A gap‐deconvolution filter with gap α is defined as the prediction error operator with prediction distance α. A spike‐deconvolution filter is defined as the prediction error operator with prediction distance unity. That is, a spike‐deconvolution filter is the special case of a gap‐deconvolution filter with gap equal to one time unit. Generally, the designation “gap deconvolution” is reserved for the case when α is greater than one, and the term “spike deconvolution” is used when α is equal to one. It is often stated that gap deconvolution with gap α shortens an input wavelet of arbitrary length to an output wavelet of length α (or less). Since an arbitrary value of α can be chosen, it would follow that resolution or wavelet contraction may be controlled by use of gap deconvolution. In general, this characterization of gap deconvolution is true for arbitrary α if and only if the wavelet is minimum delay (i.e., minimum phase). The method of model‐driven deconvolution can be used in the case of a nonminimum‐delay wavelet. The wavelet is the convolution of a minimum‐delay reverberation and a short nonminimum‐delay orphan. The model specifies that the given trace is the convolution of the white reflectivity and this nonminimum‐delay wavelet. The given trace yields the spike‐deconvolution filter and its inverse. These two signals are then used to compute the gap‐deconvolution filters and their inverses for various prediction distances. The inverses are examined, and a stable one is picked as the most likely minimum‐delay reverberation. The corresponding gap‐deconvolution filter is the optimum one for the removal of this minimum‐delay reverberation from the given trace. As a byproduct, the minimum‐delay counterpart of the orphan can be obtained. The optimum gap‐deconvolved trace is examined for zones that contain little activity, and the leading edge of the wavelet following such a zone is chosen. Next, the phase of the minimum‐delay counterpart of the orphan is rotated until it fits the extracted leading edge. From the amount of phase rotation, the required phase‐correcting filter can be estimated. Alternatively, downhole information, if available, can be used to estimate the phase‐correcting filter. Application of the phase‐correcting filter to the spike‐deconvolved trace gives the required approximation to the reflectivity. As a final step, wavelet processing can be applied to yield a final interpreter trace made up of zero‐phase wavelets.
Spatiotemporal neural characterization of prediction error valence and surprise during reward learning in humans