In the forest to the south of Karlsruhe in Germany, there are still
remains of a fortification wall built in the 18th century.
It was constructed in 1707 during the so-called War of the Spanish
Succession to stop the French army invading from the southwest.
The line of defense consisted of ramparts and trenches and was
fortified in parts with palisades.
The line was nevertheless quickly overcome and the French troops then
directly leveled the ramparts and trenches in many places.
In few places, however, the fortification is still visible.
For example here, close to the Messerholz-Karlsruhe near Einstetten.
Can one still find the old structures beneath the earth?
Where they are leveled and where on the surface no evidence can be
seen of them?
The inversion of complete seismic waveforms makes this and other
things possible.
Hello and welcome.
In this video, I will explain, using the example of the Ettlinger
Line, how seismic waveforms can provide information about
heterogeneous subsurface structures.
Here I will show you the technique of full waveform inversion, FWI.
Using the example of the Ettlinger Line, I will show you how to
identify the level trench within the ground.
You will learn how the full waveform inversion works, how it is
implemented technically using a computer and what previous knowledge
of the subsurface is necessary for this.
This is an aerial view of the study area.
You can see the course of the Ettlinger Line here and here in the
forest.
The straight line is interrupted by V-shaped bulges and square
entrenchments, which are also referred to as redoubts.
Outside the forest, the structure is not visible.
In 2016, seismic measurements were intended to show whether the filled
-in trenches in particular were still visible after 300 years.
For this purpose, a seismic profile was established transversely over
the presumed course of the trench.
This consisted of 48 geophones.
Hammer impacts on a metal plate were used as the seismic sources.
These were executed several times and then stacked vertically to
increase the signal-to-noise ratio.
Here you can see a recording.
The aim of the evaluation was to obtain background information from
the recorded waveforms.
For this purpose, the FWI uses the entire waveform.
Unlike other methods that analyse travel times of individual wave
types or limit themselves to reflected or refracted waves, FWI uses
both the amplitudes as well as the phase information from all the
waves produced.
In this method, it is not necessary to determine single wave types
such as P, S or surface waves.
Instead, on the basis of three-dimensional equations of motion, the
complete seismogram is analysed and the entire wave field in a
subsurface is calculated.
The equation of motion relates temporal and spatial oscillations with
each other.
I shall write them here.
Rho times d2u i to dt squared.
This is the second temporal derivative of the displacement vector or
the first derivative of the velocity vector.
This is how the particle velocity changes with time.
On the other side of the equation stands d sigma ij to dxj plus Fi.
Sigma is the stress tensor, which here is differentiated with respect
to the local variable x.
Fi is the externally applied force.
The first term on the right-hand side of the equation therefore
describes the spatial variation of the stress field.
You can see here that the spatial and temporal derivatives must be
calculated.
And indeed, at many locations in the subsurface and at many points in
time.
This can only be done numerically with the help of high-performance
computer processing.
One way to implement this task is to use the finite difference method.
In order to calculate the wave field in a subsurface, the subsurface
is divided into an equidistant grid.
The finite difference method approximates the derivatives of the
equation of motion using differential quotients at the equidistant
grid points.
Both the spatial and temporal derivatives are then calculated in
equidistant time steps.
This is now, so to speak, the framework upon which the calculations
are based.
But how does that work?
The objective of inversion in geophysics is always to deduce the
material properties of the subsurface from the observed measurement
data.
Several different methods can be used to seek out a subsurface model
that can satisfactorily explain the measured data.
One approach is to try out as many models as possible and to see which
one is best.
This approach requires a great deal of computational power and time,
because each randomly selected model needs to be calculated
individually.
A more efficient way is to systematically refine an existing starting
model of the subsurface.
In this approach, the starting model must include all the information
available and resemble the actual subsurface as closely as possible.
This will ensure at the end that a realistic subsurface model can be
calculated.
Here you can see an example of a starting model for the subsurface in
a surveyed area.
The top image shows color-coded the two-dimensional distribution for
the shear wave velocity v s below the profile.
The values range from almost 150 m per second at surface to about 350
m per second at 20 m depth.
The velocity of the P-waves in the lower image is set in the upper
part to about 300 m per second and in the lower part to approximately
2300 m per second.
The lower image shows the density distribution.
The inversion procedure now produces an improved model through
skillful optimization, which can satisfactorily explain the
measurement data.
This is the path taken by the full waveform inversion.
It uses the entire phase and amplitude information from different
seismic wave types.
For this purpose, it is a fundamental issue which criterion should be
optimized to find a starting model, and I will show this to you now.
The transformation matrix G is used to link the model vector of the
subsurface properties M with the measured data D.
D equals G times M.
This is called forward calculation.
Here, with the help of the finite difference method, the entire wave
field is calculated in a specified subsurface model.
G is specified by the equation of motion using additional boundary
conditions.
The reverse calculation, which is the path that we want to use, namely
from the measured data to a subsurface model, is not trivial.
This process is called inversion.
Full waveform inversion uses a procedure which minimizes the deviation
between the observed measurement data and the data synthetically
calculated by the subsurface modelling.
In practice, synthetic data is forward calculated from the starting
model and compared with the observed data.
The inversion algorithm then adjusts the model and the synthetic data
is then recalculated.
This iterative process is repeated until the deviation between the
observed and synthetic data falls under a previously defined
criterion.
For the example of the Ettlinger line, you can see in this application
how the fitting error, also called the misfit, gets smaller from
iteration to iteration and asymptotically approaches a minimum.
The small upward jumps in the curve represent increases in the
misfits.
This means an initially poorer adaption, since for these iteration
steps additional frequency content of the waveforms was added.
The additional data is expected to make the fit inferior.
But this improves again in subsequent steps.
After the minimization of the misfits, this final velocity model is
produced from the full waveform inversion.
Clearly, one can see in the middle of the profile a hollow with an
increased shear wave velocity.
While VS velocities of around 100 m per second prevail there, the
surrounding velocities are 150 to 200 m per second and slightly
higher.
The hollow coincides with the presumed location of the Ettlinger line.
The reduction in velocity is due to the presence of less solidified
material, with which the trench had been filled all those years ago.
For the final velocity model, this animation shows how the waves
propagate in the underground.
The top image shows P-waves and vertically polarized S-waves, while
the lower image shows horizontally polarized S-waves.
It can be clearly seen how the waves are influenced by the velocity
anomaly of the filled trench and how new waves form there.
The animation visualizes what happens in the background during the
inversion process.
The calculation of the wave field in a specified subsurface model.
During the inversion, the resulting synthetic waveforms could be
calculated at the observation points on the surface and compared with
the observational data.
In this video, I have presented how full waveform inversion has been
applied to study the Ettlinger line.
This method uses the entire recorded waveforms and calculates a wave
field through the subsurface.
The calculation is very time-consuming and you have learned that the
calculation of the wave field is technically implemented using the
finite difference method.
During the course of the inversion, a realistic starting model of the
subsurface is improved over several iteration steps until the misfit
between the observed data and the forward model data falls below a
predefined value.
In this way, it is possible with the help of full waveform inversion
to analyze even weak velocity contrasts such as those that occur
within the subsurface at the trenches of the Ettlinger line.