Far-field wavefield characteristics of downhole seismic sources

Borehole sources,whose scope goes far beyond sources in boreholes,are of extreme importance in research with active seismic sources,including deep seismic sounding,reverse vertical seismic profiling(RVSP),seismic while drilling,mining geophysics,etc.Sources used in these studies are all of cylindrical structures,which is the reason why they are called borehole sources and why their wave fields has unique characteristics.Previous studies on borehole sources are mostly based on analytical solutions obtained when small-borehole assumption(the borehole radius is significantly smaller than the characteristic wave length)and far-field assumption(the offset is greater than the characteristic wave length)are satisfied.It is still an open question whether the analytical solutions are applicable to cases that violate the two assumptions.This study is based on the synthetic seismograms computed by both analytical solutions and semi-analytical solutions.The analytical solutions used in previous studies are obtained through asymptotic analysis,while the semi-analytical solutions are computed by numerical integration.The semi-analytical solutions are of higher accuracy and therefore regarded as"true solutions".Synthetic seismograms from the analytical solutions are compared to true solutions to validate whether the analytical solutions are applicable to certain cases or not.Accuracy is crucial to the comparison.Yet the high oscillation of solutions in frequency-wavenumber domain brings out a great challenge.We developed a brand-new numerical method called Steepest Descent Integration Method(SDIM).The new method is inspired by the Method of Steepest Descent(SDM)in asymptotic analysis that is specially designed for highly oscillatory integral and is the very method used to obtain the analytical solutions.Replacing approximate integration path and approximate integrand in SDM with accurate ones,SDIM breaks the restraints of small borehole and far field and can compute seismograms at arbitrary offset and arbitrary source frequency with extremely high accuracy efficiently.We calculate the seismograms by both SDIM and SDM for a large offset(1000m,significantly large compared to borehole radius of 0.1m)and varied source frequency(0.1~1000Hz).The assumption of small-borehole is violated in high frequency cases,while farfield assumption fails when the frequency is low.The same experiment is conducted for all three basic borehole sources.The works presented in the paper can be categorized into two parts,namely the new SDIM and comparison of seismograms.The study of SDIM shows that:(1)The solutions of borehole sources problem in frequency-wavenumber domain are highly oscillatory.The oscillation depends on source frequency and offsets.High frequency sources result in severe oscillation,so as large offsets.(2)The oscillation is attributed to Hankel functions in the solutions whose exponential part account for most of it.Hence,exponential functions are used in the derivation of SDIM instead of Hankel functions,making the work much easier.(3)The only difference between SDIM and SDM is the accuracy of the steepest descent path and the integrand.SDIM uses the accurate path and integrand,while SDM uses approximate ones.In addition,four numerical examples are presented in the paper.Each is designed specifically.They demonstrate that:(1)Results from SDIM are identical to ones from SDM when small-borehole assumption and far-field assumption are satisfied,which supports the validity of SDIM.(2) When small-borehole assumption is violated,the SDM results differ much from the SDIM ones that are considered as true results.It infers that the influence from borehole might not be ignored even for far-field wave field.(3)When far-field assumption fails,the results from SDM are inaccurate as well,which means the absolute value of the offset cannot guarantee far-field.The ratio of the offset to the characteristic wave length matters.(4)The same phenomenon occurs in the wave field of all the three basic borehole sources.Obtaining accurate far-field seismograms is the key problem of borehole sources research.Yet it is challenging because of highly oscillatory integral involved.By taking advantage of the special form of analytical solutions,we developed a brand-new method for computing highly oscillatory wavenumber integration.It completely avoids the oscillation and results in numerical integration of a fully smooth function,leading to synthetic seismograms with high precision.It also allows us to compute P,S and surface waves separately,reducing their mutual interference.Numerical experiments demonstrate that the results from SDM are considerably different from ones from SDIM,in both amplitude and phase when the small-borehole assumption or the farfield assumption fails.Therefore,the SDM has its constraint and SDIM is a better alternative if accurate wave-field information is needed.