With the prospects of seismic equipment being able to measure rotational ground motions in a wide frequency and amplitude range in the near future, we engage in the question of how this type of ground motion observation can be used to solve the seismic source inverse problem. In this paper, we focus on the question of whether finite-source inversion can benefit from additional observations of rotational motion. Keeping the overall number of traces constant, we compare observations from a surface seismic network with 44 three-component translational sensors (classic seismometers) with those obtained with 22 six-component sensors (with additional three-component rotational motions). Synthetic seismograms are calculated for known finite-source properties. The corresponding inverse problem is posed in a probabilistic way using the Shannon information content to measure how the observations constrain the seismic source properties. We minimize the influence of the source receiver geometry around the fault by statistically analyzing six-component inversions with a random distribution of receivers. Since our previous results are achieved with a regular spacing of the receivers, we try to answer the question of whether the results are dependent on the spatial distribution of the receivers. The results show that with the six-component subnetworks, kinematic source inversions for source properties (such as rupture velocity, rise time, and slip amplitudes) are not only equally successful (even that would be beneficial because of the substantially reduced logistics installing half the sensors) but also statistically inversions for some source properties are almost always improved. This can be attributed to the fact that the (in particular vertical) gradient information is contained in the additional motion components. We compare these effects for strike-slip and normal-faulting type sources and confirm that the increase in inversion quality for kinematic source parameters is even higher for the normal fault. This indicates that the inversion benefits from the additional information provided by the horizontal rotation rates, i.e., information about the vertical displacement gradient.

The inversion for kinematic finite-source models of earthquakes plays an important role in the research of earthquake dynamics and the general understanding of earthquakes. Finite-source inversion of earthquakes is non-unique due to the infinite-dimensional inverse problem, which we try to solve with a finite number of observations. A discretization of the fault area by subfaults converts this problem to a finite size inversion, but this does not completely remove the inherent non-uniqueness of the problem. Besides that, noisy data, sparse geographical coverage of seismic stations, the non-linearity of the forward problem, and possible unrealistic simplifications in the parametrization of the fault also contribute to the non-uniqueness.

According to classical elasticity theory, if

With the potential availability of seismometers measuring rotational ground
motions, the goal of this study is to test the effect of incorporating
rotational ground motion data into finite-source inversions for different
fault mechanisms. Following the approach by

We directly compare the results with and without incorporation of rotational
ground motion data for a pure strike-slip earthquake and for a dip-slip
earthquake; i.e., the strike also propagates in vertical direction. Because
rotations around the horizontal axes (i.e.,

To better assess the quality differences we keep the number of seismograms used in the inversion the same for all scenarios. Because using three translational and three rotational ground motion recordings doubles the number of seismograms, we half the number of seismic stations when using the so-called 6C (six-component) data, as compared to only three translational seismograms (3C/three-component data). In the following chapter, we present the Bayesian inversion scheme. Next, we explain the parametrization of the synthetic earthquakes, explain the spatial distribution of the seismic stations, and define the different inversion scenarios. We will then present the inversion results with and without incorporation of the rotational ground motion data for the two different earthquake mechanisms.

Although computation time is drastically increased when applying
probabilistic inversion schemes, it is able to quantitatively analyze
drawbacks such as regularization or falling into local minima in the
iterations during the process of minimizing the misfit between model and
data. We use the basic ideas of Bayesian inversion for a finite fault
developed by

The prior pdf

To quantify the posterior pdf compared to the prior pdf, i.e., the relative
information gain, we use the so-called Shannon's measure of
information gain: the relative information gain

The generation of the databases, in which Green's functions are stored,
is done with the program Instaseis

We illustrate receiver and fault setup in Fig.

44 receivers, marked as gray (only used without rotation data) and
red triangles (used for all inversions), are set up around the source with a
spacing of

The parametrization of the fault and the rupture itself is used in the same
manner as in

2-D illustration of the fault neglecting constant third dimension. The fault is broken up into 24 subfaults, each consisting of 64 point sources. Each subfault will rupture appropriately delayed in time, depending on the distance to the hypocenter. The hypocenter (marked as a yellow star) is set to be exactly between four subfaults for easier computation of the rupture times of the subfaults.

The complete seismic response

Note that there are two parameters (i.e., rupture velocity and rise time) that
are not linearly related to displacement or velocity, contrary to the 24 slip
amplitudes. We therefore invert for 26 kinematic source parameters.

Illustration of the parameter space for the inversion. Rupture
velocity and rise time are set to be homogeneous across the fault. There are
24 slip amplitudes, one for each subfault across the fault. The values range
from

We compare the inversions for two different fault mechanisms. For Scenario I we choose a pure left-lateral strike-slip event (rake parallel in strike
direction). For Scenario II, its dip (angle between the fault and the
horizontal plane) is changed from 90 to

For the first inversions in each scenario, we choose to use three-component velocity seismograms from all 44 stations. To reasonably compare the inversions with and without rotation rate data, we use only half of the stations (22) for the second inversion. With six-component (three velocity, three rotation rate) stations, the same number of seismograms (132) will be used for both inversions. The stations in the second inversions have been selected regularly (every second one) to achieve the lowest azimuthal gap possible.

Our third analysis consists of a statistical evaluation of 200 inversions, each with a randomly selected set of receivers, for both mechanisms. Since our previous results are achieved with a regular spacing of the receivers, we try to answer the question of whether the results are dependent on the spatial distribution of the receivers. Especially the different energy ratios of the rotation components suggest that there should be differences in the inversion results with varying station distribution.

We illustrate the results for both inversions for the 24 slip amplitudes in
Scenario I with a regular station distribution in Fig.

Inversion results for slip amplitudes for a strike-slip fault. Each
panel represents a subfault with the same arrangement as in
Fig.

Increase in information gain for 6C inversions compared to 3C inversions in percent. Subfaults are analyzed in the three horizontal layers (varying depth). The largest ratio of information gain increase for the slip amplitude between the 6C and the 3C inversion is in the deepest subfault layers. Rupture velocity and rise time benefit from rotation rate data for both types of scenarios, but the improvement is larger for the dip-slip scenario.

We illustrate the posterior pdf's for the two parameters that are not
linearly related to displacement, i.e., rupture velocity and rise time, in
Fig.

Inversion results for rupture velocity

In Fig.

Inversion results for slip amplitudes for a dip-slip fault. Each
panel represents a subfault with the same arrangement as in
Fig.

Inversion results for rupture velocity

The increase in information gain for rupture velocity (

The 6C inversions in the previous chapters are carried out with a manually predefined station selection.

The unequal distribution of energy ratios of rotation rate components might
lead to the assumption that there is a higher information gain for the 6C
inversion in Scenario II, because the regular receiver distribution leads
to larger number of receivers lying north and south of the fault than east
and west of the fault. These receivers feature a higher energy ratio of the
horizontal rotation components. To minimize this factor we run 200 6C
inversions with 22 random stations each inversion, for both faulting types,
and we show the results in Fig.

Histograms for the information gain for 200 inversions with random station distribution. The results for a strike-slip fault are shown in the left column and for a dip-slip fault in the right column. We add the information gain for the slip parameters row-wise to better assess the inversion quality dependence with depth. Slip (in gray), rupture velocity (in blue), and rise time (in green) are colored differently to distinguish between the three types of parameters. The statistical analysis validates the previous results. Independent of the choice of receivers (due to the random selection for each inversion) the information gain increase for the 6C inversion decreases with increasing depth; rupture velocity and rise time seem to benefit from the rotation rate data as well. Overall, the information gain increase is higher for Scenario II, even when randomizing the stations for the 6C inversion.

The results for a strike-slip fault are shown in the left column and for a
dip-slip fault in the right column. We look on the slip for the top, middle,
and bottom row individually to better assess the inversion quality dependence
with depth for the different scenarios. Therefore there are five histograms for
the information gain for the 200 6C inversions. We mark the information gain
value for the 3C inversion for each scenario with a red vertical line in the
histogram plots. Slip (in gray), rupture velocity (in blue), and rise time (in
green) are all colored differently to distinguish between the three types of
parameters.

For the top layer of the subfaults, the information gain for a dip-slip fault (right) reaches the highest values, which is also the result from the previous inversion. There are some receiver selections that even increase the information gain compared to the results from the inversion with regular receiver spacing.

The middle subfault layer already features a smaller variance in the information gain distribution for both faulting types. We also see a decrease in information gain increase, compared to the top layer, which is to be expected from the results before. Additionally, there is a higher information gain increase for a dip-slip fault (right).

The deepest layer of subfaults shows the same information gain for 3C and 6C inversions for a strike-slip fault (left). The figure shows that including rotational ground motion does not contribute to improving the results for the slip on the bottom row. This is different for a dip-slip fault (right). Every inversion increases the information gain for slip amplitudes in the bottom layer. We see some inversions in which the information gain doubles compared to the 3C inversion. This major difference in the two scenarios is one of the key aspects of our results. The ability to invert for the slip amplitudes at higher depth is something that we only achieve with the incorporation of rotational ground motion rate data into the inversion. The statistical evaluation of 200 inversions with varying receiver consolidates this.

The inversions for rupture velocity result in a higher information gain for a dip-slip fault, even with random station distribution. Although inversions for rupture velocity for a strike-slip event (Scenario I) already benefit from rotation rate data, the improvement is even higher for a dip-slip (Scenario II) event.

The incorporation of rotation rate data only slightly improves the
information gain for a strike-slip fault when inverting for rise time. The
randomization of the receiver distribution decreases the information gain
compared to the results from the inversion with a regular receiver spacing
(

We compare the results for both types of faulting in
Fig.

For the dip-slip mechanism, the inversion quality depends much more on the
spatial distribution of stations (larger spreading in
Fig.

Histograms for maximum amplitudes of the three rotation components
at all stations. The top row shows the distribution for the strike-slip
fault, the bottom row for the dip-slip fault. The logarithmic scale for the
rotation rates helps to get a sense for the amplitudes' order of magnitude.
The histograms show how all 44 maximum amplitude values (for all 44
receivers) range between

With the range of calculated rotational ground motions between

We expect to have higher energy in the rotational horizontal component
signals for the dip-slip fault compared to the strike-slip fault. This would
strengthen our assumption that this is a main reason for the increase in
information gain for a fault that also slips in vertical direction. If we
compute the total energy in all three rotational components measured at all
stations, the energy of all rotation rate signals in the horizontal
components makes up a very large part of the total energy measured at all
stations for the dip-slip fault (57 % of the total energy), whereas the
strike-slip fault features the highest energy ratio in the vertical component
signals (40 % of the total energy for rotations around both horizontal
axes), but horizontal rotation only contributes 20 % of the total energy.
This makes sense since we created our strike-slip earthquake to solely
rupture in horizontal direction. To better assess the relevance of this study
we check whether the theoretically predicted rotational motions could be observed
in the field by state-of-the-art measurement devices. We therefore calculate
the maximum amplitude for all 44 receivers for each of the three rotation
rate components. They are shown in Fig.

We show a summary of the improvement of the information gain between the
different scenarios and parameters in percent in Table

Rotation data seem to be most beneficial for inversions of kinematic
parameters of sources that fracture at least partly in vertical direction.
Additionally, the increase in information gain for rupture velocity and rise
time suggests possible improvements in inversions for other seismological
research areas. These parameters are important for rheology and friction laws
used in dynamic rupture simulations

A successful decrease in non-uniqueness in probabilistic finite-source
inversion suggests manufacturing portable rotation sensors with the mentioned
sensitivities. The noise level, which was set to be equal to the noise level
for translational ground motions sensors of 10 %, should not exceed this
to successfully improve inversion results

Overall, non-uniqueness, which is an imminent problem in inversion theory, has been significantly reduced by the incorporation of synthetic rotational ground motion data.

The results from this study clearly suggest that seismological studies can benefit from the ability of seismometers to also measure rotational ground motions in addition to the conventional three translational components. This applies not only to finite-source inversions, which are a crucial part of seismological research, but probably to all work related to earthquake source inversions that relies on data recorded by seismic receivers.

We successfully showed how rotation rate measurements can improve the quality of seismological kinematic source inversions. This is most prominent for an earthquake which also fractures in vertical direction, due to the higher energy in the horizontal rotation components. It is very likely that the vertical displacement gradient, included in these components, leads to the observed decrease in non-uniqueness since this information can not be obtained with translation recordings on the Earth's surface.

All data is freely available for download via the following link:

We would like to thank Lion Krischer for the support in professional code
optimization. The authors acknowledge funding from the ERC advanced project
ROMY (