Reversibility of three-parameter W transform

The three ⁃ parameter W transform(TPWT)is an effective tool for analyzing non ⁃ stationary sig⁃ nals,and it has been successfully used in oil and gas reservoir identification. However,the reversibility of TPWT has not been discussed in detail in previous studies. Therefore,this paper first reviewed the basic principles of TPWT and theoretically analyzed the reversibility of TPWT. Both the theoretical analysis and numerical calculation results show that:①TPWT not only solves the problems of low time resolution at low⁃ frequencies and shift of the distribution centroid of the time⁃frequency spectral energy toward higher frequen⁃ cies in S transform(ST)but also overcomes the amplitude splitting phenomenon of the time⁃frequency spec⁃ trum at the dominant frequency in W transform(WT),which can characterize hydrocarbon reservoirs more accurately and is more beneficial for seismic interpretation. ② In theory,TPWT is not strictly reversible but an approximately reversible transform tool,which is different from Fourier transform(FT)and ST. ③ The numerical calculation results of a synthetic seismogram and a real seismogram show that compared with origi⁃ nal seismic data,the relative errors of seismic data reconstructed by inverse TPWT are 11. 47%~21. 35%, or in other words,the theoretical irreversibility leads to significant reconstruction errors,seriously affecting the application range of TPWT. TPWT is not applicable in fields that require data reconstruction such as de⁃ noising and high⁃resolution processing. © 2024 Editorial office of Oil Geophysical Prospecting. All rights reserved.