Geophys. J. Int. (1997) 129, 269-280
Wide-angle seismic velocities in heterogeneous crust
John Brittan and Mike Warner
Department of Geology, Imperial College, London SW7 2BP, UK. E-mail: [email protected]. uk
Accepted 1996 December 10. Received 1996 December 5; in original form 1996 September 6
SUMMARY
Seismic velocities measured by wide-angle surveys are commonly used to constrain
material composition in the deep crust. Therefore, it is important to understand how
these velocities are affected by the presence of multiscale heterogeneities. The effects
may be characterised by the scale of the heterogeneity relative to the dominant seismic
wavelength (A); what is clear is that heterogeneities of all scales and strengths bias wideangle velocities to some degree. Waveform modelling was used to investigate the
apparent wide-angle P-wave velocities of different heterogeneous lower crusts. A constant composition (50 per cent felsic and 50 per cent ultramafic) was formed into a
variety of 1- and 2-D heterogeneous arrangements and the resulting wide-angle seismic
velocity was estimated. Elastic, 1-Dmodels produced the largest velocity shift relative to
the true average velocity of the medium (which is the velocity of an isotropic mixture of
the two components). Thick (width>>A) horizontal layers, as a result of Fermat’s
Principle, provided the largest increase in velocity; thin (width << 2) vertical layers produced the largest decrease in velocity. Acoustic 2-D algorithms were shown to be
inadequate for modelling the kinematics of waves in bodies with multiscale heterogeneities. Elastic, 2-D modelling found velocity shifts (both positive and negative) that
were of a smaller magnitude than those produced by 1-D models. The key to the
magnitude of the velocity shift appears to be the connectivity of the fast (and/or slow)
components. Thus, the models with the highest apparent levels of connectivity between
the fast phases, the 1-D layers, produced the highest-magnitude velocity shifts. To
understand the relationship between measured seismic velocities and petrology in the
deep crust it is clear that high-resolution structural information (which describes such
connectivity) must be included in any modelling.
Key words: crust, inhomogeneous media, seismic velocities
INTRODUCTION
The interior of the Earth is clearly not a simple assemblage of
concentric shells whose compositions and organisations are
homogeneous. Surface exposures of crustal material suggest
that the interior of the Earth is a heterogeneous composite, the
general definition of heterogeneity being a spatial variation of
a particular physical characteristic (Macbeth 1995). The
scale length of the heterogeneities present in the Earth’s crust
range from the megascale (tectonic features such as plates or
orogenic belts of hundreds of kilometres extent) down to below
the microscale (the minerals that compose these features whose
characteristic size is often less than 1 mm).
The information that a seismic wave carries about a
medium it has traversed is dependent upon the size, strength
and arrangement of the heterogeneities of which the medium
is composed. If the surface observer is to interpret this
information correctly, it is vital to understand the effects of
0 1997 RAS
heterogeneity. In particular, it is important to discover whether
certain types of heterogeneity can unduly dominate the
response to the seismic waves and thus mislead an interpreter
who has no apriori knowledge of the actual structure. This is
particularly important in the case of the exploration of the
continental and marginal lower crust for two reasons. First,
the information provided by seismic experiments over these
areas is the highest-resolution in situ information available,
and second, the results from such experiments suggest a diverse
pattern of heterogeneity, particularly in the lower crust
(Sadowiak, Meissner & Brown 1991; Mooney & Meissner
1992). The evidence from reflection profiles clearly indicates
that wide-angle seismic waves traverse areas of lower crust
that are highly heterogeneous. In this paper we investigate the
effect of multiscale heterogeneities upon the traveltimes of
waves traversing such regions, and, in particular, we discuss
the wide-angle apparent velocities of heterogeneous arrangements of lower-crustal material. The apparent velocity
269
210
J. Brittan and M . Warner
[sometimes known as the effective velocity (Brittan & Warner
1996)] is defined as the gross P-wave seismic velocity the
heterogeneous medium would have if it was interpreted to be
isotropic and homogeneous. Our approach to this problem is
first to give an overview of how different scales of heterogeneity
modify the wavefield and then to use modelling to quantify the
effects of these modifications on the velocities measured by
wide-angle, deep-crustal seismic surveys. Finally, we discuss
’ the implications of the results for studies of the composition of
the lower continental crust.
The effect of the size and strength of a heterogeneity upon
the seismic wavefield will depend upon the characteristics of
the wavefield itself [in particular the dominant wavelength R
and thus the dominant wavenumber k (=27t/l)]. Once the
effect of the size and strength of individual heterogeneities is
quantified, the spatial distribution of the heterogeneities can be
taken into account and a realistic model of the subsurface can
be created. Fig. 1 illustrates a classification of the methods
used to quantify the effects of heterogeneity in seismology. The
phenomena are classified according to ka, where a is the scale
length of the heterogeneity, and kL, where L is the path length
travelled by the waves through the inhomogeneous body. The
dimensionless number ka measures the ‘roughness’ of the
material within one wavelength. If a>>L then the wave is
passing through a homogeneous body and the effect upon the
wavefield is relatively easy to quantify. When a 5 L the body is
heterogeneous. To further subdivide the classification of wave
propagation in heterogeneous media it is possible to use the
1000
100
10
1
0.1
0.01
pathlength kL
Figure 1. Classification of approaches to quantifying the effects of
heterogeneity upon the seismic wavefield (redrawn from Aki &
Richards 1980). The effects are classified according to ka: the wavenumber x the heterogeneity scale length; and kL: the wavenumber x
the travel distance. Details of the subdivisions are given in the text. The
shaded area marks the range of seismic path lengths traditionally
utilized in wide-angle seismic studies of the continental lower crust.
wave parameter D (Aki & Richards 1980). The parameter D is
the ratio of the size of the first Fresnel Zone to the scale length
of the heterogeneity, and is given by
If D < 1 (i.e. in the area above the line D = 1 on Fig. l), then the
heterogeneities are, in principle, resolvable using geometrical
ray theory; the heterogeneous medium may be thought of as
piecewise homogeneous. When ka is large enough for the
individual heterogeneities to be resolved by geometrical ray
theory, then the paths the seismic rays follow will be defined by
Fermat’s Principle of Least Time :
il ray that starts from a point A and goes to a point B will
foUow the path between the two points which requires the
shortest travelling time’.
Thus, for a medium with large heterogeneities the seismic ray
will follow a path in which proportionally more time is spent in
the faster heterogeneities than the slower heterogeneities.
When D > 1 then the waves are multiply scattered and it is
difficult to describe analytically the gross effect upon the
wavefield. Attenuation and dispersion are high in this region
(Macbeth 1995) and much of the seismic energy may become
focused in one part of the heterogeneous medium.
When the characteristic scale length of the heterogeneity is
much smaller than the seismic wavelength, the body behaves as
a homogeneous equivalent medium with properties averaged in
some manner from its composite heterogeneities. For a body
composed of small, I-D heterogeneities (is. thin layers), the
elastic properties of the equivalent medium are described
by Backus averaging (Backus 1962). For a body composed
of small, 3-D heterogeneities, the minimum and maximum
seismic velocities that are possible for the equivalent medium
are given by theVoigt-Reuss bounds (Walpole 1981).Kerner &
Pratt (1997) found that the maximum and minimum velocities
predicted by Backus averaging were close or identical to the
Voigt-Reuss bounds calculated for a particular layered
sequence. They concluded that the velocity anisotropy of the
equivalent medium is strongest when the heterogeneities are
arranged in a 1-D, thin-layer arrangement.
The limit of the range of validity of equivalent medium
theory is still not clear; analytically it has been described as
when the fractional ‘loss’ of energy due to the heterogeneities
( A Z / I , where I is energy, given by eqs 13.55 of Aki & Richards
1980) is less than 10 per cent. Experimental studies of waves
travelling parallel to the axis of symmetry in a 1-D stack
(Marion, Mukerji & Mavko 1994) suggested that the transition
between the equivalent medium approximation and the piecewise homogeneous approximation occurs over a narrow
range when kaw0.5. At this transition the apparent velocity
measured parallel to the axis of symmetry is intermediate
between that measured for very thick layering (which gives the
maximum apparent velocity) and that for very thin layering
(which gives the minimum apparent velocity). Of particular
note from the modelling of Marion er al. (1994) is a large
decrease in the amplitude of the transmitted wave; this
decrease in amplitude is likely to be symptomatic of layering in
the transitional scale.
To summarise, Fermat’s Principle implies that in the
presence of large-scale heterogeneities ( k a >z 1) the apparent
velocity should be higher than the average velocity of the
medium, as the seismic wave will travel proportionally more
01997 RAS, GJI 129,269-280
~
Seismic velocities in heterogeneous crust
271
of its path-length within the high-velocity material. At the
other end of the scale, small-scale heterogeneities can lead to
apparent velocities larger or smaller than the average velocity
of the material; the direction of the bias is dependent on the
organisation of the heterogeneities relative to the dominant
direction of wave propagation. At intermediate scales the effect
of the velocity field is suggested to be intermediate between
that of the two endmembers (Marion et al. 1994).
(where x is the offset from the source and t is the unreduced
picked time). A velocity of the lower crust was then derived
using the general form of Dix's equation (Dix 1955).
Fig. 2 illustrates the passage of a seismic ray through a twolayer medium. Dix (1955) showed that the angle of incidence of
the seismic ray PI at an offset x1 is given by
MODELLING RESULTS
where VI is the velocity of the upper layer, T ( x ) is the arrival
time at offset x and VH(x1) is the hyperbolic velocity at offset
X I . The hyperbolic velocity is defined as the square root of the
inverse slope of the x2-t2 graph at x= X I . From a knowledge of
PI,the distances {and the corresponding traveltimes) of the
segments SG and F H may be calculated, thus the time spent in
the upper layer may be taken from the total traveltime to give
the corrected travel time (rx)c.
Analytically this is given by
In this paper we present the results of modelling seismic
propagation through simple 1-D and 2-D crusts whose lower
halves have highly contrasting structures. The aim of this
modelling was to recover the average velocity of the heterogeneous lower crust; this is the velocity that the whole volume
of material would have if it was an isotropic mixture of its
components. In order to demonstrate clearly the effect of a
change in the structure upon the inferred velocity of the
lower crust (the apparent velocity), the lower-crustal composition was kept constant. The lower crust was composed of
a single material that could be split into two distinct components. Although hampered by sampling bias, which may
underestimate the proportion of intermediate rocks in crustal
sections [e.g. the exposed lower crust in the Fiordland of New
Zealand is extensively metagabbroic diorite (Oliver & Coggon
1979)], laboratory studies of lower-crustal rock samples
suggest a bimodal distribution of P-wave velocities between
felsic {slow) and mafic/ultramafic (fast) lithologies (Rudnick
& Fountain 1995). A clinopyroxene (diopside) was chosen to
represent the mafic/ultramafic component and a plagioclase
(albite) was chosen to represent the felsic component. The
reaon for choosing such a composition was simply to have
tRo components with highly contrasting elastic properties,
thus representing an extreme case in terms of real crustal
composition. In the section on lower continental crust
we will discuss in greater detail how the magnitude of
heterogeneity-related velocity shifts relates to differences in
crustal composition.
The seismic velocities and densities of the two component
materials are taken from laboratory studies of single-crystal
elastic properties (collated in Brittan & Warner 1996). The
velocities have been normalized to standard lower-crustal P-T
conditions (0.8 GPa and 400 "C).The albite is modelled to
have a P-wave velocity of 6.36 km s-', an S-wave velocity of
3.68 km SKI
and a density of 2.62 g cm-3 at lower-crustal
conditions; the diopside is modelled to have a P-wave velocity
of 7.82 km s-', an S-wave velocity of 4.55 km s-I and a
density of 3.28 g cm-3 at lower-crustal conditions.
Lower-crustal velocity is usually determined by using the
amplitude and traveltime behaviour of either the strong Moho
reflection, PmP, or the lower-crustal diving waves. The lowercrustal diving waves are, on most wide-angle data, of relatively
small amplitude and thus of less use than the P m P phase. In
all models in this study, there are no velocity gradients within
the lower crust and consequently no diving waves. Velocity
gradients within the lower crust complicate the interpretation
process and may lead to unrealistic levels of lateral heterogeneity. From each of the synthetic seismic responses produced
by numerical modelling, the apparent wide-angle velocity was
estimated. The arrival times of the PmP phase were picked
from the synthetic section and plotted on an x2-t2 graph
01997 RAS, GJI 129,269-280
(3)
where dl is the depth to the base of layer 1 (i.e. the length of
segment SA). Similarly, the horizontal distance travelled in
the upper layer (equivalent to 2AG) may be calculated and
subtracted from the total offset to give the corrected offset xc.
Hence, the corrected offset, XC, is given by
xc=x-2dl tanp, .
(4)
Plotting (TJ: against x; and taking the inverse of the gradient
gives the squared velocity of the second layer (in the following
models, the lower crust), V2.This is the apparent velocity of the
second layer.
This technique assumes that the depth and velocity of the
overlying upper crustal section is known. In most crustal
experiments the nature of the upper crust is known to a considerably higher degree than that of the lower crust, therefore
this is a plausible assumption. The errors quoted for each
apparent velocity are the standard error from a linear leastsquares fit to the ( T x ) ~ - plot
x ~ (over a given range of offsets).
In reality, the errors may well be larger due to uncertainties in
dl and Vl.
1-D MODELS
Details of the 1-D modelling of heterogeneous lower crusts
have been given previously (Brittan 1995; Brittan & Warner
1996). In this section we shall simply report the results
S
X
A
B
D
Figure 2. Passage of a seismic ray through a plane two-layer medium
(after Dix 1955).The ray travels a horizontal distance x from S to F and
arrives with an incident angle PI. The upper layer has velocity V I and
the lower layer has velocity Vz.
272 J. Brittan and M. Warner
important to this study. Three simple 1-D models were tested.
The first model had a homogeneous lower crust; this is similar
to the structure assumed in most forward modelling of wideangle data. The second model had a lower crust with the
bimodal composition arranged into horizontal layers. The
layer width was of the same order of magnitude or larger than
the investigating seismic wavelength, that is ka>>1. The third
model also has the bimodal composition, in this case arranged
in' vertical layers. In contrast to the second model, the layer
width is much less than the investigating seismic wavelength,
that is k a c 1. The thin layers were replaced by an anisotropic
equivalent medium. The geological justifications of models
such as these are given later in this paper.
The synthetic seismic response of the first model showed
two main phases on the seismogram, the wide-angle reflections
from the top of the lower crust (labelled PcP) and the wide-angle
Moho reflection (labelled PmP).The observed PmP traveltimes
were picked, and using the technique described above an
apparent lower-crustal velocity of 7.02 (kO.01) km s-' was
derived (over an offset range of 100-400 km). This is identical
to the average (Voigt-Reuss-Hill) velocity of the lower crust
in the model.
The thick horizontal layers of the second crustal model affect
the seismic wavefield in a significantly different manner. For
oblique angles of incidence, the seismic wave will spend proportionally more of its travel path in the high-velocity layers
and thus the resultant time-averaged velocity will move
towards the higher velocity. In a thickly layered medium, the
apparent velocity will not correctly reflect the average velocity.
In the case of the thick layers it was very difficult to fit a straight
line to the xc(Tx): graph at all offsets, that is finding an isotropic lower crust that will produce the measured traveltimes at
all offsets was impossible. As heterogeneous crust of this
nature is grossly anisotropic, the apparent velocity is clearly
offset-dependent. It was shown that as the offset of the aperture
increases, the apparent velocity increases. This is an effect of
Fermat's Principle; at wide-angles, as the path length of the
wave increases, the wave will spend more of its travel path
in the fast layers. Calculating the apparent lower-crustal
velocity for this model using the traveltimes from longer
offsets (100-300 km) gave a velocity of 7.47 (k0.04) km s-'.
The amplitude response of the second model was also considerably different from that with the isotropic, homogeneous
lower crust. The amplitude of the Moho reflection decreases
rapidly with offset as a large proportion of the energy is
scattered incoherently by the large heterogeneities-this
energy from the lower crust appears on each trace between the
reflected waves from the Moho and from the top of the lower
crust. Such reverbatory data have often been seen in wide-angle
seismic experiments and have been modelled as a function of
lower-crustal heterogeneity (e.g. Sandmeier & Wenzel 1986,
1990; Larkin & Levander 1995).
The seismic response of the model with thin vertical layers
gave a traveltime of the PmP phase that was different from that
of the corresponding phase in both previous 1-D models.
Calculating the apparent lower-crustal velocity shows that
signals reflected from the Moho appear to travel through the
lower crust with the lowest velocity of the three models,
6.83 (k0.02) km s-' (over an offset range of 100400 km).
This value confirms the results of Marion et al. (1994) as the
wide-angle waves travel close to the axis of symmetry. The
wave detected perpendicular to the dominant layering (in this
case the wave used to derive the apparent velocity) will be an
organisation of energy from multiply scattered waves and will
always have a velocity less than the average velocity of the two
composite materials (Thomsen 1986).
2-D MODELLING
In the 2-D modelling we concentrated on investigating the
seismic response of a large block of lower crust composed
of heterogeneities that are large enough (in relation to the
dominant seismic wavelength) to make equivalent media
theory invalid and yet considerably smaller than the blocks
typically modelled using ray theory (e.g. scale lengths less than
around 20 km). This, however, presented a number of practical
difficulties. In particular, adequate modelling in this regime
requires the use of a complete wave solution in two dimensions
(i.e. finite-difference or finite-element schemes) with a densely
sampled grid of nodes covering the crustal section. To model a
165x22 km crustal section using the acoustic wave equation
the model must be specified at approximately 1 . 6 ~ 1 0 'nodes,
and using the elastic wave equation at approximately 1.7X1O6
nodes. It is therefore clearly impractical to use a deterministic
approach to build the model (i.e. to individually locate
heterogeneities by hand). A more feasible alternative is to build
heterogeneous models whose characteristics (both in terms of
physical properties such as velocity or density and the spatial
organisation of these properties) can be parametrized using
statistics. In stochastic modelling the aim is to reproduce some
characteristic of the seismic response rather than to match
the data exactly; the end goal is to find the gross statistical
properties of the heterogeneous area (Bean & McCloskey
1995).
Previous studies of the nature of crustal heterogeneity
have concentrated on matching synthetic data derived from
stochastic modelling with actual field observations. For uppercrustal studies the field observations used have been well
logs and the data from seismic experiments; for deeper areas
the data modelled have tended to be earthquake data from
large, widely spaced arrays. Three main forms of spatial
distribution are commonly used to form a stochastic model:
Gaussian, exponential and self-similar (or fractal) correlation
functions. The nature and characteristics of the three distributions are discussed in detail in Frankel & Clayton (1986)
and Holliger & Levander (1992) and will only be described
briefly here. The 2-D Gaussian correlation function has the
form
(5)
where
r = / z .
The correlation length in the x-direction is a and the correlation length in the z-direction is b. The exponential spatial
distribution is described by
C(r)=e-'.
(6)
Self-similar spatial distributions are described using the 2-D
Von Karman correlation function
C(r)=
4nv2rYKV(r)
K"(0) '
(7)
01997 RAS, GJI 129, 269-280
Seismic velocities in heterogeneous crust
where K, is the modified Bessel function of order v and v is the
Hurst number, which describes the fractal nature of the
medium. For a Hurst number v = 0.5, the correlation function
simplifies to the exponential correlation function. There are
two interrelated important differences between the three
distribution functions. First, for the Gaussian and exponential
correlation functions the correlation length (a or b) is
approximately the same as the dominant wavelength of the
heterogeneities, while for self-similar media the correlation
length is in effect the upper bound upon the fractal nature of
the medium (Bean & McCloskey 1995). Heterogeneities in
self-similar media can occur with sizes larger or smaller
than correlation length, but only those smaller than it are
fractal in nature. Second, self-similar distributions by their
nature do not lose power at small wavelengths (high wavenumbers), unlike the corresponding Gaussian and exponential
distributions.
A number of studies have attempted to find which of these
three distribution functions produces the most accurate fit to
data collected from the Earth. Bean & McCloskey (1995)
argued that only a self-similar velocity distribution can
account for both the reflectivity seen from upper-crustal borehole studies and the nature of wide-angle seismic data from the
upper crust. Frankel & Clayton (1986) modelled the traveltime
fluctuations across large seismic arrays and the codas from
microearthquakes (f> 1 Hz). The best-fitting stochastic
model to these data utilized a self-similar correlation function
with a _+lo per cent standard deviation in velocity and a
horizontal correlation distance greater than 10 km. Holliger &
Levander (1992) digitized two geological sections through the
Ivrea Zone, a section of exposed lower crust. The best fit to the
spatial distribution of petrophysical properties was found to
be a self-similar Von Karman function with Hurst number 0.3
and correlation lengths of around 700 m (horizontal) and
150 m (vertical). Holliger, Levander & Goff (1993) showed that
a random lower crust with these properties and a bimodal
velocity distribution would produce a seismic signature very
similar to that often seen on deep seismic experiments. In
general, the studies suggest that the crust can be best modelled
using a self-similar correlation function. The lower crust in
particular appears to have a ratio of horizontal to vertical
correlation lengths of 3 : 1; however, the actual dominant correlation lengths present are unclear. The numbers given above
are a strong function of sampling interval.
The 2-D media used in our modelling studies were generated
using a linear stochastic process (Kerner 1994). The spread of
P- and S-wave velocities and densities are described by a
Gaussian distribution with user-specified mean and variance.
The spatial arrangement of these parameters are described by a
2-D autocorrelation function representing their distribution in
the x and z directions. The following is a simplified description
of the process of constructing a random field for a particular
parameter (e.g. P-wave velocity).
The first step is to assign a random function to the model
grid to describe the parameter distribution. This takes the
form of a Gaussian deviate with a zero mean and a variance of
one. This function must then be filtered to provide a spatial
arrangement with the required 2-D statistics. The statistics
of the spatial distribution are described by autocorrelation
functions in the x- and z-directions; the autocorrelation
of a spatial series describes the similarity between that
series and a spatially shifted version of itself. The 1-D
01997 RAS, GJI 129,269-280
213
autocorrelation functions for the exponential and Gaussian
spatial arrangements were given by a generalized Gaussian
function (Tarantola 1987)
where r(.)is the gamma function, Lp is the correlation length
and p describes the shape of the autocorrelation function.
The two shapes of autocorrelation function used were the
exponential function ( p = 1) and the Gaussian function ( p = 2).
The Von Karman correlation function (eq. 7) was used to
simulate a self-similar medium. A value of v = 0.3 was used for
the Hurst number of the distribution.
As the filtering process involves the convolution of the 2-D
spatial autocorrelation functions with the random parameter
distribution, it is computationally more efficient to Fourier
transform both functions and carry out the operation in the
wavenumber (k) domain. The resulting filtered random function is then inversely transformed into the spatial domain
and scaled as required by multiplication with the parameter
standard deviation and addition of the chosen mean value.
This process produces a continuous field in the random
variable with a parameter spread about a pre-determined mean
with the chosen standard deviation. The statistics of the field
can be entirely described in terms of the correlation length in
the x-direction, a, the correlation length in the z-direction, b,
the mean parameter value p, the standard deviation c and the
type of autocorrelation function. To combat the influence upon
the model of unrealistic parameter values from the tail ends of
the distribution, a high/low cut-off is applied; usually any grid
point whose parameter had a value greater than four times the
field standard deviation was given the value of four times the
standard deviation. A cut-off closer to the mean value than
this would adversely affect the distribution statistics and was
avoided.
A binary field (i.e. a field with only two component
parameter values) may be approximated from a continuous
field derived as above by simply applying the following criteria.
Any grid point whose randomized parameter (e.g. velocity) is
greater than the mean value is given the value of the first
component; all other grid points are given the value of the
second parameter. In the 2-D models used, the velocity components are the same as those used in the 1-D modelling, i.e. a
pyroxene and a plagioclase. Fig. 3 illustrates three binary
random fields generated using the procedures described above.
All three fields have the same input correlation lengths in the
vertical and horizontal directions; the only difference between
them is the correlation function used.
ACOUSTIC MODELLING
The first 2-D modelling runs were performed in the frequency
domain using a modified second-order, acoustic finitedifference code [originally described in Pratt & Worthington
(1990); modifications described by Jo, Shin & Suh (1996)l. The
model had dimensions of approximately 165 km by 22 km. The
width of the upper and lower crust were both just under 11 km.
The grid spacing was set to 150 m, and thus the model could
contain frequencies up to 10 Hz.
Prior to calculating the seismic response of any 2-D
heterogeneous models, a test run was conducted using a model
whose lower crust was 1-D. This was undertaken in an attempt
274 J. Brittan and M. Warner
I
5i
I
Average velocity
of lower crust
I
I
II
20
I
I
I
I
I
I
-
I
5
6
7
8
9
Velocitv (krn/s)
Von Karman
$ z correlation length
-*
x correlation length
Figure 3. Typical binary random fields used to represent the structure
of the lower crust. Each of the three fields has a correlation length in
the x direction of 3 km and a correlation length in the z direction of
0.75 km. The Von Karman model was created using a Hurst Number
of 0.3.
to understand whether an acoustic code correctly models
the kinematic response of a heterogeneous medium. The 1-D
test model is illustrated in Fig. 4. The average P-wave velocity
of the lower crust was 7.02 km SK'. Fig. 5(a) shows the
synthetic seismic response of this model calculated using the
elastic reflectivity code (Fuchs & Muller 1971) and Fig. 5(b)
illustrates the synthetic seismic response produced by the
acoustic finite-difference code.
The 1-D model was chosen to have layering on scales varying
between 300 and 1550 m; the dominant wavelength of the
seismic wave was approximately 2820 m. Thus, many of the
layers were larger than the limits of effective media theory, yet
smaller than the resolution of ray theory. The responsk of the
model to elastic waves (Fig. 5a) displays the characteristics
noted in the earlier modelling of horizontally layered media.
The data show large amounts of energy scattered within the
heterogeneous lower crust and, most importantly, the prominent reflection from the Moho (marked PmP) confirms that the
apparent lower-crustal velocity is significantly faster than the
average velocity. As in the case of horizontal layering with
either small or large layer widths, the seismic waves travel
within the fast layers and thus the apparent velocity is biased
towards higher values.
The synthetic seismograms produced by the acoustic finitedifference code are noticeably different (Fig. 5b). The lowercrustal heterogeneity does not produce strong reflections, and
Figure 4. 1-D layered model used to compare the results of acoustic
and elastic modelling of wave propagation. The widths of the layers in
the lower crust were determined randomly.
the apparent velocity derived from the first arrivals (beyond
100 km offset) is less than the average velocity of the medium.
Thus, it appears that acoustic modelling of the kinematic
response of multiscale heterogeneities could lead to erroneous
conclusions, i.e. that horizontal layering leads to negative
wide-angle velocity biases. This conclusion raises the following
important questions.
First, what causes the acoustic code to produce slow
apparent velocities for this model? When modelling wave
propagation using the acoustic wave equation, the model is
effectively a compositejuid: in this case a layered fluid. No
shear stresses act at the 'interfaces' within the fluid and thus the
boundary conditions are the same as those for a wave travelling
at normal incidence to the interface. In the case of 1-D layers
that obey the Backus criterion (i.e. are much smaller than the
seismic wavelength), the elastic properties of the equivalent
medium (and thus the apparent wide-angle velocity) are
critically dependent upon the local relationship between stress
and strain at each boundary. A thinly layered fluid does not
produce the same equivalent medium as a thinly layered elastic
solid with the same V p structure. Thus, the acoustic wave
equation does not predict correctly the kinematic response of a
1-D thinly layered stack.
The second important question is how does this conclusion
affect the results of other studies of heterogeneous media that
utilized acoustic methods? In particular, Korn (1993), Roth,
Miiller & Snieder (1993) and Muller, Roth & Korn (1992) used
ray-perturbation theory and an acoustic finite-difference code
to study the velocity shift due to heterogeneities. Fig. 6 shows
the velocity shift measured by Korn for a range of heterogeneity scales (ka). In the model illustrated in Fig. 4, the
heterogeneities had values of ka between 0.6 and 3.0 and the
arrangement produced a significant negative velocity shift
(marked on Fig. 6). The results of the acoustic modelling are
01997 RAS, GJI 129, 269-280
Seismic velocities in heterogeneous crust
215
I
1
f
e
E
-1
Y
+
0
60
120
160
1
1
-
7
2
7
7
3
4
I
7
'
5
6
ka
Distance (km)
Figure 6 . Variation of normalized velocity shift in heterogeneous
media modelled using an acoustic finite-difference code (modified
from Korn 1993). The open circles represent the average, normalized
velocity shifts modelled by Korn (1993) and the length of the line
through each circle shows the standard deviation of this velocity shift.
The solid circle represents the normalized velocity shift estimated from
acoustic modelling of the I-D horizontally layered model (Fig. 5b)
in this study. The velocity shift (6u) is normalized to the product of
the variance of the velocity of the heterogeneous medium (6') and
the medium background velocity (uo). The horizontal and vertical
dashed lines through the filled circle represent the range of values of ka
in the 1-D model used in this study and the estimated uncertainty in the
velocity shift, respectively.
1
2
3
0
40
80
7 20
160
Distance Ikm)
Figure 5 . Synthetic seismic response of the model illustrated in Fig. 4
calculated using (a) an elastic reflectivity code and (b) an acoustic
finite-difference code. The dashed lines mark the traveltimes of PmP
arrivals for a model with a homogeneous lower crust with P-wave
velocity of 7.02 km S S ' . The principal reflections from the Moho in the
layered model are labelled PmP.
broadly consistent with the results of Korn (1993); it is plain
that in the acoustic approximation the kinematic behaviour for
heterogeneities well outside the equivalent medium regime is
correctly predicted. However, acoustic studies of smaller
heterogeneities do not produce the correct response, and in
extreme cases, such as horizontal layering, give the opposite
effect (e.g. a negative velocity bias in the place of a positive
one).
To summarize, if the heterogeneities are wavelength size or
smaller, the acoustic wave equation reproduces the kinematic
response of the elastic medium incorrectly. Therefore, the
following section describes the modelling of 2-D heterogeneities
using the elastic wave equation.
ELASTIC MODELLING
The 2-D elastic modelling was carried out using a secondorder, time-domain, central finite-difference scheme (Kelly
et al. 1976; Kerner 1989).The model dimensions were the same
as for the acoustic modelling; however, to avoid grid dispersion
10 grid points per wavelength were used (as opposed to the
acoustic modelling which used four). Thus, the grid spacing
was set at 45 m. In addition problems were encountered if
01997 RAS, GJI 129,269-280
the time-step was set at the stability limit of the algorithm.
Consistent results required the time-step to be set well below
this limit. The source was placed on the left-hand edge of the
model; this artificial boundary had a symmetry condition. All
other edges were A2 absorbing boundaries (Clayton &
Engquist 1977)-boundaries such as these use the paraxial
approximation to the elastic wave equation to separate the
inward- and outward-going wavefields. This reduces the
magnitude of spurious reflections from the edge of the domain
of computation. Numerical instabilities in the results occurred
when 2-D heterogeneities were placed against the right-hand
boundary, and thus a small portion of the model closest to this
edge and was always kept homogeneous. The effect of this area
upon the resulting seismograms was negligible.
As with the acoustic modelling, the technique was tested
using a homogeneous model and a model with 1-D heterogeneities. The process gave the expected average velocity for
the homogeneous model, and the 1-D model (Fig. 7a) produced a synthetic seismogram very similar to that calculated
using the elastic reflectivity code (Fig. 5a). Thus, in contrast to
the acoustic formulation, elastic finite-difference modelling
yields a positive velocity shift from a medium with horizontal
layering. This algorithm was then applied to models with 2-D
heterogeneities in the lower crust.
Fig. 7(b) shows the synthetic seismogram produced for a
model whose lower crust is a 2-D random medium. This
medium has an exponential autocorrelation function and a
correlation distance of 9.0 km in the x-direction and 0.36 km
in the z-direction. It is clear that the PmP arrivals have been
shifted to earlier times by the heterogeneities; the traveltimes were picked and, using the method described earlier,
an apparent lower-crustal velocity of 7.16 (k0.03) km SKI
was calculated (over an offset range of 45-160 km). This
276 J. Brittan and M. Warner
1
0
-1
-2
-3
0
40
80
120
160
1
Distance (km)
r
E
Y
0
(0
c.
rn
n
8
3
-1
TI
E
-2
E
i=
0
10
80
120
-3
160
Distance (kml
Figure 7. Synthetic seismic response of (a) the model illustrated in
Fig. 4 calculated using an elastic finite-differencecode and (b) a model
with 2-D heterogeneities calculated using an elastic finite-difference
code. The correlation length of the heterogeneities in the x-direction
is 9.0 km and the correlation length of the heterogeneities in the
z-direction is 0.36 km. The data are plotted trace-normalized. In both
sections the dashed line marks the PmP traveltimes for the equivalent
model with a homogeneous lower crust of velocity 7.02 km s-l and the
solid line marks the picked PmP reflection.
80
120
180
Distance (km)
procedure was carried out for a number of different correlation
lengths in the x- and z-directions and for different autocorrelation functions (i.e. exponential, Gaussian and Von
Karman). The shape of the autocorrelation function had n o
noticeable effect on the velocity shifts recorded; the only
important difference between the synthetic data from the three
functions utilized was that the Gaussian functions produced
seismograms lacking in power at high frequencies (Fig. 8).
Fig. 9(a) shows the synthetic seismogram that resulted from
a model whose correlation lengths in the two directions
were equal and Fig. 9(b) illustrates the results when the correlation length in the z-direction was much greater than that in
the x-direction (i.e. the structure is tending towards the thin
vertical layers of the I-D modelling). These results indicate that
the nature of the velocity shift changes dramatically with the
change in correlation lengths. Fig. 10 shows the apparent
velocities measured for the 2-D models as a function of the
Figure 8. Details of the synthetic seismogram derived from models
whose lower crusts were random media with different autocorrelation
functions (Von Karman, exponential, Gaussian) but had the same
correlation lengths in both directions ( a i b=4).
ratio between the correlation length in the x-direction (a)
and the z-direction (b). As the 1-D modelling suggested, the
highest positive velocity shift occurs when the correlation
length parallel to the dominant propagation direction (i.e. a)
is much larger than the orthogonal correlation length (b).As
the two correlation lengths become equal, and thus the
heterogeneities tend towards circular geometries, the velocity
shift tends towards zero. At the other extreme, when the
correlation length in the z-direction is much larger than that in
the x-direction, there appears to be a small negative velocity
shift.
01997 RAS, GJI 129, 269-280
Seismic velocities in heterogeneous crust
277
7.3
t
.
E
7.2
u1
m
;-1
Y
g
Y
._
m
2c
2
7.1
F
a
n
\verage velocity
a
3
0
40
80
120
160
Distance (km)
7.0
6.9
0.0001
0.0010
0.0100
0.1000
1.0000
10.0000
D
f
h/a
?
E
-1
Figure 10. Plot of heterogeneity aspect ratio versus apparent lowercrustal velocity. The aspect ratio of the heterogeneities is defined as the
ratio of the correlation length in the z-direction, b, to the correlation
length in the x-direction, a.
F
2
3
40
80
I20
780
Distance (km)
Figure 9. (a) Synthetic seismic response of a model with 2-D heterogeneities calculated using an elastic finite-difference code. The correlation length of the heterogeneities in the x-direction is 0.36 km and the
correlation length of the heterogeneities in the z-direction is 0.36 km.
(b) Synthetic seismic response of a model with 2-D heterogeneities
calculated using an elastic finite-difference code. The correlation length
of the heterogeneities in the x-direction is 0.36 km and the correlation
length of the heterogeneities in the z-direction is 1.125 km. The dashed
line on both sections marks the PmP traveltimes for the equivalent
model with a homogeneous lower crust of velocity 7.02 km s-'.
DISCUSSION
The main conclusion that can be drawn from the modelling is
that a lower crust with a constant average composition (in this
case a bimodal mixture of felsic and ultramafic materials) can
have a wide range of apparent seismic velocities depending
upon the way the material is spatially arranged. Fig. 11 shows
the apparent velocity measured at wide angles plotted against
the correlation length of the heterogeneities in the different
arrangements investigated. The correlation length is taken to
be the x-direction. We have used the algorithms of Shapiro
& Hubral (1994) as a comparison to our results. Shapiro &
Hubral (1994) provide a general description of the kinematic
response of an elastic, 1-D, randomly layered stack across the
whole frequency domain and for a variety of incident wave
angles. The algorithms are based upon wavefield localization
and self-averaging techniques. The velocities measured at
wide angles range from 6.83 (f0.02) km s-l for the model
01997 RAS, GJI 129,269-280
with the lower crust composed of thin vertical layers, to
7.47 (k0.04) km s-' for the model with the lower crust composed of thick horizontal layers. All models have the same
composition-the only difference between them is their local
arrangements. All the velocities can be compared with the
velocity measured from the model with the lower crust composed of an isotropic mixture of the two component phases,
7.02 km SK'. The 1-D inhomogeneous structures produce
significant biases in the velocity measured at wide angles; the
thick layering, in this case, producing over twice as much
velocity shift (with respect to the average velocity) as the thin
layering. If either of the 1-D inhomogeneous physical
situations modelled (thick, horizontal or thin, vertical layers)
were interpreted in terms of a homogeneous layer, the velocity
attributed to this layer and the composition inferred by such a
velocity would be significantly wrong. The 2-D models produced apparent velocities that were intermediate between the
end-member 1-D models. These results are generally in good
agreement with the work of Mukerji et al. (1995). However,
these authors incorrectly asserted that the apparent velocity in
three dimensions is faster than in two dimensions, which is, in
turn, faster than in one dimension. This is true only for 1-D
wave propagation parallel to the axis of symmetry. It has been
clearly shown in this work that waves travelling in 1-D media
perpendicular to the axis of symmetry undergo a velocity shift
of a greater magnitude than is seen in 2-D media.
LOWER C O N T I N E N T A L CRUST
A number of authors have argued that some process must work
to remove a significant proportion of mafic material from the
continental crust during its evolution (Nelson 1991, 1992; Kay
218
J. Brittan and M . Warner
1
7.6 -7-
t l
.. 8.ctruv..
6.0
I
s
r
0
z
8
..
I
0
s
0
--7---7---T--0
0
0
0
0
0
9
v
9
s
8
z
9
0
0
F
&a
Figure 11. Plot of apparent velocity of the lower crust measured at
wide angles against the dominant correlation length of the heterogeneities in the lower crust (normalized to the wavenumber of the
investigating seismic wave). Marked on the diagram are the results of
the numerical modelling of 1- and 2-D models plus the analytical
results calculated using the algorithms of Shapiro & Hubral(l994) and
Backus (1962). The angles on the analytical results refer to propagation
angles relative to the symmetry axis of the layering. The numbers refer
to the results from 1-D models with thick horizontal layers (1) and thin,
vertical layers (2) and the 2-D models that had ratios between their x
and z correlation lengths of (3) 0.006 (4)0.04 ( 5 ) 0.16 (6) 1 and (7) 3.13.
& Kay 1991). These suggestions are based upon the different
geochemical (Pearcy, DeBari & Sleep 1990) and geophysical
signatures of island arc crust (Suyehiro et al. 1996) and bulk
continental crust (Christensen & Mooney 1995). It is possible
to hypothesize that the mismatch in geophysical signature
occurs because the seismic velocities measured by wide-angle
surveys over the continental crust are consistently biased
towards slow values, and thus towards felsic compositions, by
the presence of heterogeneity. We can use the results of our
modelling studies to investigate this hypothesis.
To test the validity of the hypothesis, three inter-related
subhypotheses must be found to be true. These are: (1) there are
arrangements and scales of heterogeneities that lead to large
reductions in apparent velocity in wide-angle seismic experiments; (2) that such arrangements are geologically feasible;
and (3) that there is extensive evidence for such structures
in the lower-crustal seismic record. This paper has dealt
extensively with the first subhypothesis. As is clearly shown by
Fig. 11, the largest negative velocity shift occurs when the
heterogeneities are arranged as thin (a<<A), vertical layers.
Increasing the scale length of the heterogeneities, while
retaining the vertical alignment, leads to a decrease in the
magnitude of the negative velocity shift. Changing the aspect
ratio of the heterogeneities towards a more spheroidal nature
leads to the velocity shift tending towards zero. Any structure
with horizontally extended heterogeneities leads to a positive
velocity shift for wide-angle waves.
As the first subhypothesis appears to be true, the second
subhypothesis must be considered; is an arrangement of thin
vertical layers, alternating in composition between fast and
slow rock types, geologically feasible? Compositionally, this
arrangement seems plausible. As previously mentioned,
experimental studies of lower-crustal rock types indicate that
the lower crust is a combination of felsic (slow) and mafic/
ultramafic (fast) lithologies (Rudnick & Fountain 1995).
Structurally, such an arrangement would appear to be the
logical end-product of the intrusion of mafic melt into
extending felsic crust. Sheet-like mafic intrusions tend to form
perpendicular to the principal regional stress; thus intruded,
extending crust would be expected to be riddled with vertical,
mafic dykes. However, as explained by Holliger & Levander
(1994), the stress regime during continental extension is not
constant and vertical dykes tend to transform locally into
horizontal sills in zones of rheological weakness. Thus, the
lower crust could be expected to contain a mixture of vertical
and horizontal heterogeneities (as in Fig. 3 of Holliger &
Levander 1994). Wide-angle waves travelling through such
crust would undergo both positive and negative velocity shifts
and it is likely that such arrangements would have little overall
effect on the apparent seismic velocity. Therefore, in testing the
second subhypothesis, thin, vertical layers are geologically
feasible but physical modelling suggests that large areas of
associated horizontal layers are likely to be present and that
these areas will considerably reduce any negative velocity shift.
Finally, attention must focus on the third subhypothesisthat there is evidence for extensive, thin, vertical layering in the
lower crust on the seismic record. Emmerich (1992) modelled
the near-normal-incidence seismic response of a lower crust
whose heterogeneities have a variety of ratios of x correlation
length ( a ) to z correlation length (b). It was shown that when
the heterogeneities are arranged as thin, vertical layers (i.e.
b>>a),the reflectivity of the lower crust drops to virtually zero.
This is in sharp contrast to the response of the horizontally
extended heterogeneities(a>>b) and isotropic heterogeneities
(a=b), which both show high levels of coherent reflectivity.
Such high levels of reflectivity also characterize the continental
lower crust-this suggests that the third subhypothesis is false.
In summary, evidence from near-normal-incidence seismic
data and physical modelling suggests that the continental
lower crust is dominated by structures that would cause a
positive velocity shift (if any). Thus, it is concluded that the
apparent loss of mafic material during the evolution of the
lower continental crust cannot be explained solely in terms of
velocity shifts due to the presence of a heterogeneity. In addition, the results of the modelling clearly indicate that the task
of estimating rock composition from seismic velocities is
severely complicated by the effects of multiscale structure.
As noted in the introduction to the modelling, the model
chosen to illustrate the effects of heterogeneity on seismic
velocity was an extreme case in terms of lower-crustal composition and thus is likely to misrepresent the true scale of
velocity shifts in the deep crust. In Table 1 we show the results
of modelling the lower crust using a number of different
bimodal compositions. The velocity shift quoted is that calculated for the case of a wide-angle survey over thick, horizontal,
1-D layers-this represents the magnitude limit of the velocity
shift possible due to heterogeneity (e.g. Fig. 11). As argued
above, it is likely that any velocity shifts present in real studies
of the lower continental crust have this polarity (i.e. positive).
01997 RAS, GJI 129,269-280
Seismic velocities in heterogeneous crust
~
Table 1. Comparison of the velocity-shift magnitude for different
bimodal lower-crustal compositions. The shift magnitude measured
is that for a model with large, 1-D heterogeneities. All models
comprise the two components in equal proportions. The plagioclase/
clinopyroxene mixture is that used in the modelling throughout this
paper; the other mixtures are some of those applicable to the lower
crust suggested by the work of Christensen & Mooney (1995). The
values for each composition are taken from their Table 4 assuming a
depth of 30 km and an average lower-crustal temperature of 467 "C.
Component 1
Component 2
plagioclase
tonalitic gneiss
felsic granulite
rnafic granulite
clinopyroxene
mafic granulite
mafic granulite
mafic garnet
granulite
Average Modelled Velocity
Velocity Velocity
Shift
km sK1
km s-'
km s-l
7.02
7.45
+0.43
6.46
6.57
+0.11
6.52
6.60
+0.08
6.86
6.91
+0.05
As is to be expected, Table 1 shows that the largest velocity
shifts occur for models whose components have highly contrasting properties, for example in a lower crust composed of
equal mixtures of mafic granulite and mafic garnet granulite
(as suggested by Christensen & Mooney 1995) the velocity shift
is likely to have the same magnitude as the errors in the velocity
estimates.
CONCLUSIONS
The modelling work in this study has clearly indicated the
importance of heterogeneity when interpreting the velocities
derived from wide-angle seismic experiments. The 1-D models
explicitly illustrated that large-scale heterogeneities lead to
apparent velocities that overestimate the average velocity of
the heterogeneous medium and that small-scale heterogeneities
can either over- or underestimate the average velocity,
depending upon their arrangement relative to the direction of
wave propagation. In particular, for any negative velocity
bias to be present in a wide-angle experiment, the heterogeneities must be arranged so that their axis of symmetry is
predominantly parallel to the dominant direction of wave
propagation (i.e. tending with offset towards the horizontal).
Introducing 2-D arrangements of heterogeneities leads to
velocity biases that are smaller than those seen in the 1-D
models, although the geometrical effect is the same (more
vertical heterogeneities tend towards negative velocity shifts).
The key to understanding the magnitude and direction of
any velocity shift is the connectivity of the phases (for a useful
description of connectivity in the crust see Goff & Levander
1996). The largest magnitude shift in velocity is seen for
the model whose fast-velocity heterogeneities are the most
uninterrupted in the dominant direction of wave propagation,
the thick-layered 1-D model. Conversely, the smallest
magnitude shift was found in the model whose fast velocity
heterogeneities existed as spherical inclusions, i.e. were almost
entirely unconnected. Any models that showed a negative
velocity shift had well-connected fast heterogeneities whose
orientation was perpendicular to the dominant direction of
wave propagation. It is therefore an important step to
model accurately the connectivity of the phases present in the
deep crust when trying to understand the velocities derived
from seismic surveys. The work of Goff & Levander (1996)
provides a method of simulating sinuous connectivity in
01997 RAS, GJI 129,269-280
219
stochastic models of the deep crust; models with high levels
of connectivity should show velocity shifts depending upon
the relative orientations of connected phases and wave
propagation.
Finally, the comparison between acoustic and elastic
modelling techniques of small heterogeneities has indicated the
need to use elastic modelling if the waves are propagating
at angles other than parallel to the heterogeneity axis of
symmetry.
ACKNOWLEDGMENTS
JB was supported through the duration of this work by a
scholarship from the Shell International Petroleum Company.
We would like to thank Claudia Kerner and Gerhard Pratt
for their contributions to useful discussions on the modelling,
and Nikolas Christensen and Dave Smythe for their helpful
reviews.
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