Generating low frequencies: a comparison of prediction filters and well log frequency replacement

Heather Jeanne Eileen Lloyd, Gary F. Margrave

Accurate bandlimited acoustic impedance inversion is impaired by the missing low-frequencies in recorded data. There are several methods for restoring these missing frequencies, they can be recorded in the field, estimated by model inversion, borrowed from well logs or predicted using the available frequencies in the spectra. We examine the last of these. Two methods of prediction filters were explored in solving for the missing frequencies. The first method is the one-lag prediction method which predicts the needed samples using one-lag at a time (Figure 1). The second method uses multiple-lags to create the needed samples (Figure 2). These methods were tested with a simple 3-layer model and a more complicated 12-layer model, where it was evident that the method was sensitive to four main parameters. To create the filters and predict the new samples, the prediction methods need a reliable band of frequencies that must be constant amplitude and span a larger frequency range than the low-frequency gap. The length of the filter also needs to be longer than the number of layers in the model. The third sensitivity was the effect of noise in the signal, where low signal to noise affected the character of the inversion. The fourth sensitivity was the number of layers that the method could accommodate. For a realistic synthetic example using a synthetic trace created from well logs and deconvolved, both the prediction methods failed when compared with the BLIMP (Band-limited impedance) method. This was partially due to the natural roll-off of low frequencies when there are a large amount of layers in the model and the complexity of the frequency spectrum. This study has not conclusively determined which prediction method is preferable, but the one-lag method is faster and often has less error when compared with the multi-lag method.