Seismic Data Reconstruction Using Curvelets

When smoothing a function with high-frequency noise by means of optimal cubic splines, it is often not clear how to choose the number of nodes. The more nodes are used, the closer the smoothed function will follow the noisy one. In this work, we demonstrate that more nodes mean a better approximation of Fourier coefficients for higher frequencies. Thus, the number of nodes can be determined by specifying a frequency up to which all Fourier coefficients must be preserved. A comparison of the corresponding smoothing results with those obtained by filtering using a moving average of corresponding length and a lowpass with corresponding high-cut frequency show that optimal cubic splines yield better results as they preserve not only the desired low-frequency band but also important high-frequency characteristics.