any layer within the model, and the

computations for the two direct approaches can be modified

to save the intermediate results for every layer providing the

basis for the rapid computation

of

the indirect technique.


6/13

474

G.

E. Randall

4

I

I

8-

-

-

-

6+

- -

--

-

- -

5-

I

. c

Depth km)

Figure

2 c).

P- nd S-wave velocities versus depth for the crust and upper mantle m odel used for dem onstratio n purpo ses.

1 , L - I

/ '

R D

L

Figure

3.

Schematic diagram of reverberation operators for the top

down case for

D'.'-

as described in equation 4). The figure shows

the addition

of

a layer at the bottom

of a

stack of layers in the

top-down progression. The direct arrival and a single reverberation

are schematically shown in layer L -

1,

with R representing the

reflectivity for the region above layer L - 1, and D',L-'

representing the receiver function for the region above layer L

-

1.

/

N-1

/

N

Figure 4. Schematic diagram of reverberation operators for the

indirect formulation of the synthetic receiver function calculation.

The direct arrival and a single reverberation are schematically

shown in layer

L ,

with all opera tors as defined in Figs 1 and 3.


7/13

Differential seismograms fo r lithospheric receiver functions

475

When differential seismograms are computed, only a

single layer at a time is allowed to have perturbed

parameters, and some form

of

difference approximation is

used, such as

(10)

as [t

+ I

sSy [ t4.1

+ 6 4 z ) l ssy [4

I

a d z

W Z

where 6 a z ) represents the perturbed parameter in the

velocity model. If layer

L

is perturbed a careful analysis

of

the receiver function shows that all the computations above

layer L (DlSL nd R k L ) , and below layer L ( R g N and

TG.N) ,

depend on layer

L

only through the interface

reflection and transmission matrices for the top and bottom

of layer L. This may be seen by examining equations (6) and

(7) for the region above layer L, and equations (A6) and

(A14) in the Appendix for the region below layer L.

Intermediate results saved in the bottom-up and top-down

computations are combined with the perturbed interface

reflection and transmission matrices to complete the

calculations D l S L ,RLL

R k N

nd

TG N

for the perturbed

model. These computations are less expensive than a normal

layer reverberation because they can use previously

computed partial results. The propagation phase delay

matrix for layer

L

also needs to be recomputed. Then the

indirect formulation can rapidly form the reverberation

using the modified layer L values. This provides a new

receiver function for the modified velocity model, and the

difference between this and the receiver function for the

unperturbed model represent the differential change in the

receiver function for a differential change in layer L. In Fig.

5

examples of perturbed and differential seismograms for a

perturbation of the velocity model of Fig. 2(c) (layer 4 was

perturbed by a

0.5

per cent P-wave velocity increase) are

shown.

For a single differential seismogram, the perturbed

receiver function is computed by using the saved values of

the transmission and reflection operators above and below

the perturbed layer from the direct techniques in a

Comparison

of

Unperturbed

.vs.

Perturbed Synthetics

ennett

v 7.95

.............

kennett

vp

7.95

0.2

-

c

E

-s

0.0

Q

0

.-

0.2

20

30

40 50 60

Time (sec)

Figure 5. Comparison

of

unpertu rbed and perturbe d synthetics computed with Kennett s m etho d, and comparison of the differential

seismograms computed with Haskell s method and Kennett s meth od. In the perturbed and differential seismograms, the variat ion

of

the

P-wave velocity in layer 4 of the velocity model (show n in Fig. 2c) is

0.5

per cent from

8.0

to 8.04 km s-'.

All

traces have been low-pass filtered

with a zero phase (non-cau sal) filter with the corner frequ ency at 1Hz. T he comparison of unperturbed an d perturbed seismograms in 5(a) and

5(c) shows the effect of phase velocity. In 5(a) the phase velocity

of 7.95

km s-' means that P-wav es are evanescent in layer 4, and minimal

energy tunnels up or down through layer 4 as a P-wave. T he arrivals after about 45 s in 5(a) show amplitude changes, but minimal traveltime

chang e. In contrast, the phase velocity in 5(c)

is 8.15

k m -

s,

and the later arrivals

in

5(c) occur e arlier becau se the slightly increased velocity in

layer

4

reduces the travel time for P-waves travelling through layer 4. The compar ison of differential seismo grams comp uted with Kennett s

and Haskell s methods a re shown in 5(b) and 5(d) for phase velocities

of 7.95

and 8.15 km s- , respectively.

For

the purposes

of

comparison,

all differential seismograms have been normalized. This demonstrates the equivalence of the two techniques.


8/13

b)

Comparison

of

Kennett .vs. Hoskell Differentiol Seismograms

- 1

1 .o

0.5

L

c

W

g 0.0

- _

0.5

n

9

1.0

- 1.0

0.5

c

Y

I

c3

W

N

.-

E 0.0

z

-0.5

.o

I

I

I

Time (sec)

C)

Comparison of Unperturbed

.vs.

Perturbed Synthetics

0.2

c-

20

30

..

I

I

I

40 50 60

Time (sec)

Figure

5. (Conrinued)


9/13

Difere ntial seism ograms for tithospheric receiver functions

477

d )

Comparison of Kennett JS. Haskel l Dif ferential Seismogroms

1 .o

0.5

-

Q

g

0.0

.- -0.5

-=

L

-1.0

L

1.0

0.5

Lc

- -

n

* -

-

g

0.0

0

z

-0.5

-1.0

20 30 40 50 60

Time (sec)

Figure 5. (Cont inued)

computation for a perturbed layer reverberation. This

means the expense

of

computing a perturbed receiver

function for each of the N layers is nearly the expense for N

reverberation operators, which is nearly the same expense

for a single synthetic receiver function for an N-layer model.

This is the basis for the claim that the N differential

seismograms for an N-layer model can be computed in about

the time for three seismograms, when the cost for the two

direct computations are also considered. As discussed

earlier, the time required for a direct computation grows

linearly with the number of layers. Clearly as the number of

layers increases, this indirect formulation becomes more and

more attractive.

Additional computational reduction is achieved because

the perturbed reflection and transmission matrices for the

interfaces are independent of frequency for elastic media,

and only weakly dependent on frequency for attenuating

media. Thus, the perturbed reflection and transmission

matrices for the interfaces can be computed once for a fixed

slowness, and then used for all frequencies in a synthesis.

Computational efficiency has been achieved by increasing

the storage of intermediate results, and it is important to

note the storage required. All the computations discussed

above are for a single value of frequency and a single value

of slowness. Storage is required for the four matrices

(DIsL,

R L L ,RkN nd T f y N ) , or each of the N layers. Each matrix

is four complex numbers, typically requiring 8 bytes

per complex number, so N (layers) x 4 (matrices) x 4

(elements) x

8

(bytes) is required for a single frequency

and slowness. For a 32-layer model, the total storage for

intermediate results would be 4 kilobytes. Additional

storage for the perturbed interface reflection and transmis-

sion matrices is nearly twice the previous storage. There are

four complex 2 X 2 matrices for each interface, and these

must be computed for a variation in the layer parameters

above and below each interface, which results in additional

8 kilobytes for a 32-layer model. Some storage can be saved

by exploiting the symmetries for the transmission matrices

discussed in the Appendix. At worst, the total intermediate

storage would be

only

12 kilobytes. The storage for the

NF

complex frequency values of the synthetic and N differential

seismograms would easily be much more important than the

intermediate storage. For example, if N F is 1024 and N is

32, then the complex frequency domain storage is 33

(waveforms)

X

1024 (frequencies) x 8 (bytes) or 264 kilo-

bytes. For teleseismic receiver function modelling, only a

single slowness is typically required. If a spectral synthetic

with multiple slownesses were required, the order of the

numerical slowness integration technique would control the

number of stored frequency domain values at different

slownesses that had to be stored. A simple trapezoidal

integration or a first order Filon (Frazer & Gettrust 1984)

integration would allow the integral to be formed as a

running sum with no intermediate storage. In any case, the

storage for intermediate results would not grow.

The FORTRAN code implementing this theory runs on a


10/13

478 G. E. Randall

SUN

3/50,

modest 32-bit microprocessor based work-

station w ith 4 megabytes

of

mem ory, and is easily contained

in memory without using virtual memory to dynamically

page the program. The SUN 3/50 had the optional floating

point coprocessor (MC68881). The synthetics (vertical and

radial), and differential seismograms (vertical and radial),

with respect to P-wave velocity in each

of

the layers were

computed for two models. In the first case the computations

were for a 27-layer velocity model using a sampling interval

of 0.025s (20Hz Nyquist), and 2048 t ime points. The

frequency domain conjugate symmetry for real time

functions is exploited to permit evaluation of the transforms

of the signals at 1025 frequency samples. This is a high

resolution example, and is excessively detailed for the

analysis of teleseismic data, demonstrating a loose upper

bound for computation time. The SUN 3/50 performed the

frequency domain computations in 1550

s

and a SUN 4/280

performed the same computations in 170s. Both codes

required modest additional time

for

inverse Fourier

transforms and file i/o of the synthetic and differential

seismograms. A more typical computation, representing

parameters used for analysis of teleseismic

SV

waveforms,

with a velocity model of

18

layers using a Nyquist of 2 H z

and 512 time samples took 256 s on a SUN 3/50 and 28 s on

a SUN 4/280.

4 D I S C U S S I O N

This paper has described three techniques for computing

synthetic seismograms

of

the response

of

the lithosphere to

teleseismic waves. Each technique emphasizes reverberation

within different parts of the velocity model, and these

different viewpoints are used to exploit the intermediate

results of the first two techniques in a computationally

simple third technique. No single viewpoint for the

computation of the synthetic receiver functions is as

effective at minimizing the recomputation

of

results.

After the theory was developed, a comp uter program was

written to calculate differential seismograms, and the results

were verified by comparison with a brute force computation

using a Haskell-Thomson formulation. T he drama tic speed

improvement motivated the incorporation of these algo-

rithms into an existing inverse modelling code (Owens,

Zandt & Taylor 1984) that had been based on ray-theory

synthetics. The first results using the modified modelling

program have already been presented (Priestley

,

Zandt

Randall 1988) for P -wave da ta.

The substantial reduction of computation time of

differential seismograms for the teleseismic modelling

presented here should serv e as an incentive fo r other inverse

modelling studies such as refraction waveform modelling.

The multiple reformulation of the forward problem can

clearly reduce the computation time of differential

seismograms in complicated models involving many layers.

The analysis

of

intermediate storage discussed above should

apply to the refraction modelling case as well. A vectorized

algorithm (Phinney, Od om Fryer 1987) would require

storage of intermediate results at every frequency, but the

interface matrices could be stored as frequency independ-

ent. Fo r a synthetic with 1024 frequencies, the interm ediate

storage would then be abou t 4 megabytes. A mor e thorough

analysis should be the topic

of

a p ape r specifically discussing

the linearization of the reflectivity problem.

5 C O N C L U S I O N S

The use

of

multiple synthetic seismogram formulations, and

storage of intermediate results, can be combined t o create a

much faster computation of differential seismograms for

inverse modelling studies. The substantial speed increase

makes it realistic to appraise the non-uniqueness of

inversion solutions by comparing the results from a suite of

inversions with different starting models, and different

model parametrizations. The increased accuracy of a

complete synthetic, as opposed to a finite ray approxima-

tion, is a substantial additional benefit when laterally

homogeneous media are modelled.

A C K N O W L E D G M E N T S

This research was supported at SUNY at Binghamton by

National Science Foundation grant NSF-EAR-8508125 and

in part by grant NSF-EAR4306562

for

computational

facilities.

Additional support was provided at Lawrence

Liverm ore National Laboratory through the D epartm ent of

Energy Contract W-7405-ENG-48.

I would like to acknowledge the careful review of the

work leading to this manuscript by D r Geo rge Zan dt and D r

Steven R. Taylor. Dr Taylor pointed out the earlier work by

Fernandez.

1

would also like to acknowledge thoughtful

reviews of this manuscript by Charles Ammon, Dr Keith

Nakanishi,

Dr

Howard Pat ton, and Dr Norman Burkhard.

I

would especially like to acknowledge the incisive com men ts

of

an anonymous reviewer.

R E F E R E N C E S

Dunkin, J W., 1965. Computation

of

modal solutions in layered,

elastic media at high frequencies, Bull.

seism. SOC.A m . ,

55,

Fernandez, L. M., SJ., 1965.

Spectrum

of P

Waves ,

p. 172, Saint

Louis University, St

Louis,

Missouri.

Frazer, L. N. Gettru st, J. F., 1984. On a generalization

of

Filons

method and computation

of

oscillatory integrals of seismology,

Geophys.

J.

R .

astr. S OC .

7 6 ,

461-481.

Haskell,

N .

A ,, 1962. Crustal reflection

of

plane P a nd SV waves,

1.

geophys. Res. , 67, 4751-4767.

Kennett , B. L. N., 1983.

Seismic Wave Propagation in Stratified

Media,

Cambridge University Press, Cambridge.

Kind, R. , 1978. Th e reflectivity meth od for a buried s ource , J.

Geophys . , 44,603-612.

Langston, C. A., 1977. The effect of planar dipping structure on

source and receiver responses for constant ray parameter,

Bull.

seism. SOC.A m . , 6 7 ,

1029-1050.

Menke, W., 1984.

Geophysical Data Analysis: Discrete Inversion

Theory, Academic Press, New

York.

Owens, T . J . , Zandt , G. Taylor , S. R., 1984. Seismic evidence

for

an ancient r if t beneath the Cumberland Plateau,

TN:

a

detailed analysis of broadband teleseismic P-waveforms, J.

geophys. Res. ,

89 7783-7795.

Phinney, R. A. , Odom,

R.

I. Fryer , G. J. , 1987. Rapid

generation of synthetic seismograms in layered media by

vectorization of

the algorithm, Bull.

se ism. SOC. A m. , 77 ,

Priestley, K. F., Zan dt, G . Randall, G.

E.,

1988. Crustal

structure in Eastern Kazakh,

U . S . S . R

rom teleseismic receiver

functions, Geophys. Res. Let t . ,

15,

613-616.

Shaw, P. R. Orcutt ,

J.

A., 1985. Waveform inversion of seismic

refraction data and applications to young Pacific crust,

Geophys.

J.

R .

mtr.

SOC. ,

2 , 375-414.

Zandt , G . Randall , G. E., 1985. Observation of shear-coupled P

Waves,

Geophys. Res. Lett.,

12 565-568.

335-358.

2218-2226.


11/13

Differential seismograms

f o r

lithospheric receiver functions

479

a n d t ,

G., Taylor,

S.

R. Ammon,

C.

J.,

1987. Analysis of

teleseismic waveforms for structure beneath Medicine Lake

Volcano, Northern California, Seisrn. Res. Lett., 58, 34.

A P P E N D I X

Introduction

This Appendix is intended to serve as a brief, heuristic

introduction to K ennett s notation and technique. Th e

interested r ead er is encouraged to consult K ennett s

monograph for detailed mathematical derivation

of

t he

results

The formulations presente d below are all directed towa rds

solving the wave equation in horizontal plane layered elastic

or attenuating media.

I

have chosen to present only the

P-SV matrix form ulation, and omit the simplification to an

analogous set of scalar eq uations for

SH.

The solutions will

be in the frequency-slowness domain, and can be inverse

transformed from the frequency domain to the time dom ain,

at a fixed slowness, to model a seismogram for a teleseismic

arrival.

Reflection and transmission at an intedace

At an interface between two homogeneous layers, reflection

matrices are calculated that transform a wave vector

of P

an d SV incident amplitudes in to reflected

P

and SV

amplitudes. In the notation for a reflection, a subscript D

denotes an incident wave vector from above the interface,

with energy directed downward and reflected energy

directed upward. Similarly, a subscript

U

denotes incident

wave vector from below an interface, with energy

propagating upward, and reflected energy directed down-

ward. In Kennett s notation:

The superscripts in the matrix elem ents refer to the incident

and reflected wave type with the first letter being the

reflected wave and the second letter being the incident

wave. For example, an

SP

uperscript represents an incident

P-wave, mode converted to an SV-wave by reflection.

A similar notation applies for transmission matrices with a

D subscript representing incident energy from above an

interface with incident and transmitted energy both d irected

downward. Similarly, a

U

subscript de notes incident energy

from below an interface with both incident and transmitted

energy directed upward. Again the second superscript

of

the

matrix elements refers to the incident wave type, and now

the first superscript refers to the transmitted wave type. In

Kennett s notation:

T,=[

T g p TLs]

and

T u = [ T ; ~F1

TS,PTF ' TCp T E

Two additional matrices are needed for reflection at the

free surface and representation of physical free surface

displacements. The additional matrix for reflection of

upward directed energy at a free surface is also defined by

. G P P

rips.

K T

= ( R s P

R )

with the usual meaning for the superscripts. T he m atrix that

transforms P and SV amplitudes at the free surface into

physical displacements is

W = ( ~ R Pvp wRSvs)

where the second superscript deno tes the incident wave type

and the first superscript denotes either vertical

(V),

or radial

(R)

displacement at the free surface.

Kennett has chosen a normalization for wave vectors that

for a unit amplitude wave vector in a perfectly elastic

medium corresponds to unit energy flux in the depth

direction. This leads

to

symmetry properties for the

reflection and transmission matrices:

where the superscript

T

denotes the matrix transpose

operation. These sym metry relations will also hold iR a more

general context. Co mp utation and storage can both be saved

by exploiting these symmetry relations.

A point that may seem trivial now, but will simplify

reading of the following equations, is the notion that the

equations are understood from right to left . This is a

consequence

of

the matrix algebra and column vector

notation used by Kennett. The right to left notation is

already seen in the superscript notation for matrix elements,

with incident on the right and transmitted or reflected on t he

left. Finally, the reflection and transmission matrices are

independent of frequency for elastic media, and only weakly

dependent on frequency for attenuating media. The

frequency independence can result in a substantial savings of

computation because the reflection and transmission

matrices can be computed once for a fixed slowness, and

then used for all frequencies in a synthesis. In the synthetic

seismograms for teleseismic w aveforms, only one slowness is

needed but many frequencies must be computed, making

the savings significant.

Reflection and transmission in layered media

The next major a rea to develop involves the construction of

matrix reflection and transmission operators for a layered

region, not just a single interface. This involves combining

the results of the previous section with propagation through

layers for a net result that accounts for reverberation within

layers bounded by interfaces. The simplest cases are for a

single layer bounded above and below by infinite homo-

geneous half-spaces. From the results for these cases,

the general problems

of

multiple layers are solved by

recursion. Also, at this point the approximations for partial

reverberation are developed. A heuristic presentation will

be used to develop insight into the equations that Kennett

derives from a theoretical basis. O pe rat ors for reflection of

energy incident from above a layered region and

transmission

of

energy incident from below a layered region

are developed here, and used in the m ain body of the pape r.

For a layer between two half-spaces denoted

A

B, and C

from top to bottom and incident energy from above,

Kennett has shown that the reflection operator for the

region is

Rgc

=

R gB

+

TGBPBREcPB(I

-

R G B P B R ~ c P B ) - * ~ D E

A6)


12/13

480

G.

E . Randall

where the matrix superscripts refer to the layers bounding

the interface for that matrix. The matrix P , is a diagonal

matrix of the vertical component

of

the delays for

propagation of

P-

and S-waves through layer

B .

This is

simply represented in the frequency domain using the

vertical slowness and layer thickness by the following:

where h is the layer thickness,

w is

the radian frequency, p

is the horizontal slowness, and the square root terms are

P -

and S-wave vertical slownesses. respectively. The signs of

the square roots are chosen such that the imaginary part of

the square root is positive, which will result in a stable,

exponentially decaying representation for evanescent waves.

If t he other possible convention for Fourier transform sign

was used, a correspondingly different convention for the

signs of the square roots would be required to ensure

stability.

The phase delay matrix is guaranteed stable by

construction, and this formulation causes Kennett s

technique to be stable even

for

high frequencies. This

formulation is possible because the overall energy transfer is

directed. For example, in transmission problems net energy

propagates in only one direction, either up or down, for

each problem solved. Intermediate reverberations tem-

porarily reverse direction, but the net propagation is still

directed either up or down. Similarly, in reflection, the

direction of net energy propagation is always reversed. This

allows only phase delays to be considered, and they may be

constructed to be stable.

This is in contrast to the Haskell-Thomson approach that

uses a matrix propagator formalism that has both upward

and downward travelling waves, that must have different

phase delay sign conventions. In an evanescent wave

problem, one set of the phase terms will be stable, and the

other will of necessity be unstable. Theoretically, the

instabilities should cancel algebraically in the solution of

many problems, but numerically the problem is still unstable

because of finite precision arithmetic.

The expression for

R

is difficult to interpret directly but

fortunately a convenient expansion is available. The matrix

identity

I

X)-l=

+

x +X*(I x)-l

A81

can be expanded repeatedly to produce terms

of

a matrix

geometric series, and a final remainder term. Providing the

series converges,

m

c X

I

-X)-l

= O

an approximation to the result is then just the partial sum

I

+

x

+

x2+.

.+

X = I x)-'

where is taken as large as required for an approximation

of a specified accuracy. The remainder term is then

Use of the partial sum here will correspond to partial

reverberation.

If the first two terms in the partial sum expansion,

corresponding to n = 1, are used to approximate the matrix

inverse, a simple interpretation appears involving a single

reverberation within layer B . Expanding,

R;= R; + G BP BR gC P B( I R G BP BR Ec PB) G B (A12)

and collecting terms

The three terms in the sum on the right-hand side can be

analysed separately by reading each term from right to left.

The first term, RG , is just the reflection from the interface

separating layer A and B . The second term,

T;'UBPRREcPBFDB,epresents transmission down through

the interface between

A

and

B ,

propagation through

B ,

reflection from the interface between B and

C ,

propagation

up through B ,

and finally transmission back up through the

interface between A and

B.

The third term is more complicated and represents the

first internal reverberation within the layer B. The third

term,

T , H ~ R ~ g c ~ B ~ ~ B ~ B ~ ~

epresents trans-

mission down through the

AB

interface, propagation

through

B ,

reflection from the

B C

interface, propagation

through

B ,

and now a reflection from the

A B

interface, and

another sequence

of

propagation through

B

reflection from

the B C interface, and propagation through B followed by a

final transmission back up through the AB interface.

These three terms combine to

form

an approximation to

the wavefield reflected from the region, here allowing only a

single internal reverberation in layer

B . As

higher order

terms are included in the partial sum approximation, more

internal reverberations are included in the approximate

wavefield. In the original expression using 1-

R$BPBREcPB)-' all the terms in the infinite

series

are

included, and therefore this term is really a reverberation

operator for the layer

B ,

including all internal reverbera-

tions. A simple schematic diagram illustrating the

reverberation process is shown

in

Fig. Al(a).

A similar analysis for the operator representing

Figure Al.

Schematic diagrams for reflectivity and transmissivity.

In the top half the individual reverberations are shown for the

reflectivity as described in equation (A6)

of

the Appendix. In the

bottom half, the reverberations fo r the transmissivity a s described in

equation (A14) of the Appendix are shown.


13/13

Diflerential seismogram s for lithospheric receiver functions 481

transmission upward from C through B into

A

can be done. that

only

contains two terms that can be interpreted

A simple schematic for this reverberation sequence is shown separately. The first term, PUBPBTEc, is merely the direct

in Fig. Al(b). The complete form

of

the result is: transmission from

C

through B to

A ;

the second is the term

corresponding to a single reverberation within layer B

A14) before transmission into A. Here the reverberation

The result for the corresponding expansion for a single operator, (I

-

PBRgCPBRflB)-', is just the transpose of the

reverberation is reverberation operator for the reflection case analysed

above. This relation can also be used to save significant

e

PuBPBTEc

+

~ u B P B R ~ c P B R < B P B T ~ c

A15)

computations in each layer.

ec

eB l

P ~ R ~ ~ P ~ R ; ~ ) - ~ P ~ T F .

geophys. j. int. 1989 randall 469 81

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