Shaping regularization in geophysical estimation problems |

Smoothness is controlled by the choice of the regularization operator and the scaling parameter .

Figure 1 shows the impulse response of the regularized
smoothing operator in the 1-D case when is the first difference operator. The impulse response has exponentially
decaying tails. Repeated application of smoothing in this case is
equivalent to applying an implicit Euler finite-difference scheme to
the solution of the diffusion equation

exp
Left: impulse response of regularized
smoothing. Repeated smoothing converges to a Gaussian bell shape. Right:
frequency spectrum of regularized smoothing. The spectrum also converges to
a Gaussian.
Figure 1. |
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As far as the smoothing problem is concerned, there are better ways to
smooth signals than applying
equation 5. One example is triangle
smoothing (Claerbout, 1992). To define triangle
smoothing of one-dimensional signals, start with box smoothing, which,
in the -transform notation, is a convolution with the filter

Triangle smoothing is more efficient than regularized smoothing, because it requires twice less floating point multiplications. It also provides smoother results while having a compactly supported impulse response (Figure 2). Repeated application of triangle smoothing also makes the impulse response converge to a Gaussian shape but at a significantly faster rate. One can also implement smoothing by Gaussian filtering in the frequency domain or by applying other types of bandpass filters.

tri
Left: impulse response of triangle
smoothing. Repeated smoothing converges to a Gaussian bell shape. Right:
frequency spectrum of triangle smoothing. Convergence to
a Gaussian is faster than in the case of regularized smoothing. Compare to
Figure 1.
Figure 2. |
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Shaping regularization in geophysical estimation problems |

2013-07-26