The stacking of long-offset seismic data requires a nonhyperbolic traveltime function that depends on two-way traveltime, normal moveout (NMO) velocity and effective anellipticity. Based on a standard fractional approximation, a new parameterization in slowness-squared parameters provides optimal sampling of the NMO velocity and anellipticity. The automatic nonhyperbolic velocity analysis is performed with a normalized bootstrapped differential semblance (BDS) coherency estimator that leads to enhanced resolution in velocity spectra compared to differential semblance. Reflection wavelet centering inside time gates results in improved estimates of the two-way time and reduced bias in estimates of the NMO velocity and anellipticity. Generalized Dix equations give estimates of apparent interval thickness, velocity and anellipticity. The interval parameters will fit a homogeneous transversely isotropic medium with a vertical symmetry axis (a VTI medium) or an isotropic layer with a linear velocity gradient. The algorithm is implemented in a two-step strategy. A coarse hyperbolic velocity analysis that identifies events in the gather and estimates a velocity law for applying thetruncation is followed by a dense nonhyperbolic search to infer the physical parameters required for time processing of PP-wave data. The algorithm outputs an automatic stack and later-ally varying moveout velocity and anellipticity maps that can be used for subsequent time processing. Two attribute maps, the BDS map and its derivative, are also computed. These contain the fingerprints of the key reflectors and can be used in structural interpretation. Automatic nonhyperbolic velocity analysis is tested on a synthetic gather and a real data set from the North Sea. Nonhyperbolic parameter search shows an en-hanced estimate of the processing parameters, velocity and anellipticity, and improved quality of the stacked section com-pared with the result from hyperbolic search. The interval moveout velocity maps demonstrate a good match with the reflector positions in the obtained sections and show a great correlation when compared with the results of more advanced processing. The interval anellipticity map is also important for enhanced time processing and resolving the time-depth conversion problem, but the parameter is meaningless when the anisotropy is important or when the aperture is small, mainly for deep reflectors.

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