1. Full-waveform Inversion of GPR Data and its Frequency
Domain Approach
Good afternoon everyone. I’m very glad I have this opportunity to be here and give a presentation of my work last year. It’s content two parts. First I will give a overview of the time domain of FWI and it’s application to a real data from Boise data set. And then I will talk about the development of frequency domain FWI.
X. Yang, A. Klotzsche, G.A. Meles, J. Bikowski,
P. Kumbhar, H. Vereecken, J. van der Kruk

2. What is Ground-Penetrating Radar (GPR)
Crosshole GPR
Surface GPR
Civil engineering Hydrogeophysics Glaciers Geology
GPR has been widely used for a wide range of applications. this method enables a quick and effective mapping of the soil properties and subsurface objects.
Two commonly used setups for measurement are cross-hole GPR and surface GPR.

3. Ray-based methods Waveform methods
Input data:
First arrival times
First cycle amplitudes
Input data:
Significant parts of wavefields
Inversion based on Maxwell’s Eq.
Observed data
Receiver depth [m]
Time [ns] 80 90
100
110
120
130
140
150
160 3 4 5 6 7 8 9
first-cycle amplitude
first-arrival
Depth [m]
full-waveform
Traditionally, ray-based methods are used for tomographic inversion of crosshole GPR data, where as input FAT (indicated by the red arrows and circle) and FCA (indicated by green arrows and circle) are used. Due to that only a small amount of the data is used, only a coarse structures can be obtained and events like with high amplitudes that are presented at later arrivals (show waveguide) are ignored. Further, no quantitative values for the conductivity can be obtained,
On the other hand, full-waveform methods use all information (blue arrow and circle) or at least significant parts from the data, including secondary events like reflections and refraction. Note also structures like this are in cooperated. Therefore, higher resolution images can be obtained for permittivities and conductivities, but of course its more expensive to compute.
first arrival
times
full-waveform
first cycle
amplitudes
Time
Time [ns]
inexpensive, coarse structures
detailed sub-wavelength
structures, expensive 3

4. Outline Principle of full waveform inversion (FWI) of GPR data
A crosshole synthetic model
Compare with time domain FWI results
A four-sided synthetic model
High contrast parameters (nonlinear problem)
A preliminary surface GPR synthetic model
Summery

5. ? Full waveform inversion principle Measurement True model Forward
Initial model from ray-based inversion
Cost function

6. ? Full waveform inversion principle Measurement Forward Cost function
goes down
Inversion
True model
10 iterations results

7. ? Full waveform inversion principle True model
Measurement
Forward
Cost function
goes down
Inversion
True model
Final iterations results

8. Why frequency domain? Time domain (TD) Frequency domain (FD)
Forward method
FDTD is not efficiency
when the observed time
is long
FDFD is efficiency for the
multiple-source locations setup
Sampling
Whole observed traces to inversion
Can select limited number
of frequencies to inversion
Cost function
A limited selection of
misfit functions
A wide range of misfit functions can be easily
implemented in frequency domain
Nonlinear problem
Easy to trapped into local minimum
(Meles, 2011)
From low to high frequencies,
can easily reduce the
non-linearity problem
In frequency domain we can just select several frequencies to achieve FWI, it can save much time, meanwhile start from low frequency to high frequency can easily help us to improve the non-linearity problem in FWI. 8

9. Frequency domain full-waveform inversion
Forward
Use frequency-domain finite-difference (FDFD) method.
Use MPI and MUMPS to get a parallel version code and can directly run on the supercomputer.
Inversion
Use Gauss-Newton method which means we need to calculate approximate Hessian matrix.
Using vector character of electromagnetic wave and a simultaneous update scheme.
Use linear/parabolic approach to get the optimized step length.
We are already have the frequency domain FWI code. The forward part use FDFD method which is very mature and stable. We also parallel the code use MPI technique and use direct solver to let the code working on our supercomputer in Juelich. For the inversion part we use Gauss-Newton method which means we need to calculate approximate Hessian matrix. And we also use parabolic approach to get the optimized step length. Comparing to time domain FWI use these technique making the inversion more efficiency.
Ernst (2007a,b), Meles (2010, 2011), Rumpf (2004), Operto (2004, 2006)

10. FWI crosshole synthetic model in TE mode
Time domain
Frequency domain
Gradient approach
Conjugate gradient
Gauss-Newton
Steplength Calculation
Linear
Number forward model cells
250
Cell size (m)
0.04
Time/frequency sampling
Time sampling: 85 ps using 2944 timesteps
7 frequencies:
152, 200, 248, 296, 352, 400, 448 MHz.
Forward model calculation time for all times/frequencies
7min 50 s
2 min
CPU number 24
Here is a table show the detail of the different between two methods. I will give the first synthetic model to compare them.

11. Full waveform inversion of two small pipes in TE mode

12. Full waveform inversion of two small pipes in TE mode

13. Full waveform inversion of two small pipes in TE mode

14. Full waveform inversion of two small pipes

15. Full waveform inversion of two small pipes
Time domain FWI
Frequency domain FWI
CPU Time
1h 11 min 31s
(by factor 5.4)
13 min 23s
Memory
400MB
1GB (by factor 2.5)
Number of Iterations 40
10 per frequency
Reduce of cost function
0.0302
0.0431
Forward calculation times needed per iteration 4
Total forward models calculations
160
287

16. Four-sided TE & TM FWI of two crosses with high contrast

17. Four-sided TE & TM FWI of two crosses with high contrast

18. Four-sided TE & TM FWI of two crosses with high contrast
(TE mode)

19. Four-sided TE & TM FWI of two crosses with high contrast
(TE mode)
(TE mode)

20. Four-sided TE & TM FWI of two crosses with high contrast
(TE mode)
(TE mode)

21. Primary result of surface GPR FWI
Surface GPR set up
Multichannel system (1 transmitter
with 5 receivers)
65 times measure for one way
6 frequencies were used
50 iterations for every frequency
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.

22. Surface GPR model
Permittivity true model
Permittivity initial model

23. Surface GPR FWI Permittivity true model
Permittivity final result (use 7 frequencies and 10 iterations) per frequency

24. Summary
Achieve full waveform inversion in frequency domain and got preliminary results.
Vectorial full wave field
Gauss-Newton method
It is important to include the vector character of electromagnetic wave and to use a simultaneous update scheme
Compared with time domain FWI frequency domain FWI has several advantages.
Select optimum frequencies
faster
Disadvantage
Large memory requirement

25. Thank you for your attention!