geophys. j. int. 1989 randall 469 81
TRANSCRIPT
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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
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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
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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
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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
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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.
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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.
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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.
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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)
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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
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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.
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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)
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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.
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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 .