Trace reconstruction by Parabolic Radon Transform



Due to some difficulties during seismic data acquisition, like natural obstacles (high voltage electricity cable, bad coupling of geophones with the ground) some of the traces cannot be recorded. Since bad traces make the final stack unclear, usually bad traces go mute while processing. The final image of the earth’s crust is highly dependent of the quality and resolution of acquired data and muting these traces may cause lack of resolution. In this paper, parabolic radon transform is utilized to restore data. Radon transform is a method in which data is transferred to t-q domain from t-x domain. One of the remarkable features in this domain is that data with irregular spacing can be used as input. If these data transfer to t-q domain and transfer back to t-x domain, they will be partially restored. If we carry out this process in an iterative algorithm, the entire missed data will be reconstructed. This method uses an interpolation and extrapolation approach so that it predicts the wavelength and amplitude of each missed trace using adjacent traces.
There are some algorithms for which we do not need pre-information in order to make weighted coefficients as these coefficients are defined automatically. The algorithm offered here uses this approach and weighted coefficients metrics are defined using the Haber norm. Based on this method, this equation should be solved for each frequency component, meaning that this method utilizes the iterative least square approach. Our experience shows that solving the equation forward and backward, maximum 10 times restores the missed traces.
Some assumptions have been made in order to simplify the question. We assumed that there is no lateral velocity variation in layers. Moreover, the length of the receiver array is small compared with the depth of the target. With this assumption we can approximate the events to hyperbola. To apply the parabolic transform, we need to approximate the hyperbolic events to parabolic events. Thus, we applied a partially NMO correction on the data. The data will be corrected to the original hyperbolas, the same amount of initial NMO correction right after the reconstruction. The algorithm is run on a couple of synthetic models with various locations missed traces. We modeled parabolic and hyperbolic CMP gathers with 50 traces in which 11 traces are missed in near offset as well as in middle offset. After running the algorithm on the model, the traces were restored very well. However, far offset missing data cannot be extrapolated completely. We applied a white noise in the middle offset; the result was in agreement with the original wiggle synthetic CMP gather. Since the parabolic transform is used, the data is fully restored providing the events are completely parabolic.
The reconstruction algorithm is applied on real marine data afterwards. This CMP gather contains 51 traces irregularly spaced and sampled by 4ms rate. Some of the traces from the middle and near offsets were muted arbitrarily. After applying a set of forward and inverse Radon transform, the data were restored remarkably and concentration of energy in semblance panel became much better.
This method makes no artifact as this is interpolation and/or extrapolation of existing hyperbolic events. Although hyperbolic algorithm is our convention (since the events are hyperbolic), this is not applicable due to computational difficulties. It is possible to perform parabolic Radon transform in frequency domain quite fast. Since the L matrix (inverse radon transform matrix) contains full information about traces and their distribution, lack of a trace or irregular spacing of them does not play an important role.