Estimation of the seismic focal spot imaging point spread function

The seismic Rayleigh wave focal spot imaging technique utilizes ambient seismic noise fields to estimate local Rayleigh wave phase velocities. Records of scattered and diffuse wavefields in dense seismic arrays are now routinely used to obtain virtual surface waves propagating between stations by cross-correlation, which supports ambient noise tomography. Rayleigh waves that refocus on the virtual source form the spatial autocorrelation field or focal spot. The shape of the narrow-band focal spot is used to obtain the local Rayleigh wave phase velocity at each virtual source or sensor in an array. It has been demonstrated that the lateral resolution of focal spot imaging depends on the data range on the order of one wavelength that is used to constrain the Bessel function model from the focal spot data. This can be observed as lateral spreading or averaging of velocities in inhomogeneous velocity distributions. Here we conjecture that the spreading effect is similar to the blurring effect observed in optical images, where the blurring is quantified by the point spread function that is the operator describing how the imaging device affects the image. Undoing the blurring in conventional images caused by the imaging device point spread function can be achieved by deconvolution methods. In seismic imaging, however, the exact properties of the focal spot imaging point spread function remain unknown. Determining the focal spot imaging point spread function properties has the potential to yield better resolved focal spot images. Experimental determination of the microscope point spread function is a routine task in microscopy, allowing for sharper images of near-diffraction limit scale objects through deconvolution. In microscopy, the empirical point spread function is determined by imaging sub-diffraction limit scale fluorescent beads acting as point sources. We adopt a similar approach to determine the empirical seismic focal spot imaging point spread function by imaging known velocity structures in synthetic focal spot imaging configurations using two-dimensional acoustics simulations in a reverberating cavity. Point-like velocity distributions are imaged to obtain empirical point spread functions. The empirical point spread functions are validated by deconvolving blurred synthetic images where the original velocity structure is known. As image deconvolution is an ill-posed problem, regularization methods are used to stabilize the solution. We utilize traditional spectral filtering methods such as truncated singular value decomposition and Tikhonov regularization, and total variation regularization to reconstruct the original velocity distribution using the empirical point spread function. Updated results of the empirical seismic focal spot imaging point spread function are presented.