... the vertical component f z of a conservative field F , the x - and y -components of the first-order Riesz transform of f z were the horizontal components f x and f y of F . Hence, the local amplitude of the monogenic signal of f z is the 3D analytic signal amplitude of the scalar potential of F and the local...

... of the monogenic signal are the two first-order Riesz transforms, i.e., the x - and y -derivatives of the vertical integral of the input function f ( x , y ) . See Table  1 for the definition of the input function and the components of the 2D and 3D vectors in each of the three cases...

... . In this particular case, the three components of the monogenic signal will be given by the original data function, and its first-order Riesz transform; i.e., M NS ( u , v ) = [ F i u u 2 + v 2 F i v u 2 + v 2 F ] . (7...

... efficiency, which is mandatory, in particular, for multisensorial data in several space dimensions. Moreover, the second step, the Hilbert transform, also needed to be generalized to extend the analytical signals to arbitrary dimensions. Based on ideas using Clifford algebra and Riesz transforms from...

...., 2013 ; Zhu and Harris, 2014 ). Many numerical methods are used to study the viscosity of wave absorption and attenuation with the fractional-order seismic wave equation. When m = 2 , the average of three points from equation  7 is approximated as the Riesz derivative and a fourth-order...

... and Van Loan, 1983 ; Riesz and Nagy, 1955 ; Aster et al., 2005 ). This approach has proven to be very useful in climate dynamics and other areas ( Vautard and Ghil, 1989 ; Yiou et al., 2000 ). Our choice of the KL transform is based on the concept of minimizing the average approximate error according...

..., frequencies, or amplitudes, we have analyzed the array acoustic logging signals by the fractional Fourier transform and the Choi-Williams distribution. First, we apply the fractional Fourier transform on an array acoustic logging waveform with proper α , then the Choi-Williams distribution analysis method...

..., the conjugate gradient method can be applied even in this case. For an underdetermined problem (where N m > N d ), the conjugate gradient iterations (6) converge to the minimum norm solution m min : (8) Expression (8) is also known as the Riesz representation formula ( Parker, 1994...

... = h z ⋆ m , (33) with the generalized 3D Hilbert (Riesz) transform filters: h x = − 1 π 2 x ( x 2 + y 2 + z 2 ) 2 , h y = − 1 π 2 y ( x 2 + y 2 + z 2 ) 2...

...Hanming Chen; Hui Zhou; Ying Rao; Pengyuan Sun; Jianlei Zhang; Yangkang Chen ABSTRACT We have developed a matrix-transform method (MTM) to numerically solve the fractional Laplacian constant- Q viscoacoustic wave equation. The new method is based on a matrix representation of the fractional...