Introduction
Since the pioneering work of , regional tomographic models
have had a considerable impact on our understanding of the upper mantle.
A tremendous increase in data quantity and improvements in inversion
techniques have greatly enhanced tomographic resolution power in recent
years. The quality of the models is however fully dependent on the quality of
the travel time residuals the models are based upon: better measurements will
ultimately translate to better models, and good routines for travel time
measurements are therefore crucial.
Relative travel times, in most cases defined as relative to the mean arrival
time of each event, or in some cases relative to a reference station, are
usually measured using a cross-correlation method. These methods exploit the
similarity of seismic waveforms measured at different stations to estimate
precise travel times in a seismic network
e.g.. Inherent in
all cross-correlation methods is a risk of cycle skipping during analysis
e.g., meaning that an earlier or later
wiggle in the seismogram is interpreted as the arrival of the phase of
interest. Initial alignment of traces is therefore necessary such that
a narrow window can be used in the search for the maximum of the
cross-correlation function. Predictions from theoretical calculations of
arrival times e.g. are usually not sufficient to solve this
issue e.g., as the travel time residuals are commonly
larger than or similar to wave periods in the highest frequency range.
Picking of approximate phase arrival times is therefore usually performed
manually before cross-correlation analysis.
recently proposed an alignment procedure to avoid manual
picking. This Iterative Cross-Correlation and Stack algorithm (ICCS) and
other recent tools e.g., are
however tailored for measuring high-frequency signals. With the development
of finite-frequency theory , it has become
essential to measure travel times in several narrow frequency bands. We find from experience
that the measurement of travel times in more narrow high-frequency bands
increases the risk of cycle skipping during analysis in both the
and algorithms, especially for lower
quality events from great epicentral distances (>75∘). An attempt
to remedy this problem in finite-frequency measurements of P-wave arrivals
has been presented by . By introducing a weak constraint
that arrival times should be close to the theoretical ones in a reference
model, they eliminate the data with, among other problems, cycle skipping.
This might however discard extreme but good data. We therefore rather aim at
correcting and using the data suffering from cycle-skipping problems.
We propose here a very simple approach that manages to remedy this problem to
a large extent. Inspired by waveform inversion of surface waves
, we measure arrival times in the lowest frequency band
first and then in successively higher bands, after adjusting for the arrival
times found in the previous frequency band. We apply the ICCS algorithm
followed by the classical Multi-Channel Cross-Correlation method (MCCC) of
along with several rounds of automated data rejection
that increase the robustness of the initial stack in the ICCS algorithm. The
application of the MCCC ensures robust results and provides quantitative
uncertainty estimates important in the further tomographic inversion.
In ACH-type body-wave tomography (ACH refers to the authors in ), the structure of the crust is usually not inverted for, but
its influence on the travel times is taken into account by a crustal
correction applied to each station individually
e.g.. Crustal corrections are usually computed
using ray theory e.g., providing frequency-independent
results. It is well-established that the reverberations in the crust affect
the travel times at different frequencies in different ways, and that crustal
delay times of high-frequency and low-frequency data are significantly
different . As finite-frequency tomography
plays on the different sensitivity of high- and low-frequency signals to the
structure, it is essential that the differences between the high-frequency
and low-frequency residuals are not biased by inadequate crustal corrections.
and have shown the importance of this
frequency-dependence for P waves with applications mainly in thin oceanic
crustal environments. We analyse it here for the continental crust and in the
framework of relative travel time tomography, where only relative residuals
within the network are important. We analyse the discrepancy between the
crustal corrections calculated with ray theory and those calculated with
a reflectivity algorithm that takes into account the frequency-dependent
effect of the crust on the travel times. We find that the discrepancy is
particularly significant in the presence of sedimentary layers and that,
depending on the geological context and used frequency range, one should
always consider the need to apply frequency-dependent corrections.
Our residual measurement procedure and the frequency-dependence of the
crustal corrections are illustrated using data from a network in southwestern
Scandinavia .
Measurement of travel time residuals
We apply our analysis to seismological data recorded by the temporary MAGNUS
network between September 2006 and June 2008 , by the
temporary DANSEIS network from April 2008 to June 2008, by the temporary
CALAS stations and by permanent stations in the study
area (Fig. ). Direct P and S phases were studied from
earthquakes with epicentral distances of 30–95∘
(Fig. ). To have an even azimuthal coverage, we used events
with smaller magnitude in the southern and western quadrant, decreasing the
data quality and increasing the need for a robust algorithm for travel time
measurement.
Topographic map of the study area with location of seismological
stations and national borders. Numbers xx are short for stations NWGxx in
the MAGNUS network.
Maps of earthquake sources for P- and S-wave travel times. The events
displayed have given travel time residuals in at least one frequency band.
Left: events for P waves. Right: events for S waves.
In short, the data-processing sequence to measure residuals is performed in
four steps. A major step to make the procedure robust to cycle skipping is to
perform the processing first on the lowest frequency band, and then on
successively higher frequency bands. Corrections for ellipticity and crustal
structure are calculated after step 4 and will be presented in the next
section. We use data from a 6.4 M event at the Andreanoff Islands
(Δ=68∘, back azimuth 5∘), 15 April 2008, to
illustrate the data processing.
High-frequency P-wave traces before and after alignment by ICCS.
(a) Seismograms after alignment by theoretical arrival times from
AK135 and data rejection in step 2 (Sect. ).
(b) Seismograms after alignment by ICCS and additional rejection of
incoherent traces in step 3 (Sect. ). The colours are used to
distinguish the different traces better and no significance is attached to
them.
Step 1: preprocessing
This step, which is detailed in , consists of filtering,
calculation of theoretical phase arrival time, and windowing according to the
frequency content of the band and the phase in question. Frequency bands for
P waves are 0.03–0.125 Hz (33–8 s) and
0.5–2 Hz (2–0.5 s) to isolate the secondary noise
peak in southern Norway around 0.2 Hz (5 s). S waves are
bandpass-filtered in the ranges 0.03–0.077 Hz
(33–13 s) and 0.077–0.125 Hz
(13–8 s). These frequency ranges are similar to those used
generally in finite-frequency tomography
, with the exception that S
waves are usually measured up to slightly higher frequencies (up to
0.5 Hz in ). note that
their highest frequency for S waves, 0.4 Hz, produces few data.
P-wave stack and individual traces after alignment by ICCS.
(a) Low-frequency band. (b) High-frequency band. The array
stack is coloured blue, the accepted traces are red and green traces are the
ones that will be rejected from analysis before step 4
(Sect. ).
Step 2: data rejection based on the envelope of data
The first round of data rejection excludes traces based on an iterative
comparison of their envelopes with the envelope of the stack (see
for details). This ensures that
strong outliers do not ruin the computation of the array stack before
applying the ICCS algorithm in the next step.
Step 3: ICCS algorithm tailored for several frequency bands
To avoid cycle skips in the Multi-Channel Cross-Correlation analysis (MCCC,
step 4), it is essential that traces are initially aligned, either by manual
picking or by an automatic procedure , and that
noisy or incoherent traces are excluded from the analysis. We use the
Iterative Cross-Correlation and Stack algorithm ICCS, to
initially align traces, but we do not pick the absolute arrival time of the
stack as in , since our focus is relative travel times.
The ICCS algorithm was recently developed to replace the
unavoidable initial phase-marking part of the cross-correlation procedures,
and is part of the AIMBAT tool (Automated and Interactive Measurement of
Body-wave Arrival Time) . In the ICCS algorithm, each
individual trace is correlated with the stack and the time lag associated
with the maximum of the cross-correlation function is found. Then each
individual trace is aligned according to the time lag and relative to the
stack in an iterative procedure where the stack is updated for each
iteration. This iterative alignment procedure is very similar to the adaptive
stacking in , except that use an
L3 misfit criterion to determine the time lags of the traces with respect to
the stack instead of the maximum of the cross-correlation function.
Figure shows the high-frequency P-wave traces before and
after alignment by ICCS.
After alignment by ICCS, the cross-correlation coefficient and mean spectral
coherence between each trace and the stack are calculated. A weighted average
of the two is computed and traces with a value lower than a user-defined cut-off
(usually about 0.5) are excluded. This procedure rejects data with
a significantly different shape than the array stack. Figure
shows the low- and high-frequency P-wave stacks after alignment by the ICCS
algorithm with a few green-coloured incoherent traces that will be rejected
from further analysis.
High-frequency S-wave stack and traces in the first iteration of the
ICCS algorithm. The array stack is coloured blue and the individual traces
red. (a) High-frequency traces aligned according to the theoretical
arrival time predicted with AK135. (b) High-frequency traces aligned
according to the low-frequency travel time residuals from the MCCC analysis
in step 4 (Sect. ).
Cycle skipping is a recurrent problem in cross-correlation analysis when the
period of the data and the travel time residuals are of the same order of
magnitude. For the low-frequency bands of both the P and S waves, the travel
time residuals are much smaller than the periods of the waves, and the risk
of cycle skipping in the pick of cross-correlation maxima is therefore very
low. In contrast, there is a significant risk of cycle skipping in the ICCS
algorithm in case of high-frequency low-quality data. This mostly happens in
the first iterations where the array stack is less robust, because it is
calculated from traces aligned according to the theoretical arrival time.
With a poor stack there will be little difference between local and global
maxima in the cross-correlation functions, and the algorithm might therefore
misplace a few traces, possibly permanently.
This problem of cycle skipping in the ICCS algorithm is probably enhanced in
our data set because most events have epicentral distances >70∘,
giving relatively strong attenuation of the high-frequency content of the
data. For events with shorter epicentral distances, even with magnitudes
M<6.0, the problem of cycle skipping in the ICCS algorithm is almost
non-existent.
To remedy the risk of cycle skipping in the first iterations of the ICCS
algorithm for the higher frequency bands, we use the low-frequency travel
time residuals from the MCCC analysis in step 4 as input time lag to
initially align the traces in the stack. Only reliable travel time residuals
with small standard deviations are used as input time lags. This ensures
a better initial stack in the high-frequency bands and reduces the number of
iterations in the ICCS algorithm.
The effect of the improved initial stack is especially important for the more
band-limited S-wave data. Figure shows the improvement
of the initial stack, when the low-frequency S-wave residuals are used to
align the traces of the high-frequency band, compared to using the
theoretical arrival time from AK135.
Step 4: Multi-Channel Cross-Correlation (MCCC) analysis
Finally, we use the classical Multi-Channel Cross-Correlation method (MCCC) of
to measure the relative travel times ti and
evaluate their uncertainty.
The MCCC algorithm is a robust method for relative travel time measurement
that also provides quantitative uncertainty estimates. In the MCCC method,
relative delay times Δtij between all pairs of stations
i,j are measured using the cross-correlation of seismograms. These relative
delays are then inverted in a least-squares sense for the optimized relative
arrival times, ti, at each station, under the constraint of zero mean,
i.e. ∑ti=0. The equation residuals between observed relative delays
and predicted delays are calculated as resij=Δtij-(ti-tj).
The standard deviation of the equation residuals associated with each trace,
σi, provides an estimate of the timing uncertainty associated with
the relative arrival time ti . An evaluation of the
uncertainty is particularly important in order to be able to invert P- and S-waves arrival times in a consistent manner and interpret jointly the obtained
P- and S-wave models.
If the mean value is subtracted from the final time lags determined by the
ICCS algorithm, they are approximately equal to the relative travel time
residuals calculated by the MCCC algorithm . The main reason
for applying MCCC to the traces aligned by ICCS is therefore the calculation
of uncertainty estimates, σi, from the equation residuals resij.
Working with high-frequency P waves, found that equation
residuals resij>0.5 s are usually the result of cycle skipping
in the cross-correlation analysis. For the low-frequency P waves we find that
a threshold of 0.8 s is appropriate to target the (very few) cycle
skips, and for the S waves we use larger thresholds of 2 s for the
low-frequency band and 1.5 s for the high-frequency band. The
cross-correlation functions associated with large equation residuals are
recalculated in a smaller window near the time lag predicted by the
least-squares solution (ti-tj), as in . The data set
of relative delay times (Δtij) is updated with the new
measurements before a weighted reinversion using the cross-correlation
coefficients between traces as weights. The differences between the
unweighted relative travel times and the weighted relative travel times are
very small.
Thickness of the sedimentary layer and of the crust in the study
region.
Crustal corrections
Before performing a tomographic inversion, the residuals need some
corrections. We compute relative travel time corrections for the ellipticity
of the Earth using the MATLAB implementation of
. These corrections are in general not very important as they
are very small compared to the residuals (-0.05 to 0.09 s for P
waves and -0.10 to 0.14 s for S waves in our case).
Crustal corrections are much more important e.g..
They are usually computed using ray theory e.g.,
providing frequency-independent results. It is well-established that the
reverberations in the crust affect the travel times at different frequencies
in different ways, and that crustal delay times of high-frequency and
low-frequency data are significantly different
. shows in particular that
the analysis of PP waves reflecting at oceanic
locations with a thin crust requires a frequency-dependent correction and
shows the importance of the frequency-dependence for P
waves, in particular for a rather thin oceanic crust. In the case of regional
body-wave tomographies, the absolute difference between the travel times of
the short- and long-period waves is
not an issue since the average travel time, calculated separately in each
frequency band, is subtracted. The important element here is if the
frequency-dependence differs from one station to the other.
Frequency-dependent crustal corrections
In order to verify the adequacy of ray-theory corrections, we have computed
the delays of long-period and short-period waves across the crustal
structures in our study area using both ray theory and the reflectivity
method. The reflectivity method assumes a 1-D model below each station and
does not take into account lateral variations, but does take into account all
reverberations and conversions within the stack of layers. It generates the
three-component impulse response of the layered structure, assuming a plane
wave incident from below with a prescribed incidence angle. We computed the
response of the crustal layer for P and SH waves using the reflectivity
software of and defined the frequency-dependent crustal
travel times as the time of the maximum of the impulse response after
filtering with the same zero-phase bandpass filter as applied to the data. In
order to ensure a common reference time and include the effect of the
topography, the crustal corrections are computed at each station by
calculating the travel times from 50 km below sea level and up to the
free surface, for waves with the prescribed incidence. As the relative travel
time residuals are zero mean, final corrections also have to be zero mean and
the actual corrections applied for each event are the travel times from
50 km depth minus the average correction for the selected data for
the particular event and frequency range. In order to keep a common
reference, we will however show here crustal corrections relative to the
average correction calculated with ray theory.
Our study area comprises a deep sedimentary basin above a rather thin crust
to the south, typical of younger continental regions. Most of the area to the
north is an older cratonic region with a purely crystalline crust of
typically 45 km to the east, thinning to smaller values to the west,
closer to the Atlantic continental margin. Its lateral variations are thus
representative of many settings for regional body-wave tomography. At each
station, we use a detailed local crustal model compiled in
from a range of sources, especially the work of
and . The model is
specified as a 1-D model below each station. The total thickness of the
sedimentary layers and the thickness of the crust, which are the two main
controlling factors for crustal corrections, are shown in
Fig. , and the S-wave velocity down to 50 km depth
is shown at a selection of stations in Fig. .
S-wave velocity models down to 50 km depth at a selection of
stations located along two east–west profiles.
Figure shows the impulse responses of the crustal
structures for P and SH waves incident at two different stations together
with their bandpassed versions. At
station NWG12 (left plots), located on crystalline crust, the secondary arrivals associated with the
crustal reverberations are late enough not to perturb the location in time of
the maximum of the impulse response after filtering. At station DK08 (right
plots), located on 5 km of sedimentary layers, the first arrival is
followed closely by large arrivals in both the P and SH cases, shifting the
arrival time of the maximum in the bandpassed signals significantly. The long-period waves arrive usually
earlier than predicted by ray theory.
The upper panels of Fig. shows P-wave crustal corrections
calculated by ray theory and reflectivity at short and long periods for all
stations. The examples shown here are for slownesses corresponding to
75∘ epicentral distance, but the same results are found at all
other relevant slownesses. The
corrections calculated with the reflectivity method are not significantly
different from those calculated with ray theory, except at low frequency in
Denmark, in the southern part of the study region, where the discrepancy
reaches about 0.3 s. This discrepancy is clearly associated with the
presence of sediments in the area (Fig. ). We can notice
that the crustal corrections at long periods follow rather closely the
thickness of the crust, even in the south. This simply shows, not
unexpectedly, that the relatively thin sedimentary layers are not seen by the
100 km wavelength long-period P waves. A good approximation would
therefore be to calculate the corrections at long periods with ray theory but
replace the sedimentary layers by crystalline crust in the upper part of the
model.
Impulse response of the crust for P and SH waves at stations NWG12
and DK08, without (blue) and with bandpass-filtering (red for high-frequency
band and green for low-frequency band). The incidence angle corresponds to
75∘ epicentral distance. The time axis shows propagation time from
50 km below sea level.
P (upper panels) and SH (lower panels) crustal corrections for all
stations in the network calculated with ray theory (left window),
reflectivity method in the high-frequency band (middle window) and
reflectivity method in the low-frequency band (right window). All corrections
are with respect to the average correction with ray theory.
Frequency-dependence in the sedimentary region is also apparent for the SH
waves (lower panels of Fig. ). The main difference with the P-wave case
is that we notice a discrepancy with ray theory already in the
high-frequency range, which has a central frequency of about
10 s here. The so-called high frequencies are therefore also affected by
the reverberations in the sedimentary layer. The discrepancy is however
highly non-linearly correlated with the sediment thickness: it is very small
at the stations to the west in Denmark, where we have the thickest sediments
(more than 5 km), but similar to the values of 0.4 to 0.7 s
that are relevant for the lower frequencies at the outskirts of the basin,
with sediments of 1 to 5 km thickness. In our case, the effect of the
reverberations on the SH waves is more complex than for the P waves and
cannot easily by modelled using ray theory only.
In classical continental environments, and for the frequencies that we have
used in our study, the frequency-dependence of the crustal variations is not
associated with variations in crustal thickness, but with variations in the
low-velocity layers in the upper crust. This dependence is of the same order
as the corrections themselves, and cannot usually be neglected.
Average P-wave travel time residuals in the back-azimuthal range 0
to 65∘. Left panels: uncorrected residuals; middle panels: residuals
corrected for topography and crustal structure with ray theory; right panels:
residuals corrected for topography and crustal structure with reflectivity.
The upper panels are for the high-frequency range and the lower panels are for
the low-frequency range.
Residuals before and after crustal corrections
Figure shows high-frequency and low-frequency P-wave
residuals before and after crustal correction. The residuals have been
measured according to the procedure described in Sect.
before crustal correction. In order to have a good station coverage, we do
not show residuals from a particular event but average residuals for
a cluster of events in back azimuths from 0 to 65∘. Averages in
other back azimuths are shown in and show the same features
as those presented here concerning the frequency-dependency of the residuals.
The variation of travel time residuals with back azimuths provides preliminary
information on the spatial distribution of velocity anomalies prior to
inversion, as discussed in . We focus here only on the
effect of the crustal corrections on the residuals.
The dominant pattern in the residual variation is not strongly affected by
the crustal corrections, showing that the signal in this example study is
dominated by mantle heterogeneity. The residuals corrected using ray theory
(middle panels) have somewhat reduced small-scale variations compared to
uncorrected residuals, especially at long periods. Comparison of the spatial
patterns in the low- and high-frequency residuals show that the raw and
ray-corrected residuals are significantly different in the southern part of
the study area, where the low-frequency waves arrive early, whereas the
high-frequency waves arrive late. After correction with the reflectivity
method (right panels), this difference between high- and low-frequency
residuals has disappeared. This clearly show that the low frequencies arrive
earlier than the high frequencies in Denmark because they are not sensitive
to the presence of thick sediments, and this discrepancy should not be
interpreted in terms of frequency-dependent sensitivity of the travel times
to the mantle structure.
Same as Fig. but for the SH waves.
For the S waves (Fig. ), as the two frequency ranges are
closer to each other and the crustal corrections are smaller relative to the
residuals, the trend in the variation of the residuals from high to low
frequencies and from ray-corrected to reflectivity-corrected residuals is not as clear.
The residuals at high and low frequency do not get closer to each other in
Denmark after reflectivity-based corrections. As noticed already, the
reflectivity-based correction in the highest frequency range depends on the
thickness of the sedimentary layer in a very non-linear way: thin layers of 1
to 5 km lead to discrepancies of 0.4 to 0.7 s between the two
types of corrections, whereas thicker layers lead to small discrepancies.
Uncertainties in the crustal model or neglecting its lateral variations may
therefore lead to errors in the crustal corrections that are of the same
order of magnitude as the error related to using ray theory.
We have done the crustal correction after measurement of the residuals. In
the present case, it has the advantage of ensuring that the similarity of the
residuals in the two frequency bands after reflectivity-based correction is
not introduced by our measurement procedure based on measuring low-frequency
bands first. In another setting, it would make sense to introduce crustal
correction before measuring the residuals, at the same time as the correction
for theoretical arrival times, as this would improve the initial alignment
for the ICCS algorithm. One should however then also take into account the
difference between the low-frequency and the high-frequency corrections when
aligning the short-period traces according to the low-frequency residuals. As
the frequency-dependent discrepancies are smaller than the periods used, we
do not expect a large benefit from this procedure, but just a better
coherency.
Discussion and conclusion
A prerequisite for high-quality regional body-wave tomography is high-quality
measurement of the residuals. As finite-frequency body-wave tomography
requires the measurement of residuals in different frequency bands on
bandpassed traces, a number of new
issues arise in the measurement and quality assurance of residuals for this
kind of tomography. We have addressed two of these issues here: how to
measure the residuals in a robust way, and whether the crustal correction
should be made frequency-dependent.
Measuring residuals
We propose an automated processing procedure combining the Iterative
Cross-Correlation and Stack (ICCS) and Multi-Channel Cross-Correlation method
(MCCC) algorithms and tailored for estimating relative travel time residuals
in several frequency bands. Using the low-frequency travel time residuals to
initially align the high-frequency traces increases the reliability of the
automated alignment in ICCS and reduces the risk of cycle skipping in the
MCCC analysis, especially for distant low-quality data. As the only human
interference is the choice of various parameters and a quality check at the
end, there is little risk of drift in the arrival time picking and a high
degree of objectiveness in the data rejection procedures. Combining the ICCS
and MCCC algorithms, we benefit from the advantages of the two methods, the
ICCS providing a robust way of aligning traces for measuring their travel
times and the MCCC providing an estimate of the uncertainties in the
measurements.
Using the long-period residuals as a priori information to adjust the
high-frequency traces carries the risk of propagating errors due to
long-period noise into the high-frequency residuals. To reduce this risk,
only low-frequency residuals with small standard deviations (in the MCCC
sense) are used to align the high-frequency traces. The alignment at these
high-quality stations benefits measurements at all stations as it improves
the quality of the stack in the ICCS algorithm. As long as spurious cycle
skips are not introduced by bad long-period data, our procedure of shifting
traces in time does not modify the waveform or their cross-correlations, and
therefore does not reduce the independency of the low- and high-frequency
measurements.
and used basically the same database as
we do, but employed different methods for measuring travel times. The P-wave
residuals of were estimated by a combination of
cross-correlation measurements and final manual picking of high-frequency P
waves (0.125–4 Hz), whereas the S-wave travel time residuals of
were found by manual picking of S waves on magnified
waveforms in a lower frequency band (0.03–0.125 Hz). An
advantage of using the same cross-correlation procedure on both the P- and S-wave data set, is that uncertainty estimates are calculated in a consistent
manner for both data sets, facilitating a quantitative comparison of the data
sets and their joint inversion, as we have done in
with the data set used here. In a later inversion for seismic velocities,
these uncertainties can be used to weight the data objectively, suppressing
the influence of individual noisy data and low-quality events. The two data
sets can be inverted either jointly for e.g. VΦ and VS,
or for VP and VS separately. If the same
inversion technique and model parameterization is used to invert for
VP and VS, the resulting models can be compared
quantitatively as in e.g. and .
Crust-correcting residuals
Another factor contributing to the quality of the residuals is how the
influence of the non-resolved upper part of the Earth is dealt with. Station
correction terms can be used to absorb the effect of the crust. In order to
avoid absorbing a significant part of the mantle heterogeneity as well, these
terms need to be damped. If the structure of the crust is known well enough,
it is therefore an advantage to correct for the crust even if a station term
is used in the inversion. Although it is well-established that the crust
affects travel times in a frequency-dependent manner
, crustal corrections in regional body-wave
tomography are usually computed independently of frequency, using ray theory
. As finite-frequency
tomography plays on the different sensitivities of different frequencies to
the Earth's structure, and as the differences in the residuals at different
frequencies are usually not very large (35 % of the short-period
residuals for P waves in and similarly in
, less for S waves in and in the
present study), biases in the crustal corrections may harm the
finite-frequency approach or at least limit its usefulness.
We have analysed, in a typical continental setting, the discrepancy between
crustal corrections calculated with ray theory and those calculated with the
reflectivity method, which takes into account all crustal reverberations. In
our continental model of southern Scandinavia, where crustal thickness varies
from 27 to 48 km, we find that Moho depth variations do not produce
any significant discrepancy between the two types of corrections. This is
true for both P and S waves, in any of the rather conventional frequency
ranges that we have used. This is in agreement with and
who show that significant effects occur for thinner crust,
e.g. oceanic crust.
The presence of 1 to 8 km thick sedimentary layers in the upper part
of the crust has, on the other hand, large effects. We find discrepancies with
ray theory of about 0.3 s for long-period P waves and 0.4 to
0.7 s for S waves. For P waves, a valid approach in our case is to
consider that the short-period waves are sensitive to the sedimentary layers
but the long periods “do not see them” and calculate the crustal
corrections with ray theory, but using two different models: one including the
sediments, the other one with sediments replaced by upper crustal rocks. This
approach may not be usable if an intermediate frequency band is introduced,
as in . It is not usable for S waves either, which are
usually measured in frequency ranges such that they interfere in a more
complex way with the sedimentary layers. For sedimentary layers of 1 to
5 km, the S waves arrive early by 0.4 to 0.7 s compared to
ray-theory predictions in both frequency ranges 0.03–0.077 Hz
(33–13 s) and 0.077–0.125 Hz (13–8 s).
The shortest periods are consistent with ray theory for larger sedimentary
thicknesses, but the longest periods are not. This complex and non-linear
behaviour makes it a challenge to compute accurate crustal corrections for S
waves: uncertainties in the thickness of the sedimentary layers or 3-D
effects not accounted for here may lead to large inadequacies in these
corrections. Frequency-dependent station terms may in some cases help absorb
some of the potential errors introduced by ray theory or by uncertainty in
the model.
Although our main purpose is to analyse the crustal corrections required in
finite-frequency tomography, our results are also relevant for traditional
regional body waves, especially in the S-wave case. Due to their lower
frequency content and the higher noise level on the horizontal components, S
waves are usually filtered to rather low frequency in traditional body-wave
tomography. , for example, use the frequency range
0.03–0.125 Hz in their study of the same data set as ours. Their
data are therefore also affected by crustal reverberations. For data filtered
in a large frequency range, the effect of the crustal reverberations will
depend on the dominant frequency in the data and may change from event to
event. used the waveform of the direct P wave to correct the
PP waves from the crustal corrections at the reflection point, ensuring that
the correct spectrum is used for each event. Ideally, this should also be
done in regional body-wave tomography, but, as opposed to the PP case, we do
not have a reference wave that has not propagated through the crust. An
average waveform may be used as reference, and the delays may be measured by
convolving the reference waveform with the impulse responses at the different
stations and cross-correlating with a reference station. For the more
narrow-filtered data used in finite-frequency tomography,
have shown that variations in the crustal corrections due to varying spectral
content within each band can be neglected, and finite-frequency crustal
corrections might therefore be done with a simpler procedure than in the
broadband case.
We conclude that in classical continental environments, the frequency-dependence of the crustal corrections is not associated with variations in
crustal thickness, but with variations in low-velocity layers in the upper
crust. This dependence may have the same order of magnitude as the crustal
corrections themselves and cannot normally be neglected. Ray theory produces
valid corrections in continental regions free from low-velocity sedimentary
layers and non-thinned crust.