Smooth dynamic warping

Dynamic time warping is a simple classic method for aligning two sampled functions of time. The sequence of time shifts computed by this method are a globally optimal solution to a non-linear optimization problem with linear inequality constraints that may be directly related to subsurface properties. In applications to seismic traces, the time shifts may increase or decrease rapidly in time (and space). However, when related to integrals of subsurface parameters, such variations in time shifts are often smooth. A new method for smooth dynamic warping exploits this inherent smoothness to increase the accuracy of time shift estimates, especially where differences between seismic traces to be aligned are not limited to time shifts. The new method requires only a few simple modifications to the classic method, and can be easily extended to multidimensional image warping as well. In an application to registration of PP and PS images, we used smooth dynamic warping to compute times shifts that vary smoothly in both time and space, and are significantly more accurate than those computed using the classic method.