We build a framework for R(p;q)-deformed calculus, which pro- vides a method of computation for deformed R(p;q)-derivative and integration, generalizing known deformed derivatives and integrations of analytic functions defined on a complex disc as particular cases corresponding to conveniently cho- sen meromorphic functions. Under prescribed conditions, we define the R(p;q)- derivative and integration. Relevant examples are also given.
We extend the sharp weighted bound of the A2 theorem to the q‐variation norm of certain Calderón–Zygmund operators (q>2), a stronger nonlinearity than the maximal truncations that have been treated before. We obtain this result by a new non‐probabilistic approach that was independently discovered by Lerner.
Stability and efficiency are two issues of general concern in inverse Q filtering. This paper presents a stable, efficient approach to inverse Q filtering, based on the theory of wavefield downward continuation. It is implemented in a layered manner, assuming a depth-dependent, layered-earth Q model. For each individual constant Q layer, the seismic wavefield recorded at the surface is first extrapolated down to the top of the current layer and a constant Q inverse filter is then applied to the current layer. When extrapolating within the overburden, instead of applying wavefield downward continuation directly, a reversed, upward continuation system is solved to obtain a stabilized solution. Within the current constant Q layer, the amplitude compensation operator, which is a 2-D function of traveltime and frequency, is approximated optimally as the product of two 1-D functions depending, respectively, on time and frequency. The constant Q inverse filter that compensates simultaneously for phase and amplitude effects is then implemented efficiently in the Fourier domain.
A principal limitation on seismic resolution is the earth attenuation, or Q -effect, including the energy dissipation of high-frequency wave components and the velocity dispersion that distorts seismic wavelets. An inverse Q -filtering procedure attempts to remove the Q -effect to produce high-resolution seismic data, but some existing methods either reduce the S/N ratio, which limits spatial resolution, or generate an illusory high-resolution wavelet that contains no more subsurface information than the original low-resolution data. In this paper, seismic inverse Q -filtering is implemented in a stabilized manner to produce high-quality data in terms of resolution and S/N ratio. Stabilization is applied to only the amplitude compensation operator of a full inverse Q -filter because its phase operator is unconditionally stable, but the scheme neither amplifies nor suppresses high frequencies at late times where the data contain mostly ambient noise. The latter property makes the process invertible, differ...