# maurer

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```					Optimized design of frequency-
domain acoustic waveform
tomography experiments

Hansruedi Maurer and
Stewart Greenhalgh
ETH Zurich, Switzerland
Outline
• Waveform sensitivities
• Experimental design of waveform tomography
experiments
• Suitable data representations
• Choice of temporal frequencies
• Temporal frequencies vs. spatial sampling
• Tests with synthetic data
• Conclusions
Time-domain sensitivities
Acoustic waveform
inversion
Solution of the forward problem:
• finite differences

• finite elements
• spectral elements
• ...

Solution of the inverse problem:
m  ( J T J +C-1 ) 1 J T  dobs  dcalc   Jm0 
M                                  

Although not necessarily computed
explicitly, the sensitivities contained
in J are essential for the solution of
the inverse problem.
Frequency-domain inversions            Frequency-domain sensitivities

• Seismic data generally band-limited

• Computationally less demanding
compared with time-domain
inversions

• “Stationary” problem to be solved
Optimized design of frequency-domain
acoustic waveform inversions

• How can we set up an experiment, with
which maximum subsurface information
can be extracted at minimal (field efforts
AND computational expenses) costs?
Eigenvalue spectra and model resolution
m  ( J T J +C-1 ) 1 J T  dobs  dcalc   Jm0 
M                                  

mest = Rmtrue
Eigenvalue

Threshold
R  ( J T J +CM ) 1 J T J
-1

Eigenvalue index
Resolved model                     Null
space                              space         Characterizes information
that can be extracted from
inversions!
Characterizes data
information content!
Test model

P-wave velocity:            2000 m/s

Finite element mesh:        0.15 m

30 m
Inversion mesh:             0.30 m

Frequencies: 100, 200, 500, 750, 1000, 1250, 1500 Hz

Source-receiver spacings:   0.25, 0.5, 1, 2, 5 m

20 m
Data representation
• Complex-valued spectra

• Hartley spectra

• Amplitude

• Phase

Which representation is most appropriate?
Log Normalized Eigenvalue spectrum                Data representation
500 Hz                                  1500 Hz
0                                        0
complex                                        complex
Hartley                                        Hartley
-1                                       -1
Amplitude                                      Amplitude
Phase                                          Phase
-2                                       -2

-3                                       -3

-4                                       -4

a                                        b
-5                                       -5
0        0.05        0.1        0.15     0        0.05        0.1       0.15
Normalized Eigenvalue index              Normalized Eigenvalue index

If possible, full complex-valued spectra
should be considered!
Choice of temporal frequencies
• Single frequency is likely inappropriate

• Many frequencies offer better resolution, but at
computationally higher costs

• Large bandwidth may be difficult to achieve

Which combination of frequencies offer
best benefit-to-cost-ratio?
Choice of temporal frequencies

Log Normalized Eigenvalue spectrum
Log Normalized Eigenvalue spectrum

Single frequencies                                                     Cumulative frequencies
0                                                                        0
up to 100 Hz
100 Hz
up to 200 Hz
-1                      200 Hz                                           -1                           up to 500 Hz
500 Hz
up to 750 Hz
750 Hz
-2                                                                       -2                           up to 1000 Hz
1000 Hz
up to 1250 Hz
1250 Hz
-3                                                                       -3                           up to 1500 Hz
1500 Hz

-4                                                                       -4

a                                                                       b
-5                                                                       -5
0       0.1     0.2     0.3                                              0       0.1      0.2    0.3
Normalized Eigenvalue index                                              Normalized Eigenvalue index

If the bandwidth is sufficiently wide, high
frequencies are becoming particularly useful!
Single frequencies    Cumulative frequencies
200 Hz             up to 200 Hz
Model resolution of single and                   0                     0

Depth [m]
10                      10
cummulative frequencies                        20                      20
30                      30

• Model resolution of single                        0   10 20
500 Hz
0     10 20
up to 500 Hz
frequencies is                                 0                      0

Depth [m]
10                      10
substantially lower than                     20                      20

those of cummulative                         30                      30
0 10 20                0 10 20
frequencies                                           1000 Hz          up to 1000 Hz
0                      0

Depth [m]
10                      10
• Acoustic waveform                            20                      20
30                      30
inversion may be capable                           0 10 20                0 10 20
to resolve features                            0
1500 Hz
0
up to 1500 Hz

outside of the                   Depth [m]
10                      10

tomographic plane!                           20
30
20
30
0 10 20                0 10 20
Distance [m]           Distance [m]

0           0.004 0.008    0          0.03    0.06
Temporal frequencies vs. spatial sampling
Temporal frequency

up to 1500 Hz
up to 1000 Hz
up to 1250 Hz
up to 100 Hz
up to 200 Hz
up to 500 Hz
up to 750 Hz
• Frequency content is

more important than
spatial sampling
5.0 m
2.0 m
0.2
• Trade-off between                                                1.0 m
0.5 m
selection of temporal
0.25 m
0
frequencies and spatial                                             a
sampling exist
0                       0
0.06
Depth [m]

10                                   10
0.04
20                                   20
0.02
30                                   30
b 0                                      c 0                        0
10 20                  10 20
Distance [m]            Distance [m]
Inversion tests
• Cumulative frequencies
superior to single
frequency inversion
a                         b

temporal frequencies and
spatial sampling
confirmed

• Features outside of
tomographic plane
resolved
c                         d

1500   1700   1900      2100   2300   2500
[m/s]
Conclusions
• Complex-valued spectra offer substantially
representations.

• Combination of some bandwidth and high
frequencies is most useful.

• Trade-off between choice of temporal
frequencies and spatial sampling exists.

```
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 views: 44 posted: 1/24/2011 language: English pages: 15