Study And Application Analysis Of Spherical-wave Reflection Characteristics

The frequency-dependent characteristics of spherical wave reflectivity(SWR)in two semi-infinite acoustic and elastic media are investigated in this thesis.It is found that the frequency-dependence of spherical acoustic wave reflectivity(SAWR)occurs only when there exists velocity difference between the upper medium and the lower one,whereas the frequency-dependence of spherical elastic wave PP reflectivity(SEWR)is not limited to this condition.The frequency-dependent characteristics of SWR(SAWR and SEWR)are summarized and classified according to the magnitude relation between the upper compressional velocity and the lower one,and the sign of the corresponding plane-wave reflectivity(PWR).Generally,large deviation exists between the SWR and the corresponding PWR at low frequency.With increasing frequency,the SWR approaches the corresponding PWR at high frequency.Particularly when the PWR is zero,both the amplitude and phase of the corresponding SWR are non-zero.The amplitude of the SWR decreases with increasing frequency,and approaches that of the PWR at high frequency.Noticeably,the phase of the SWR approaches 90° or-90° at high frequency,meaning that there is large waveform difference between the incident spherical wave and the reflected spherical wave.Based on the Sommerfeld integral formula of SAWR,the limit when KR(K is wave number and R is wave propagation distance)approaches zero is derived,which is(m-1)/(m+1),where m is the ratio of the density of the lower medium to the upper one.This has not been reported in the current literature.In addition,it is found that the real parts and the imaginary parts of SAWR lie on an ellipse,through curve fitting for SAWR with different KR on the complex plane.Based on this,an analytical formula of SAWR with high precision is derived,which is applicable for both the far-field and nearfield.The new formula is applicable for precritical small angles.It is well known that the SAWR approaches the corresponding PWR when KR tends to be infinite.Therefore,the derived formula can construct a relation between zero frequency reflectivity and infinite high frequency reflectivity,which delineates how the spherical wave approaches plane wave from low frequency to high frequency.This can enrich our knowledge for seismic wave propagation.Finally,application analyses are made by utilizing spherical wave reflection characteristics.It is found that for Class 1-2 AVO models,great deviation between SWR and PWR exists near critical angles and at low frequencies.However for Class 3-4 AVO models,relatively large deviation between SWR and PWR exists only at very low frequency.Taking Class 1 AVO as an example,the conventional PWR-based AVO inversion is performed using low-frequency reflected spherical-wave data for only precritical small and moderate angles,which produces large errors in elastic parameters estimation.This indicates that even at small or moderate angles,the frequencydependence at low frequencies cannot be neglected,and SWR-based inversion should be considered.In addition,we find that the frequency-dependence of SWR is essentially different from the frequency-dependence caused by absorption attenuation.And the frequency-dependence of SWR can have substantial influence on conventional quality factor Q estimation for low frequency and near-surface data.Thus for more accurate Q estimation,the frequency-dependent SWR should be taken account of.At last,by combining the limit value of SWR at zero frequency and ellipse fitting for SWR at different frequencies,a novel approach to invert density ratio is proposed.The validity of the method is demonstrated by using a numerical test,which also shows the importance of low-frequency information in inversion.