Deconvolution and Regularization Demonstration

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Deconvolution and Regularization Demonstration

• Rick Aster
• CIG/EarthScope Imaging Workshop Tutorial
• Washington University, St. Louis
• 11/1/06

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Goals

• Demonstrate basic Matlab coding
• Demonstrate Frequency-domain deconvolution
• Demonstrate Time-domain devonvolution
• Demonstrate deconvolution instability in the
  presence of noise
• Illustrate fundamental smoothness vs. data fit
  tradeoffs with both freqeuncy-and time-domain
  regularization for the deconvolution inverse
  problem.

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Here is the desired deconvolution result a
simple Earth impulse response.
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Here is the convolving kernel (option 1) In the
case of an instrument response deconvolution,
this is typically a band-limited (smooth)
function which isthe recording system impulse
response. In the case of a receiver function it
is the vertical-comp seismogram.
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The data is the convolution of the previous two
time function.
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The spectral division solution is essentially
perfect if the data are noiseless and the
convolving kernel is known exactly
(double-precision Matlab default calculations).
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Next, we add a small level of Gaussian white
noise to the data vector.
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and all hell breaks loose!
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We can see why the previous deconvolution result
is so terrible to deconvolve we divide the noisy
data spectrum by the convolving kernel spectrum
(smooth curve here)
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thus producing a deconvolution for the noisy
data that is completely dominated by noise!
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The water level regularization provides one way
to stabilize the deconvolution and select a best
result, based on fitting the data and having a
small model (deconvolution result) norm.
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Here is a selected optimal regularized solution
selected using the previous tradeoff curve.
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Tikhonov Regularization Tradeoff Curve
Same problem regularized via 2nd order Tikhonov
regularization
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Here is an optimal model selected from the
previous tradeoff curve.
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Deconvolution Demo Exploration

• Noise level
• Try a more complicated convolution kernel (e.g.,
  uncomment the optional convolution kernels and
  rerun the routine).
• Try a different regularization scheme by altering
  the L roughening matrix in the time domain
  deconvolution.