A stabilized least-squares imaging condition with structure constraints

Conventional shot-gather migration uses a cross-correlation imaging condition proposed by Clarebout （1971）, which cannot preserve imaging amplitudes. The deconvolution imaging condition can improve the imaging amplitude and compensate for illumination. However, the deconvolution imaging condition introduces instability issues. The least-squares imaging condition first computes the sum of the cross-correlation of the forward and backward wavefields over all frequencies and sources, and then divides the result by the total energy of the forward wavefield. Therefore, the least-squares imaging condition is more stable than the classic imaging condition. However, the least-squares imaging condition cannot provide accurate results in areas where the illumination is very poor and unbalanced. To stabilize the least-squares imaging condition and balance the imaging amplitude, we propose a novel imaging condition with structure constraints that is based on the least-squares imaging condition. Our novel imaging condition uses a plane wave construction that constrains the imaging result to be smooth along geological structure boundaries in the inversion frame. The proposed imaging condition improves the stability of the imaging condition and balances the imaging amplitude. The proposed condition is applied to two examples, the horizontal layered model and the Sigsbee 2A model. These tests show that, in comparison to the damped least-squares imaging condition, the stabilized least-squares imaging condition with structure constraints improves illumination stability and balance, makes events more consecutive, adjusts the amplitude of the depth layers where the illumination is poor and unbalanced, suppresses imaging artifacts, and is conducive to amplitude preserving imaging of deep layers.