Hyperbolic estimation of sparse and blocky models

For image estimation we use a hyperbolic penalty function. The center of a hyperbola is parabolic like `2 norm fitting. Its asymptotes are like `1 norm fitting. A transition threshold must be chosen for regression equations of data fitting and another for model regularization. We adapt the conjugate direction method to solve this problem in a manner we call the HYCD method. We present examples (1) blocky interval velocity estimation, (2) migrating aliased data, and (3) velocity transform with strong noise. Extra nonlinearity is introduced by the hyperbolic objective function, but the convexity of the sum of the hyperbolas assures the convergence. We have had sufficiently reliable performance on the three mainstream geophysical applications shown here so that we expect the HYCD solver to become our default solver.