Subsurface reflectivity estimation from imaging of primaries and multiples using amplitude-normalized separated wavefields

Several studies have shown the benefits of including multiple reflections together with primaries in the structural imaging of subsurface reflectors. However, to characterize the reflector properties, there is a need to compensate for propagation effects due to multiple scattering and to properly combine the information from primaries and all orders of multiples. From this perspective and based on the wave equation and Rayleigh’s reciprocity theorem, recent works have suggested computing the subsurface image from the Green’s function reflection response (or reflectivity) by inverting a Fredholm integral equation in the frequency-space domain. By following Claerbout’s imaging principle and assuming locally reacting media, the integral equation may be reduced to a trace-by-trace deconvolution imaging condition. For a complex overburden and considering that the structure of the subsurface is angle-dependent, this trace-by-trace deconvolution does not properly solve the Fredholm integral equation. We have inverted for the subsurface reflectivity by solving the matrix version of the Fredholm integral equation at every subsurface level, based on a multidimensional deconvolution of the receiver wavefields with the source wavefields. The total upgoing pressure and the total filtered downgoing vertical velocity were used as receiver and source wavefields, respectively. By selecting appropriate subsets of the inverted reflectivity matrix and by performing an inverse Fourier transform over the frequencies, the process allowed us to obtain wavefields corresponding to virtual sources and receivers located in the subsurface, at a given level. The method has been applied on two synthetic examples showing that the computed reflectivity wavefields are free of propagation effects from the overburden and thus are suited to extract information of the image point location in the angular and spatial domains. To get the computational cost down, our approach is target-oriented; i.e., the reflectivity may only be computed in the area of most interest.