scholarly journals The technique of recovery of causative sources in case of strong interference of gravity fields using the continuous wavelet transform

2021 ◽  
pp. 84-89
Author(s):  
E. Utemov ◽  
◽  
D. Nurgaliev ◽  
Keyword(s):  
Inverse Problem ◽  
Wavelet Transform ◽  
Basis Functions ◽  
Wavelet Basis ◽  
Continuous Wavelet ◽  
Direct And Inverse Problems ◽  
Strong Interference ◽  
Gravity Fields ◽  
Close Relationship ◽  
Gravimetric Data

The technique of processing gravimetric data is offered in this study. Offered technique based on wavelet transform with so-called «native» wavelet basis functions. Distinctive feature of the technique is a close relationship with both direct and inverse problems of gravimetry. It was shown that the peculiarity allows to quite simply and quickly location of causative sources even under of strong interference of gravity fields. Keywords: gravimetry; wavelet transform; anomaly; inverse problem.

scholarly journals Continuous Wavelet Transform of Schwartz Distributions in DL2′(Rn), n ≥ 1

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1106
Author(s):  
Jagdish N. Pandey
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
The Real ◽  
Schwartz Distributions ◽  
Distributional Sense ◽  
Wavelet Inversion

We define a testing function space DL2(Rn) consisting of a class of C∞ functions defined on Rn, n≥1 whose every derivtive is L2(Rn) integrable and equip it with a topology generated by a separating collection of seminorms {γk}|k|=0∞ on DL2(Rn), where |k|=0,1,2,… and γk(ϕ)=∥ϕ(k)∥2,ϕ∈DL2(Rn). We then extend the continuous wavelet transform to distributions in DL2′(Rn), n≥1 and derive the corresponding wavelet inversion formula interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is defined by an element ψ(x) of DL2(Rn)∩DL1(Rn), n≥1 which, when integrated along each of the real axes X1,X2,…Xn vanishes, but none of its moments ∫Rnxmψ(x)dx is zero; here xm=x1m1x2m2⋯xnmn, dx=dx1dx2⋯dxn and m=(m1,m2,…mn) and each of m1,m2,…mn is ≥1. The set of such wavelets will be denoted by DM(Rn).

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