1. Frequency Domain Full Waveform Inversion
of GPR Data
Today I want to give a short overview of my PhD thesis
“Full-waveform inversion of surface GPR data for hydrogeological applications”
Yang, Xi
Dr. Bikowski, Jutta
Prof. Dr. van der Kruk, Jan

2. Outline Introduction Why frequency domain inversion?
Non-linearity problem in full waveform inversion
Frequency domain full waveform inversion
Synthetic model
Outlook
First I want to give a very short introduction to Ground Penetrating Radar.
Then I will use Common midpoint measurements to explain the advantages and disadvantages of standard ray based techniques for hydrogeological applications.
In the next step I will introduce full-waveform inversion of GPR data and its application.
And finally I will show preliminary results of the work I have done until today,
name the next steps
And give an outlook of the work.

3. Introduction
Ground-penetrating radar (GPR) as a non-destructive method has been widely used for imaging of the near surface for a number of applications.

4. Introduction
So looking at the applications of Ground Penetrating Radar for hydrogeological questions, this method enables a quick and effective mapping of soil’s moisture content.
Using a transmitter and receiver antenna, high frequency electromagnetic waves are send to the subsurface.
If there is a contrast in electromagnetic properties, especially in permittivity and conductivity, these waves are reflected and detected at the surface.
Thereby permittivity defines the velocity of the wave and conductivity affects the attenuation of the wave.
That means:
if we want to get information about the permittivity we have to analyse the velocity;
if we want to get information about the conductivity we have to analyse the amplitude of the wave.
Ray path

5. Introduction
One of the most promising new techniques for imaging GPR data is full waveform inversion (FWI). x
If we use CMP measurements to determine the electromagnetic properties, the velocity of the subsurface can be calculated from the picked ground wave.
Using the electromagnetic velocity of the air (0.3m/ns) and the calculated ground wave velocity, the relative permittivity of the subsurface can be derived.
And therefore, water content can be calculated for example with Topp’s equation.
But using the air wave – ground wave picking technique to analyse a CMP data set, only velocity information are used to calculate quantitative values for permittivity.
There are no quantitative values for conductivity.
Additionally standard ray-based techniques are only applicable when you can identify a clear ground wave.
Time

6. Introduction
One of the most promising new techniques for imaging GPR data is full waveform inversion (FWI).
Ernst, J. R., Hansruedi Maurer, Alan G. Green, and Klaus Holliger, Full-Waveform Inversion of Crosshole Radar Data Based on 2-D Finite-Difference Time-Domain Solutions of Maxwell’s Equations, IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, 2007, VOL. 45, NO. 9, P
Ernst, J.R., Green A.G. Maurer, H. and Holliger, K., 2007, Application of a new 2D time-domain full-waveform inversion scheme to crosshole radar data, Geophysics, 72, J53-J64.
Meles, G., Van der Kruk, J., Greenhalgh, S.A., Ernst, J., Green, A.G. and Maurer, H A new vector waveform inversion algorithm for simulataneous updating of conductivity and permittivity parameters from combination crosshole/Borehole-to-surface GPR data, IEEE Trans. Geosci. Remote Sensing. 48,
Klotzsche, A., J. van der Kruk, G. A. Meles, J. Doetsch, H. Maurer, N. Linde, Full-waveform inversion of Crosshole Ground Penetrating Radar Data to Characterize a Gravel Aquifer close to the River Thur, Switzerland, Near Surface Geophysics, 2010.
Meles, G., Greenhalgh, S.A., Van der Kruk, J., Green, A.G. and Maurer, H. Taming the non-linearity problem in GPR full-waveform inversion for high contrast media. Submitted to Journal of Applied Geophysics.
If we use CMP measurements to determine the electromagnetic properties, the velocity of the subsurface can be calculated from the picked ground wave.
Using the electromagnetic velocity of the air (0.3m/ns) and the calculated ground wave velocity, the relative permittivity of the subsurface can be derived.
And therefore, water content can be calculated for example with Topp’s equation.
But using the air wave – ground wave picking technique to analyse a CMP data set, only velocity information are used to calculate quantitative values for permittivity.
There are no quantitative values for conductivity.
Additionally standard ray-based techniques are only applicable when you can identify a clear ground wave.

7. Application to real Data – Boise: Overview
Survey layout
OBSERVED ELECTRIC FIELD:
(Receiver Gather: REC 20)
Braided river deposits by porosity:
PARAMETERS:
Grid Cells Size [m]
Forward
0.02
Inverse
0.06
Survey Area:
Width [m]
~9.0
Depth [m]
~15.6
Recording
#Transmitters 40
#Receivers 77
Transmitter Pulse
Fc [MHz]
~80
dominant [m]
~1.3
21%
26%
The survey was conducted at the Boise Hydrogeophysical Research Site in the US. The area is about 9 by 16 meters large and dominated by water-saturated braided river deposits. 40 receivers are located in the left and 77 transmitter in the right borehole.
(NEXT)
1 - A noise-filtered and windowed radar section for the central receiver and every third transmitter is shown to the right…
-> The traces are much more complicated at least partially due to the water-filled boreholes -> I therefore use a longer source wavelet to try to compensate these effects. In any case it may be hard to reconstruct these traces with what ever approach I choose… (NEXT)
23%
Data: Boise Hydrogeophysical Research Site, USA: e.g. Tronicke, J., Holliger, K., Barrash, W. and Knoll, M.D. 2004; Water Resources Research, 40.

8. Ray-based
Residual Trace RMS: 3.37e-6

9. Ernst et al.
Residual Trace RMS: 2.44e-6

10. Meeles et al.
Residual Trace RMS: 2.40e-6

11. Recently reprocessed
Residual Trace RMS: 1.93e-6

14. Time Domain – Frequency domain
Why frequency domain?
Time Domain – Frequency domain x
If we use CMP measurements to determine the electromagnetic properties, the velocity of the subsurface can be calculated from the picked ground wave.
Using the electromagnetic velocity of the air (0.3m/ns) and the calculated ground wave velocity, the relative permittivity of the subsurface can be derived.
And therefore, water content can be calculated for example with Topp’s equation.
But using the air wave – ground wave picking technique to analyse a CMP data set, only velocity information are used to calculate quantitative values for permittivity.
There are no quantitative values for conductivity.
Additionally standard ray-based techniques are only applicable when you can identify a clear ground wave.
Time 14

15. Challenges of FWI Why frequency domain? Computational efficiency
3-dimension
The nonlinearity of the waveform inversion problem
Inversion algorithm chosen (conjugate gradient, Gauss Newton, full Newton, and so on);
Complexity of the true model;
Regularization constraints applied;
Level of data noise contamination;
Choice of the initial model.

16. Why frequency domain?
In frequency domain full waveform inversion we can probably improve the inversion process as follows:
Computational efficiency
Time domain FWI considers whole trace in time which means all frequencies.
Frequency domain FWI can use selected frequencies.
Non-linearity problem
Time domain FWI can be trapped in local minima due to non-linearity.
Frequency domain FWI can select the suitable frequencies, starting with low frequencies, such that the inversion problem is only weakly non-linear. By including higher frequencies the resolution can be improved for each iteration.
More efficient misfit function
Time domain misfit function is not free to choose.
A wide range of misfit functions can be easily implemented in frequency domain.
But these conditions are not always true.
Due to precipitation and evaporation highly dynamic processes take place in the shallow subsurface resulting in Gradients in the electromagnetic properties or thin layering.
In case of a thin surface layer of high-permittivity material overlying low permittivity material dispersion occurs resulting in a series of interfering multiples.
There is clear identifiable ground wave – there is only a package interfering signals.
You cannot calculate the ground wave velocity, you cannot calculate the permittivity and therefore the water content of the subsurface.

17. Mixed time-frequency domain full-waveform inversion
Meles et al. (submitted)

18. Mixed time-frequency domain full-waveform inversion
FBID: full bandwidth initial data
PBED: progressive bandwidth expansion of data
TTT: traveltime tomography
Meles et al. (submitted)

19. Frequency domain finite differences forward modeling implementation
Field equation
(1)
Consider E field, and introducing the Green’s function we can write the solution as
(2)
In conclusion:
Standard ray-based techniques only use a pre-selective area of the data and the full waveform of the signals is not taken to account.
As long as direct and reflected waves can be easily identified this may not be a problem, but as the complexity of the subsurface arises these techniques are not applicable.
Furthermore, until now only permittivity is used to determine the water content. There are no information about conductivities.
solutions for multiple sources (i.e. multiple RHS terms) can be obtained efficiently by forward and backward substitutions
MUMPS : a parallel sparse direct solver

20. Forward modelling × × model green function f=400MHz
Now I want to present preliminary results of the work I have dine until today.
The focus of the first month was to estimate a preliminary source wavelet and inversion algorithm.
As a starting point I used an ideal dispersive CMP dataset.
In a first step these dataset has been analysed using a conventional dispersion inversion. An the results of this inversion are:
permittivity upper layer
permittivity lower layer
and thickness.
Due to the fact that this inversion only uses the phase velocity spectra are there are only quantitative values for conductivity. Additionally all amplitudes are normalized, so there is no amplitude information.
model green function f=400MHz

21. Forward modelling comparison (FDTD & FDFD 201 frequencies)

22. Frequency domain FWI Cost function/misfit function (3)
Gradient method of inversion
(4)
(5)

23. Frequency domain FWI
Current step lengths using linear estimation (Meles et al. 2010)
Planned new step lengths
(6)

24. Gauss-Newton method
Gradient methods have the disadvantage of being local. That is, they converge to the nearest local minimum (if such local minima exist) instead of converging to the global minimum. However, they have the advantage of being tremendously effective. Newton methods are derived by considering an expansion of the misfit function in eq. (3) as a Taylor series and retaining terms up to quadratic order
(7)
H is the Hessian second-derivative matrix
(8)

25. Gauss-Newton method
It shows that a better estimate is the gradient, preconditioned, or filtered by the inverse Hessian.
(9)
The exact Hessian Gauss-Newton method with the simplest form of regularization.
(10)
(11)

26. Program flow chart Start here Raybase tomography model it=1
Pre-processed data
Raybase tomography model
Loop 1 over frequency
Mono-frequency dataset
Starting 𝛆 & 𝛔 model
Compute Green functions for each receiver positions
Compute the diagonal terms of Hessian
Gradient scaling
iw=1
True
False
Loop 2 over iterations
Starting 𝛆 & 𝛔 model
Compute Green functions for each receiver position
Estimate source s
Compute residuals 𝛅E and cost function value C
Back-propagate residuals from all receiver position
Compute gradient of the cost function
Scale the gradient by the diagonal terms of Hessian
Search optimal step length 𝜸 and compute 𝛅𝛆 & 𝛅𝛔m
Updated permittivity and conductivity model 2 1
it=1

27. Program flow chart Current time domain FWI New frequency domain FWI
Simultaneous
Inversion
Initial 𝜀 0
Model 𝜎 0
Calculate
Misfit
Gradient
ε Step Length
σ Step Length
Update
Model
Model 𝜀 10 𝜎 10
After 10 iterations:
4 FDTDs×10 =
40 FDTDs a) b) c) d)
FDTD +
2 FDTD +
4 FDTDs
10 times
6 FDFDs× FDFD =
70 FDFDs
FDFD +
4 FDFD +
6 FDFDs
Hessian Matrix

28. Synthetic models

29. Outlook
Achieve simultaneous full waveform inversion in frequency domain
Crosshole setup
On-ground setup
Many possible ways to improve the frequency domain inversion:
Select optimum frequencies
Expand to 2.5D & 3D
Use different cost functions (e.g. log)
Use for each frequency a CPU on supercomputer
 Reliable high-resolution inversion results

30. Thank you for listening!

31. Raybase tomography model
Pre-processed data
Raybase tomography model
Loop 1 over frequency
Mono-frequency dataset
Starting 𝛆 & 𝛔 model
Compute Green functions for each receiver positions
Compute the diagonal terms of Hessian
Gradient scaling
Loop 2 over iterations
Compute Green functions for each receiver position
Estimate source s
Compute residuals 𝛅𝐄 and cost function value C
Back-propagate residuals from all receiver position
Compute gradient of the cost function
Scale the gradient by the diagonal terms of Hessian
Search optimal step length 𝛄 and compute 𝛅𝛆 & 𝛅𝛔
Updated permittivity and conductivity model 2 1
iw=1
True
False
it=1

32. Other kind of misfit function
Changsoo Shin (2008)

33. Frequency domain FWI (6) So the gradient can be write as (7)
(6)
So the gradient can be write as
(7)
In all commonly used standard ray-based techniques, the full waveform of the GPR signal are not taken into account.
In many cases full-waveform inversion can be applied such that all information present in the measured traces is used to find a high resolution model of the reality.
So the aim is to directly extract subsurface properties from the inversion to.
Until now there are a few applications of full-waveform inversion for GPR data.
The frechet derivatives are
(8)