maurer

Document Sample
maurer Powered By Docstoc
					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




                                        Source-receiver spacing
  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

• Trade-off between
  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
  more information than other data
  representations.

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

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

				
DOCUMENT INFO
Shared By:
Categories:
Stats:
views:44
posted:1/24/2011
language:English
pages:15