Density heterogeneities are the source of mass transport in the Earth. However, the 3-D density structure remains poorly constrained because travel times of seismic waves are only weakly sensitive to density. Inspired by recent developments in seismic waveform tomography, we investigate whether the visibility of 3-D density heterogeneities may be improved by inverting not only travel times of specific seismic phases but complete seismograms.

As a first step in this direction, we perform numerical experiments to
estimate the effect of 3-D crustal density heterogeneities on regional seismic wave propagation. While a finite number of numerical experiments may not
capture the full range of possible scenarios, our results still indicate that
realistic crustal density variations may lead to travel-time shifts of up to

Our numerical experiments suggest that waveform perturbations induced by
realistic crustal density variations can be observed in high-quality regional
seismic data. While density-induced travel-time differences will often be
small, amplitude variations exceeding

Lateral variations in density are the driving force behind mass transport in
the Earth, from crust to core

Unlike seismic velocities that can be inferred from the travel times of elastic waves, unambiguous information on density is difficult to find in most seismic observables.

Within the framework of seismic ray theory

Unlike body wave travel times, the frequency-dependent travel times of Rayleigh
waves reveal significant non-zero sensitivity to density variations

At the long-period end of the seismic spectrum, the gravest normal modes of
the Earth are sensitive to long-wavelength density structure as a result of
the gravitational restoring force

Despite these difficulties, various attempts have been made to constrain 3-D
density structure in the Earth. On the global scale, geodynamic data,
including estimates of plate motion history and the location of subducting
slabs, may be used to constrain the broad distribution of density in the
mantle

On regional scales, several authors jointly inverted body-wave travel times
and gravity data under the assumption that seismic velocities and density are
almost uniformly scaled to each other

With the steadily increasing quality of seismic data, new observables with
sensitivity to 3-D density variations are becoming sufficiently robust.

In addition to improving data quality, new opportunities may arise from the
development of full-waveform inversion techniques that are capable of
exploiting complete seismograms without being restricted to the well-known
seismic phases

As a first step towards full-waveform inversion for regional density
structure, we present a study on the imprint of 3-D density heterogeneities in
the crust on seismic wave propagation in the period range from

Following a presentation of the numerical setup, we will present detailed
analyses of travel time and amplitude variations induced by 3-D crustal density
heterogeneities. We expect scattering to be the dominant mechanism by which
density heterogeneities influence the seismic signal. Scattering is most
effective when scatterers are of similar size or smaller than the wavelength,
which is why we will study the influence of frequency, propagation distance
and medium complexity. Being focused on a future full-waveform inversion for
density, we do not consider specific seismic phases, but try to provide
ensemble estimates of waveform perturbations. Given the complexity of
regional-scale seismic waveforms at periods below

To assess the impact of 3-D density heterogeneities in the crust on seismic wave propagation, we compute numerical solutions to the elastic wave equation

The grid of receivers (black triangles) on the surface of the computational domain.
The source is located at 5 km depth; its location and orientation are indicated by the beach-ball plot.
The receiver marked by a large red triangle and at an epicentral distance of

For the numerical solution of Eq. (

Our computational domain is a spherical section that is

Realisations of random density variations. Left:

Because the true 3-D density structure of the crust is insufficiently
constrained, we use synthetic random density models in our numerical wave
propagation experiments. For this, we superimpose random velocity and density
variations with pre-defined correlation lengths in the horizontal and
vertical directions onto the crustal part of the background model, i.e. the
upper

To ensure that the amplitudes of velocity and density variations are
realistic, we combine information from tomographic models and empirical
velocity–density scalings. For this, we compute the root-mean square (rms) of
the

Comparison of synthetic seismograms for homogeneous and heterogeneous crustal densities in
the broadest frequency band from

In our numerical experiments, we use media with homogeneous crustal density
and random 3-D variations in

Since our ultimate goal is to use complete three-component seismograms to
constrain density in the Earth, we do not compare isolated and well-defined
seismic phases. Instead, we compute time- and frequency-dependent travel-time
and amplitude differences. For this, we bandpass-filter the seismograms into
a pre-defined frequency band and apply a zero-centred moving window

Lateral and vertical correlation lengths of random medium variations used to assess the influence of medium complexity.

While more information-rich quantifications of seismic waveform differences
may be constructed, for instance on the basis of wavelet transforms

In the following sections, we present a phenomenological study on the impact
of crustal density heterogeneities for media with different horizontal and
vertical correlation lengths, summarised in Table

We start with the analysis of media with

Component-wise time shifts and relative amplitude differences in the
frequency band from

Waveform differences mostly tend to increase with increasing travel time, in
accord with the expectation that (multiply) scattered waves should arrive
later than the primary waves by which they have been excited. The magnitude
of the time shifts are approximately independent of the component, reaching
around

Figure

In addition to the dependence on the random velocity and density structure,
Fig.

The same as in Fig.

A variant of Fig.

This preliminary, and mostly visual, analysis of frequency dependence indicates that waveform differences are primarily caused by scattering that transfers energy from the large-amplitude N–S component onto the smaller-amplitude E–W and vertical components. Constructive and destructive interference between primary and scattered waves may cause the wave amplitudes to deviate in both directions. An increase of amplitudes may be further supported by additional wave focusing induced by 3-D density heterogeneities. The approximate frequency independence of travel-time differences, however, can hardly be explained with basic wave propagation intuition.

Normalised histograms of time shifts (left) and relative amplitude differences (right) averaged
over five random media realisations with

To make our analysis more comprehensive and efficient, we compute histograms
of time shifts and relative amplitude differences for all 930 stations in the
receiver grid. In line with our future goal, which is to use full-waveform
inversion to constrain 3-D density variations, we do not consider specific
seismic phases, but longer time series that comprise body, surface and
scattered waves. After calculating the misfits for all of the receivers of
the grid, we stack their values into histograms, each histogram corresponding
to a different frequency band. The values that we consider in the stacking
procedure are measured up to

Relative amplitude differences for the broader frequency band, i.e. for
frequencies that are on average higher, have a considerably larger spread
than at lower frequencies. In the

Top row: Normalised histograms of frequency-dependent time shifts for a single random medium
realisation with

The dependence of time shifts on frequency is more complex, as shown in the
left panel of Fig.

To investigate this phenomenon further, we show histograms for the lowest and
highest frequency bands for a single random medium realisation and for the
three different components in the top row of Fig.

Normalised time shifts (left) and relative amplitude differences (right) for stations in
two epicentral distance ranges:

To investigate whether density-related travel time and amplitude differences
are only local effects or accumulate with propagation distance, we plot
histograms for stations in two different epicentral distance ranges:

Standard deviations of time shifts and relative amplitude differences as a function of frequency bandwidth.

For epicentral distances between

Histograms of time shifts (left) and relative amplitude differences (right) for
a complex medium with

In order to reveal the physical origin of the waveform differences, we
consider random media with different lateral correlation lengths, listed in
Table

As shown in the top row of Fig.

The histograms in Fig.

Our numerical experiments show that 3-D crustal density heterogeneities may lead to both positive and negative variations in the travel times and amplitudes of seismic waves. This indicates that 3-D density structure leaves an imprint on regional seismic wave fields that goes beyond simple scattering attenuation of the main arrivals.

Illustration of finite-frequency travel-time shifts induced by a 38 % density perturbation. Top row:
Wave field snapshots before, during and after the wavefront interacts with the density heterogeneity, marked by the dashed
box. The interaction with the density heterogeneity causes reflections and amplitude changes of the direct wave. The onset
of the wavefront, however, remains unaffected. Bottom row: Synthetic seismograms taken at the position of the black
circle in the top-row snapshots. The actual onset time of the waveforms at around

To understand the effects which play a major role in wave propagation, we look first at the misfit histograms for different frequency bands
(Sect.

Our lowest bandwidth has a peak frequency of

An increase of the peak frequency of the bandwidth in our particular case means observing less scattering for certain waves, however, it also means a proportional increase in relative propagation distance. This implies that we will observe waveform differences accumulated for bigger number of wavelengths, and the amount of large non-zero density-related misfits will increase. This is an effect that changes the histogram shape in a way opposite to moving away from resonance. The larger propagation distance could then be one of the reasons behind the broader histogram spread for the higher frequency band.

Increasing the relative propagation distance is equivalent to moving further
from the source, which is consistent with Sect.

While the results for different frequency bands are physically governed by
the amount of scattering and the length of the relative propagation distance,
and the results for different epicentral distances by the length of the
propagation distance only, for various medium complexities we observe how
important scattering is for sensitivity to density. The noticeable change in
histogram shape in Fig.

As we show, the behaviour of density-induced misfits can be most often
explained by an interplay of two physical parameters: the amount of
scattering and the relative propagation distance. However, not all of the
observed features can be interpreted on this ground. For instance, travel-time
variations do not seem to exhibit a pronounced frequency-dependence, in
contrast to amplitude variations that decay rapidly with decreasing frequency
(see Sect.

Figure

We should also take into account the possible signal processing artefacts
that may play a role in our analysis. In comparing between different
frequency bands, we are effectively changing the bandwidth used. Therefore,
some of the large time shifts that are visible in the lower frequency
histogram in Fig.

Naturally, scattering is a complex process that depends on the ratio of the
scatterer size to the wavelength, the strength of scatterers and the source
power

The density-induced waveform differences that we found in our numerical
experiments are above the noise level of many of today's regional-scale
seismic recordings. While this indicates that density heterogeneities do
leave a measurable imprint, it does not automatically imply that crustal
density structure can be easily recovered in a tomographic inversion.
Trade-offs with

Finally, we note that the amplitude of the secondary wave field scattered off
density heterogeneities may have similar or smaller amplitudes than globally
propagating waves, e.g. PcP, PcS or ScS. Therefore, care needs to be taken
when wave propagation is modelled regionally

In the absence of detailed information on crustal density structure on
regional scales, we base our numerical experiments on realisations of random
Earth models. To ensure that the random models are plausible, we translate
rms variations in

For simplicity, we assume that the amplitude spectrum of the crustal velocity
variations is white, meaning that velocity and density variations have nearly
identical power at all scales considered in this study, i.e. from

Uncertainties in velocity–density scalings are mostly caused by the natural
scatter of the velocity–density relation in natural rocks. While

In the light of these uncertainties, it must be kept in mind that the waveform variations resulting from our synthetic random density heterogeneities represent a first rough estimate. It is intended to reveal the first-order effects but not the smaller details that certainly depend on the characteristics of a specific region.

The shifts in travel time observed here as a result of density structure may
cause a bias in velocity structure obtained in tomographic models. In order
to obtain an estimate of these velocity biases, we take a simplified
approach. We consider the highest frequency band from

Based on these simplifications, we estimate that the shear velocity bias for
an epicentral distance of

Fractional attenuation bias

To quantify potential biases in attenuation induced by unknown 3-D density
structure, we adopt similar simplifications as in the previous section. In
the ray theory approximation, relative amplitude differences between
attenuated and attenuation-free waves are given by

Taking the variance of the relative amplitude differences of

We presented a series of numerical experiments to study the effect of 3-D
crustal density heterogeneities on regional seismic wave propagation in the
frequency range from

While numerical experiments can of course never be exhaustive, our series of
tests still allows us to make a limited number of general statements: for
media with

Both amplitude and travel-time variations increase with increasing epicentral
distance. This indicates that density does not only have a local effect,
which is an essential prerequisite for the applicability of tomographic
methods to constrain 3-D density in the crust. Waveform perturbations clearly
increase with increasing medium complexity. They are practically negligible
in transmission mode, i.e. when the correlation length of the medium
heterogeneities is much larger than the wavelength. However, when the
correlation length approaches the wavelength, density-induced waveform
perturbations can be observed easily. Recent regional-scale full-waveform
inversions operate in a regime where resolved heterogeneities have
characteristic sizes comparable to the wavelength

Our most important finding is that waveform perturbations induced by
realistic crustal density variations can certainly be observed in modern,
high-quality regional seismic data. While travel-time differences of typically
less than

The wave propagation package SES3D is a free open source software released under
Apache 2.0 License. It is available for download at:

The synthetic data used in this study, along with python tools for random media generation,
signal comparison and histogram stacking, are to be found at:

We generate random media with the Fourier method, widely used in
seismological research

Agnieszka Płonka performed all 3-D numerical wave propagation experiments. All three authors were involved in the design of the experiments, the interpretation of results and the manuscript writing.

The authors would like to thank Hanneke Paulssen, Ivan Pires de Vasconcelos and Jeannot Trampert for numerous interesting discussions, and two anonymous reviewers for their constructive comments. This research was supported by the Swiss National Supercomputing Center (CSCS) in the form of the GeoScale and CH1 projects, and by the Netherlands Organisation for Scientific Research (VIDI grant 864.11.008). Edited by: M. Malinowski Reviewed by: two anonymous referees