Seismic dip estimation based on the two-dimensional Hilbert transform and its application in random noise attenuation

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## Nonstationary polynomial fitting

Traditional stationary regression is used to estimate the coefficients by minimizing the prediction error between a master'' signal s( ) (where represents the coordinates of a multidimensional space) and a collection of slave signals (Fomel, 2009)

 (7)

When is 1D and , and , the problem of minimizing amounts to fitting a straight line to the master signal. Nonstationary regression is similar to equation 7 but allows the coefficients to vary with , and the error (Fomel, 2009)

 (8)

is minimized to solve for the multinomial coefficients . The minimization becomes an ill-posed problem because rely on the independent variables . To solve the ill-posed problem, we constrain the coefficients . Tikhonov's regularization (Tikhonov, 1963) is a classical regularization method that amounts to the minimization of the following functional (Fomel, 2009)

 (9)

where is the regularization operator and is a scalar regularization parameter. When is a linear operator, the least-squares estimation reduces to linear inversion (Fomel, 2009)

 (10)

where

and the elements of matrix are

compare
Figure 2.
Least-squares linear fitting compared with nonstationary polynomial fitting.

Next, we use a simple signal to simulate the variation of the amplitude of a nonstationary event with random noise (dashed line in Figure 2). In Figure 2, the dot dashed line denotes the results of the least-squares linear fitting and the solid line denotes the results of the nonstationary polynomial fitting. We compare the least-squares linear fitting and nonstationary polynomial fitting results, and we find that the nonstationary polynomial fitting models the curve variations more accurately for events with variable amplitude, particularly for .

 Seismic dip estimation based on the two-dimensional Hilbert transform and its application in random noise attenuation

Next: Synthetic data tests Up: theory Previous: Noniterative local dip calculation

2015-05-07