Subsurface Offset Behaviour in Velocity Analysis with Extended Reflectivity Images

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Migration velocity analysis with the wave equation can be accomplished by focusing of extended migration images, obtained by introducing a subsurface offset or shift. A reflector in the wrong velocity model will show up as a curve in the extended image. In the correct model, it should collapse to a point. The usual approach to obtain a focused image involves a cost functional that penalizes energy in the extended image at non-zero shift. Its minimization by a gradient-based method should then produce the correct velocity model. Here, asymptotic analysis and numerical examples show that this method may be too sensitive to amplitude peaks at large shifts at the wrong depth and to artifacts. A more robust alternative is proposed that can be interpreted as a generalization of stack power and maximizes the energy at zero subsurface shift. A real-data example is included.